SL TIPS: rithmetic-geometric

Throughout the monograph a Hilbert space H is assumed to be separable. The algebra B(H) of all bounded operators on H is a Banach space with the operator norm ·. For 1 ≤ p < ∞ let Cp(H) denote the Schatten p-class consisting of (compact) operators X ∈ B(H) satisfying Tr(|X| p) < ∞ with |X| = (X∗X)1/2, where Tr is the usual trace. The space Cp(H) is an ideal of B(H) and a Banach space with the Schatten p-norm Xp = (Tr(|X| p))1/p. In particular, C1(H) is the trace class, and C2(H) is the Hilbert-Schmidt class which is a Hilbert space with the inner product (X, Y )C2(H) = Tr(XY ∗) (X, Y ∈ C2(H)). The algebra B(H) is faithfully (hence isometrically) represented on the Hilbert space C2(H) by the left (also right) multiplication: X ∈ C2(H) → AX, XA ∈ C2(H) for A ∈ B(H). Standard references on these basic topics (as well as unitarily invariant norms) are [29, 37, 77]. In this chapter we choose and fix positive operators H, K on H with the spectral decompositions H = H 0 s dEs and K = K 0 t dFt respectively. We will use both of the notations dEs, EΛ (for Borel sets Λ ⊆ [0, H]) interchangeably in what follows (and do the same for the other spectral measure F). Let λ (resp. µ) be a finite positive measure on the interval [0, H] (resp. [0, K]) equivalent (in the absolute continuity sense) to dEs (resp. dFt). For instance the measures λ(Λ) = ∞ n=1 1 n2 (EΛen, en) (Λ ⊆ [0, H]), µ(Ξ) = ∞ n=1 1 n2 (FΞ en, en) (Ξ ⊆ [0, K]) do the job, where {en}n=1,2,··· is an orthonormal basis for H. We choose and fix a function φ(s, t) in L∞([0, H] × [0, K]; λ × µ). For each operatorX ∈ B(H), the algebra of all bounded operators on H, we would like to justify its “double integral” transformation formally written as Φ(X) = H 0 K 0 φ(s, t) dEsXdFt (see [14]). As long as X ∈ C2(H), the Hilbert-Schmidt class operators, desired justification is quite straight-forward and moreover under such circumstances we have Φ(X) ∈ C2(H) with the norm bound Φ(X)2 ≤ φL∞(λ×µ) × X2. (2.1) In fact, with the left multiplication π and the right multiplication πr, π(EΛ) and πr(FΞ ) (with Borel sets Λ ⊆ [0, H] and Ξ ⊆ [0, K]) are commuting projections acting on the Hilbert space C2(H) so that π(EΛ)πr(FΞ ) is a projection. It is plain to see that one gets a spectral family acting on the Hilbert space C2(H) from those “rectangular” projections so that the ordinary functional calculus via φ(s, t) gives us a bounded linear operator on C2(H). With this interpretation we set Φ(X) = H 0 K 0 φ(s, t) d(π(E)πr(F)) X. (2.2) Note that the Hilbert-Schmidt class operator X in the right side here is regarded as a vector in the Hilbert space C2(H), and (2.1) is obvious. In applications of double integral transformations (for instance to stability problems of perturbation) it is important to be able to specify classes of functions φ for which the domain of Φ(·) can be enlarged to various operator ideals (such as Cp-ideals). In fact, some useful sufficient conditions (in terms of certain Lipschitz conditions on φ(·, ·)) were announced in [14] (whose proofs were sketched in [15]), but unfortunately they are not so helpful for our later purpose. More detailed information on double integral transformations will be given in §2.5. 2.1 Schur multipliers and Peller’s theorem We begin with the definition of Schur multipliers (acting on operators on H). Definition 2.1. When Φ(= Φ| C1(H)) : X → Φ(X) gives rise to a bounded transformation on the ideal C1(H) (⊆ C2(H)) of trace class operators, φ(s, t) is called a Schur multiplier (relative to the pair (H, K)). When this requirement is met, by the usual duality B(H)=(C1(H))∗ the transpose of Φ gives rise to a bounded transformation on B(H) (i.e., the largest possible domain) as will be explained in the next §2.2. The next important characterization due to V. V. Peller will play a fundamental role in our investigation on means of operatorsTheorem 2.2. (V.V. Peller, [69, 70]) For φ ∈ L∞([0, H]× [0, K]; λ× µ) the following conditions are all equivalent : (i) φ is a Schur multiplier ; (ii) whenever a measurable function k : [0, H] × [0, K] → C is the kernel of a trace class operator L2([0, H]; λ) → L2([0, K]; µ), so is the product φ(s, t)k(s, t); (iii) one can find a finite measure space (Ω, σ) and functions α ∈ L∞([0, H] ×Ω; λ × σ), β ∈ L∞([0, K] × Ω; µ × σ) such that φ(s, t) = Ω α(s, x)β(t, x)dσ(x) for all s ∈ [0, H], t ∈ [0, K]; (2.3) (iv) one can find a measure space (Ω, σ) and measurable functions α, β on [0, H] × Ω, [0, K]× Ω respectively such that the above (2.3) holds and Ω |α(·, x)| 2dσ(x) L∞(λ) Ω |β(·, x)| 2dσ(x) L∞(µ) < ∞. A few remarks are in order. (a) The implication (iii) ⇒ (iv) is trivial. (b) The finiteness condition in (iv) and the Cauchy-Schwarz inequality guarantee the integrability of the integrand in the right-hand side of (2.3). (c) The condition (iii) is stronger than what was stated in [69, 70], but the proof in [69] (presented below) actually says (ii) ⇒ (iii). Unfortunately Peller’s article [69] (with a proof) was not widely circulated. Because of this reason and partly to make the present monograph as much as self-contained, the proof of the theorem is presented in what follows. Proof of (iv) ⇒ (i) Although this is a relatively easy part in the proof, we present detailed arguments here because its understanding will be indispensable for our later arguments. So let us assume that φ(s, t) admits an integral representation stated in (iv). For a rank-one operator X = ξ ⊗ηc we have π(EΛ)πr(FΞ )X = (EΛξ) ⊗ (FΞ η)c so that from (2.3) we get Φ(X) = H 0 K 0 Ω α(s, x)β(t, x) (dEsξ) ⊗ (dFtη) c dσ(x) = Ω ξ(x) ⊗ η(x) c dσ(x) with ξ(x) = H 0 α(s, x) dEsξ and η(x) = K 0 β(t, x) dFtη. (2.4) More precisely, the above integral can be understood for example in the weak 2 Double integral transformations (Φ(X)ξ , η ) = Ω ((ξ(x) ⊗ η(x) c)ξ , η ) dσ(x) = Ω (ξ , η(x))(ξ(x), η ) dσ(x). (2.5) The above ξ(x), η(x) are vectors for a.e. x ∈ Ω as will be seen shortly. We use Theorem A.5 in §A.2 and the Cauchy-Schwarz inequality to get Φ(ξ ⊗ ηc)1 ≤ Ω ξ(x) ⊗ η(x) c1dσ(x) = Ω ξ(x)×η(x) dσ(x) ≤ Ω ξ(x)2dσ(x) 1/2 Ω η(x)2dσ(x) 1/2 . (2.6) Since ξ(x)2 = H 0 |α(s, x)| 2d(Esξ, ξ) with the total mass of d(Esξ, ξ) being ξ2, we have Ω ξ(x)2dσ(x) = H 0 Ω |α(s, x)| 2dσ(x) d(Esξ, ξ) ≤ Ω |α(·, x)| 2dσ(x) L∞(λ) × ξ2 (2.7) by the Fubini-Tonneli theorem. A similar bound for Ω η(x)2dσ(x) is also available, and consequently from (2.6), (2.7) we get Φ(ξ ⊗ ηc )1 ≤ ξ×η × Ω |α(·, x)| 2dσ(x) 1/2 L∞(λ) × Ω |β(·, x)| 2dσ(x) 1/2 L∞(µ) . Therefore, we have shown Φ(X)1 ≤ Ω |α(·, x)| 2dσ(x) 1/2 L∞(λ) × Ω |β(·, x)| 2dσ(x) 1/2 L∞(µ) × X1 (2.8) for rank-one operators X. Note that (2.7) (together with the finiteness requirement in the theorem) shows ξ(x) < ∞, i.e., ξ(x) is indeed a vector for a.e. x ∈ Ω. Also (2.8) guarantees that Φ(X) = Ω ξ(x) ⊗ η(x)cdσ(x) falls into the ideal C1(H) of trace class operators. We claim that the estimate (2.8) remains valid for finite-rank operators. Indeed, thanks to the standard polar decomposition and diagonalization technique, such an operator X admits a representation X = n i=1 ξi⊗ηc i satisfying X1 = n i=1 ξi×ηi. Then, we estimateAssume that a measurable function k on [0, H] × [0, K] is the kernel of a trace class operator R : L2(λ) → L2(µ), i.e., (Rf)(t) = H 0 k(s, t)f(s) dλ(s) for f ∈ L2(λ). The assumption implies in particular that k(s, t) and hence φ(s, t)k(s, t) are square integrable with respect to λ × µ so that the latter is the kernel of a Hilbert-Schmidt class operator. We prove under the assumption (i) that φ(s, t)k(s, t) is indeed the kernel of a trace class operator. Define X ∈ C1(H) by composing R with the orthogonal projection PH1 as follows: H PH1 −→ H1 ∼= L2(λ) R −→ L2(µ) ∼= H2 → H . Then (i) yields Φ(X) ∈ C1(H). For each Λ ⊆ [0, H] and Ξ ⊆ [0, K] we have Φ(X) m EΛξm , n FΞ ηn = m,n Φ(X),(EΛξm) ⊗ (FΞ ηn) c C2(H) = m,n X, Φ∗ ((EΛξm) ⊗ (FΞ ηn) c ) C2(H) = m,n X, H 0 K 0 φ(s, t) d(πl(Es)πr(Ft))((EΛξm) ⊗ (FΞ ηn) c) C2(H) = m,n X, Λ Ξ φ(s, t) (dEsξm) ⊗ (dFtηn) c C2(H) = m,n Λ Ξ φ(s, t) (XdEsξm, dFtηn) = Λ Ξ φ(s, t)k(s, t) dλ(s) dµ(t) because of 2.1 Schur multipliers and Peller’s theorem 13 m,n (XEΛξm, FΞηn)=(RχΛ, χΞ )L2(µ) = Λ Ξ k(s, t) dλ(s) dµ(t). We thus conclude that φ(s, t)k(s, t) is the kernel of the trace class operator L2(λ) → L2(µ) corresponding to Φ(X)|H1 : H1 → H2. Proof of (ii) ⇒ (iii) This is the most non-trivial part in Peller’s theorem, and requires the notion of one-integrable operators (between Banach spaces) and the Grothendieck theorem. Assume that φ satisfies (ii) and define an integral operator T0 : L1(λ) → L∞(µ) by (T0f)(t) = H 0 φ(s, t)f(s) dλ(s) for f ∈ L1(λ). What we need to show is that T0 falls into the operator ideal I1(L1(λ), L∞(µ)) consisting of one-integral operators in the space of bounded operators L1(λ) → L∞(µ). Our standard reference for the theory on operator ideals on Banach spaces is Pietsch’s textbook [72] (see especially [72, §19.2]). It is known (see [72, 19.2.13]) that I1(L1(λ), L∞(µ)) is dual to the space of compact operators L∞(µ) → L1(λ). Thanks to [72, 10.3.6 and E.3.1], to show T0 ∈ I1(L1(λ), L∞(µ)), it suffices to prove that there exists a constant C such that |trace(T0Q)| ≤ CQ (2.9) for finite-rank operators Q : L∞(µ) → L1(λ) of the form Q = l k=1 ·, hkgk with gk ∈ L1(λ) and hk ∈ L1(µ). Here, ·, · denotes the duality between L∞(µ) and L1(µ) and trace(T0Q) = n k=1 T0gk, hk for T0Q = l k=1 ·, hkT0gk. To show (2.9), one may and do assume that gk, hk are finite linear combinations of characteristic functions, say gk = m i=1 αkiχΛi , hk = n j=1 βkjχΞj where A = {Λ1,...,Λm} and B = {Ξ1,...,Ξn} are measurable partitions of [0, H] and [0, K] respectively. For p = 1, 2, ∞ write Lp(A, λ) for the (finite-dimensional) subspace of Lp(λ) consisting of A-measurable functions (i.e., linear combinations of χΛi ’s) and Lp(B, µ) similarly. The conditional expectation EB : Lp(µ) → Lp(B, µ) is given by EBf = n j=1 µ(Ξj ) −1 Ξj f dµ χΞj . Set Q˜ = Q|L∞(B,µ) : L∞(B, µ) → L1(A, λ) so that we have Q = Q˜ ◦ EB. 14 2 Double integral transformations According to [61, Theorem 4.3] (based on the Grothendieck theorem) together with [61, Proposition 3.1], we see that Q˜ admits a factorization L∞(B, µ) M˜2 −→ L2(B, µ) R˜ −→ L1(A, λ), (2.10) where M˜ 2 is the multiplication by a function ˜η ∈ L2(B, µ) and R˜ is an operator such that η˜L2(µ) = 1 and R˜ ≤ KGQ˜ (2.11) with the Grothendieck constant KG. Apply [61, Theorem 4.3] once again to the transpose R˜t : L∞(A, λ) → L2(B, µ) to get the following factorization of R˜t : L∞(A, λ) Mˆ1 −→ L2(A, λ) Sˆ −→ L2(B, µ), where Mˆ 1 is the multiplication by a function ˜ξ ∈ L2(A, λ) and Sˆ is an operator such that ˜ξL2(λ) = 1 and Sˆ ≤ KGR˜t = KGR˜. (2.12) Hence R˜ is factorized as L2(B, µ) S˜=Sˆt −→ L2(A, λ) M˜1=Mˆ t 1 −→ L1(A, λ), (2.13) where M˜ 1 is again the multiplication by ˜ξ. Combining (2.10) and (2.13) implies that Q is factorized as L∞(µ) EB −→ L∞(B, µ) M˜2 −→ L2(B, µ) S˜ −→ L2(A, λ) M˜1 −→ L1(A, λ) → L1(λ). Let S = SE˜ B : L2(µ) → L2(B, µ) → L2(A, λ) ⊆ L2(λ) and M1 : L2(λ) → L1(λ), M2 : L∞(µ) → L2(µ) be the multiplications by ˜ξ, η˜ respectively. Since Q = M˜ 1S˜M˜ 2EB = M˜ 1SE˜ BM2 = M1SM2, we finally obtain a factorization of Q as follows: L∞(µ) M2 −→ L2(µ) S −→ L2(λ) M1 −→ L1(λ) with S = S˜ ≤ KGR˜ ≤ K2 GQ˜ = K2 GQ (2.14) thanks to (2.11) and (2.12). Notice that M2T0M1 : L2(λ) → L2(µ) is the integral operator (M2T0M1f)(t) = H 0 φ(s, t)˜ξ(s)˜η(t)f(s) dλ(s). Since ˜ξ(s)˜η(t) is obviously a kernel of a rank-one operator L2(λ) → L2(µ), the assumption (ii) implies that M2T0M1 is a trace class operator. Now, it is easy to see that 2.1 Schur multipliers and Peller’s theorem 15 trace(T0Q) = trace(T0M1SM2) = Tr(M2T0M1S) (2.15) with the (ordinary) trace Tr for the trace class operator M2T0M1S on L2(µ). For every ξ ∈ L2(λ) and η ∈ L2(µ), the assumption (ii) guarantees that one can define a trace class operator A(ξ,η) : L2(λ) → L2(µ) by (A(ξ,η)f)(t) = H 0 φ(s, t)ξ(s)η(t)f(s) dλ(s); in particular, M2T0M1 = A(˜ξ, η˜). Write C1(L2(λ), L2(µ)) for the Banach space (with trace norm ·C1(L2(λ),L2(µ))) consisting of trace class operators L2(λ) → L2(µ). Lemma 2.3. There exists a constant C˜ such that A(ξ,η)C1(L2(λ),L2(µ)) ≤ C˜ξL2(λ)ηL2(µ) (2.16) for each ξ ∈ L2(λ) and η ∈ L2(µ). Proof. For a fixed ξ ∈ L2(λ) let us consider the linear map A(ξ, ·) : η ∈ L2(µ) → A(ξ,η) ∈ C1(L2(λ), L2(µ)), whose graph is shown to be closed. We assume ηn −→ η in L2(µ) and A(ξ,ηn) −→ B in C1(L2(λ), L2(µ)). Choose and fix f ∈ L2(λ), and notice A(ξ,ηn)f − BfL2(µ) ≤ A(ξ,ηn) − BB(L2(λ),L2(µ))fL2(λ) ≤ A(ξ,ηn) − BC1(L2(λ),L2(µ))fL2(λ) −→ 0. From these L2-convergences, after passing to a subsequence if necessary, we may and do assume ηn(t) −→ η(t) and (A(ξ,ηn)f)(t) −→ (Bf)(t) for µ-a.e. t. We then estimate |(A(ξ,ηn)f)(t) − (A(ξ,η)f)(t)| ≤ H 0 φ(s, t) ηn(t) − η(t) ξ(s)f(s)dλ(s) ≤ |ηn(t) − η(t)|×φ∞ × H 0 |ξ(s)f(s)| dλ(s). The last integral here being finite (due to ξ, f ∈ L2(λ)), we conclude (Bf)(t)=(A(ξ,η)f)(t) for µ-a.e. t. This means Bf = A(ξ,η)f ∈ L2(µ) 16 2 Double integral transformations and the arbitrariness of f ∈ L2(λ) shows B = A(ξ,η) as desired. Therefore, the closed graph theorem guarantees the boundedness of A(ξ, ·), i.e., A(ξ, ·) = sup A(ξ,η)C1(L2(λ),L2(µ)) : η ∈ L2(µ), ηL2(µ) ≤ 1 < ∞, A(ξ,η)C1(L2(λ),L2(µ)) ≤ A(ξ, ·)×ηL2(µ). (2.17) We next consider the linear map A : ξ ∈ L2(λ) → A(ξ, ·) ∈ B(L2(µ), C1(L2(λ), L2(µ))). To show the closedness of the graph again, we assume ξn −→ ξ in L2(λ) and A(ξn, ·) −→ C in B(L2(µ), C1(L2(λ), L2(µ))). We need to show A(ξ, ·) = C ∈ B(L2(µ), C1(L2(λ), L2(µ))), i.e., A(ξ,η) = C(η) ∈ C1(L2(λ), L2(µ)) (η ∈ L2(µ)). For each fixed f ∈ L2(λ) (and η ∈ L2(µ)), we have A(ξn, η)f → C(η)f in L2(µ). From this L2-convergence and the fact η ∈ L2(µ), after passing to a subsequence, we have (A(ξn, η)f)(t) −→ (C(η)f)(t) and |η(t)| < ∞ for µ-a.e. t. We estimate |(A(ξn, η)f)(t) − A(ξ,η)f)(t)| ≤ H 0 φ(s, t)η(t)(ξn(s) − ξ(s))f(s)dλ(s) ≤ φ∞ × |η(t)| × H 0 |(ξn(s) − ξ(s))f(s)| dλ(s) ≤ φ∞ × |η(t)|×ξn − ξL2(λ)fL2(λ). Therefore, we have (A(ξ,η)f)(t)=(C(η)f)(t) for µ-a.e. t, showing A(ξ,η)f = C(η)f ∈ L2(µ) (f ∈ L2(λ)) and A(ξ,η) = C(η) ∈ C1(L2(λ), L2(µ)) (for each η ∈ L2(µ)). Thus, the closed graph theorem shows the boundedness A(ξ, ·) ≤ C˜ξL2(λ) for some C, ˜ which together with (2.17) implies the inequality (2.16). We are now ready to prove (iii). By combining the above estimates (2.15), (2.16), (2.11), (2.12) and (2.14) altogether, we get |trace(T0Q)|≤A(˜ξ, η˜)SC1(L2(µ)) ≤ C˜˜ξL2(λ)η˜L2(µ)S ≤ CK˜ 2 GQ, proving (2.9) with a constant C = CK˜ 2 G (independent of Q). Thus, T0 ∈ I1(L1(λ), L∞(µ)) is established. The following fact is known among other characterizations (see [72, 19.2.6]): a bounded operator T : L1(λ) → L∞(µ) belongs to I1(L1(λ), L∞(µ)) if 2.1 Schur multipliers and Peller’s theorem 17 and only if there exist a probability space (Ω, σ) and bounded operators T1 : L1(λ) → L∞(Ω; σ), T2 : L1(Ω; σ) → L∞(µ)∗∗ such that L1(λ) T −→ L∞(µ) → L∞(µ)∗∗ T1 ↓ ↑ T2 L∞(Ω; σ) → L1(Ω; σ) is commutative. Therefore, we can factorize T0 as follows: L1(λ) T1 −→ L∞(Ω; σ) → L1(Ω; σ) T2 −→ L∞(µ), where (Ω, σ) is a finite measure space and T1, T2 are bounded operators. Indeed, L∞(µ) is complemented in L∞(µ)∗∗, and this T2 is the composition of a projection map (actually a norm-one projection due to M. Hasumi’s result in [35], and also see [76, p. 148, Exercise 22 and p. 299, Exercise 10]) L∞(µ)∗∗ → L∞(µ) and the preceding T2 : L1(Ω; σ) → L∞(µ)∗∗. Thanks to Lemma 2.4 below applied to the preceding bounded operators T1, T2, there exist α ∈ L∞([0, H]×Ω; λ×σ) and β ∈ L∞([0, K]×Ω; µ×σ) such that (T1f)(x) = H 0 α(s, x)f(s) dλ(s) for f ∈ L1(λ), (T2g)(t) = Ω β(t, x)g(x) dσ(x) for g ∈ L1(Ω, σ). Therefore, we have (T0f)(t) = Ω H 0 α(s, x)β(t, x)f(s) dλ(s) dσ(x) = H 0 Ω α(s, x)β(t, x) dσ(x) f(s) dλ(s) for f ∈ L1(λ), which yields (iii) and the proof of Theorem 2.2 is completed. The next result can be found in [47] as a corollary of a more general result (see [47, §XI.1, Theorem 6]), and a short direct proof is presented below for the reader’s convenience. Lemma 2.4. Let (Ω1, σ1) and (Ω2, σ2) be finite measure spaces. For a given bounded operator T : L1(Ω1; σ1) → L∞(Ω2; σ2) there exists a unique τ ∈ L∞(Ω1 × Ω2; σ1 × σ2) satisfying (T f)(y) = Ω1 τ(x, y)f(x) dσ1(x) for f ∈ L1(Ω1; σ1). Proof. Choose and fix a measurable set Ξ ⊆ Ω2. For each f ∈ L1(σ1) we observe the trivial estimate 18 2 Double integral transformations |Tf,χΞσ2 | ≤ σ2(Ξ) × T fL∞(σ2) ≤ σ2(Ξ) × T ×fL1(σ1) (with the standard bilinear form ·, · σ2 giving rise to the duality between L∞(σ2) and L1(σ2)), showing the existence of hΞ ∈ L∞(Ω1; σ1) satisfying hΞL∞(σ1) ≤ T and Tf,χΞσ2 = σ2(Ξ) × hΞ, fσ1 for f ∈ L1(σ1). Let Π denote the set of all finite measurable partitions of Ω2, which is a directed set in the order of refinement. For every π ∈ Π we set τπ(x, y) = Ξ∈π hΞ(x)χΞ (y), (x, y) ∈ Ω1 × Ω2, so that a net {τπ}π∈Π in L∞(σ1 × σ2) satisfies τπL∞(σ1×σ2) ≤ T and Tf,χΞσ2 = τπ, f × χΞσ1×σ2 for f ∈ L1(σ1) for each π-measurable Ξ (i.e., π refines {Ξ,Ω2 \ Ξ}). Thanks to the w*- compactness of φ ∈ L∞(σ1 × σ2); φL∞(σ1×σ2) ≤ T one can take a w*-limit point τ of {τπ}π∈Π. Then it is easy to see that Tf,χΞσ2 = τ,f × χΞ σ1×σ2 = Ξ Ω1 τ(x, y)f(x) dσ1(x) dσ2(y) for each f ∈ L1(Ω1; σ1) and each measurable set Ξ ⊆ Ω2. This implies the desired integral expression, and the uniqueness of τ is obvious. 2.2 Extension to B(H) We assume the condition (iv) in Theorem 2.2 (i.e., φ(s, t) admits the integral expression (2.3) with the finiteness condition described in (iv)) and will explain how to extend Φ(·) to a bounded transformation on B(H) by making use of the duality B(H) = C1(H)∗ via (X, Y ) ∈ C1(H) × B(H) → Tr(XY ) ∈ C. To do so, we first note that the roles of the variables s, t (and those of dEs and dFt) are symmetric. Thus, the function φ˜(t, s) = φ(s, t) = Ω β(t, x)α(s, x) dσ(x) gives rise to the following transformation on C1(H): Φ˜(X) = K 0 H 0 φ˜(t, s) dFtXdEs. 2.2 Extension to B(H) 19 We consider its transpose Φ˜t on B(H) = C1(H)∗, that is, Tr(XΦ˜t (Y )) = Tr(Φ˜(X)Y ) for X ∈ C1(H), Y ∈ B(H). (2.18) Let us take X = ξ ⊗ ηc here. Then, the left side of (2.18) is obviously the inner product (Φ˜t(Y )ξ,η). On the other hand, we have Φ˜(X) = Ω ˜ξ(x) ⊗ η˜(x) c dσ(x) with ˜ξ(x) = K 0 β(t, x) dFtξ and ˜η(x) = H 0 α(s, x) dEsη (2.19) (see (2.4), but recall that the roles of α and β were switched). We claim that the right side of (2.18) (when X = ξ ⊗ ηc) is Ω(Y ˜ξ(x), η˜(x))dσ(x). In fact, for vectors ξ , η we have (Φ˜(X)Y ξ , η ) = Ω (Y ξ , η˜(x))(˜ξ(x), η ) dσ(x) = Ω (˜ξ(x), η )(ξ , Y ∗η˜(x)) dσ(x) thanks to (2.5). Let {en}n=1,2,··· be an orthonormal basis for H. Since Φ˜(X)Y ∈ C1(H), from the preceding expression we get Tr(Φ˜(X)Y ) = ∞ n=1 (Φ˜(X)Y en, en) = ∞ n=1 Ω (˜ξ(x), en)(en, Y ∗η˜(x)) dσ(x) (see [29, Chapter III, §8]). Here, we would like to switch the order of ∞ n=1 and Ω, which is guaranteed by the Fubini theorem thanks to the following integrability estimate: Ω ∞ n=1 |(˜ξ(x), en)(en, Y ∗η˜(x))| dσ(x) ≤ Ω ∞ n=1 |(˜ξ(x), en)| 2 1/2 ∞ n=1 |(en, Y ∗η˜(x))| 2 1/2 dσ(x) = Ω ˜ξ(x)×Y ∗η˜(x) dσ(x) ≤ Y Ω ˜ξ(x)×η˜(x) dσ(x) < ∞ (see (2.6) and (2.7)). Hence, we get Tr(Φ˜(X)Y ) = Ω ∞ n=1 (˜ξ(x), en)(en, Y ∗η˜(x)) dσ(x) = Ω (˜ξ(x), Y ∗η˜(x)) dσ(x) = Ω (Y ˜ξ(x), η˜(x)) dσ(x). 20 2 Double integral transformations Therefore, the claim has been proved, and (for X = ξ ⊗ ηc) (2.18) means (Φ˜t (Y )ξ,η) = Ω (Y ˜ξ(x), η˜(x)) dσ(x) (2.20) with the vectors ˜ξ(x) and ˜η(x) defined by (2.19). When Y = ξ ⊗ ηc, the right side of (2.20) is Ω (˜ξ(x), η )(ξ , η˜(x)) dσ(x) = Ω K 0 β(t, x)dFtξ,η ξ , H 0 α(s, x)dEsη dσ(x) = Ω ξ, K 0 β(t, x)dFtη H 0 α(s, x)dEsξ , η dσ(x) = Ω (Y (x)ξ,η) dσ(x) with the rank-one operator Y (x) = H 0 α(s, x) dEsξ ⊗ K 0 β(t, x) dFtη c . But, notice that the two involved vectors here are exactly those defined from ξ and η according to the formula (2.4). Therefore, we have shown Φ˜t (Y ) = Ω Y (x) dσ(x) = Φ(Y ) (2.21) for a rank-one (and hence finite-rank) operator Y . For a general Hilbert-Schmidt class operator Y , we choose a sequence {Yn}n=1,2,··· of finite-rank operators tending to Y in ·2. Since the convergence is also valid in the operator norm and Φ˜t (being defined as a transpose) is bounded relative to the operator norm, we have Φ˜t (Y ) = ·- limn→∞ Φ˜t(Yn). On the other hand, we know Φ(Y ) = ·2- limn→∞ Φ(Yn) = ·2- limn→∞ Φ˜t (Yn) thanks to (2.1) and (2.21). Therefore, we conclude Φ˜t(Y ) = Φ(Y ) so that Φ˜t is indeed an extension of Φ (originally defined on C2(H)). The discussions so far justify the use of the notation Φ(Y ) (for Y ∈ B(H)) for expressing Φ˜t (Y ), and we shall also use the symbolic notation Φ(Y ) (= ΦH,K(Y )) = H 0 K 0 φ(s, t) dEsY dFt (for Y ∈ B(H)) in the rest of the monograph. 2.3 Norm estimates 21 Remark 2.5. (i) The map Φ : X ∈ B(H) → Φ(X) ∈ B(H) is always w*-w*-continuous (i.e., σ(B(H), C1(H))-σ(B(H), C1(H))-continuous) because it was defined as the transpose of the bounded transformation Φ˜ on C1(H). (ii) From (2.19) and (2.20) we observe (Φ(Y )ξ,η) = Ω (Y β(K, x)ξ,α(H, x) ∗η) dσ(x) = Ω (α(H, x)Y β(K, x)ξ,η) dσ(x) with the usual function calculus α(H, x) = H 0 α(H, x) dEs and β(K, x) = K 0 β(t, x) dFt. Therefore, Φ(X) (for X ∈ B(H)) can be simply written as the integral Φ(X) = Ω α(H, x)Xβ(K, x) dσ(x) in the weak sense. Remark that the integral expression (2.3) for ϕ(s, t) is far from being unique. Nevertheless, there is no ambiguity for the definition of Φ(X). Indeed, the definition of Φ˜(X) (= Φ˜ |C1(H) (X)) for X ∈ C1(H) (⊆ C2(H)) does not depend on this expression (see (2.2)), and Φ(X) (for X ∈ B(H) = C(H)∗) was defined as the transpose. (iii) From the expression in (ii) we obviously have f(H)(Φ(X))g(K) = Φ(f(H)Xg(K)) for all bounded Borel functions f,g. 2.3 Norm estimates We begin by investigating a relationship between the two norms Φ(∞,∞) = sup{Φ(X) : X ≤ 1}, Φ(1,1) = sup{Φ(X)1 : X1 ≤ 1}. To do so, besides Φ and Φ˜ we also make use of the following auxiliary double integral operator: Φ¯(X) = H 0 K 0 φ(s, t) dEsXdFt.Proposition 2.6. (M. Sh. Birman and M. Z. Solomyak, [16]) For a Schur multiplier φ ∈ L∞([0, H] × [0, K]; λ × µ) we have Φ(1,1) = Φ(∞,∞). Proof. For X ∈ C2(H) we easily observe Φ¯(X∗)∗ = Φ˜(X) and hence Φ˜(1,1) = Φ¯(1,1) by restricting the both sides to C1(H) (⊆ C2(H)). On the other hand, Φ(∞,∞) = Φ˜(1,1) is obvious from the definition, i.e., Φ was defined as a transpose. Therefore, to prove the proposition it suffices to see Φ(1,1) = Φ¯(1,1). One expresses H and E in the direct integral form as follows: H = ⊕ [0,H] H(s) dλ(s), EΛ = ⊕ [0,H] χΛ(s)1H(s) dλ(s) for Borel sets Λ ⊆ [0, H]. Note that it is the central decomposition of the von Neumann algebra {EΛ : Λ ⊆ [0, H]} over its center {EΛ : Λ ⊆ [0, H]} ∼= L∞([0, H]; λ). (See [17, Chapter 7, §2] for more “operator-theoretical description”.) Similarly, one can write H = ⊕ [0,K] H˜(t) dµ(t), FΞ = ⊕ [0,K] χΞ(t)1H˜(t) dµ(t) for Borel sets Ξ ⊆ [0, K]. A standard argument in the theory of direct integral shows that C2(H) is represented as the direct integral C2(H) = ⊕ [0,H]×[0,K] C2(H(s), H˜(t)) d(λ × µ)(s, t) with the Hilbert-Schmidt class operators C2(H(s), H˜(t)) from H(s) into H˜(t). Take an X = ⊕ [0,H]×[0,K] X(s, t) d(λ × µ)(s, t) in C2(H). Since EΛXFΞ = ⊕ [0,H]×[0,K] χΛ×Ξ (s, t)X(s, t) d(λ × µ)(s, t) for Borel sets Λ ⊆ [0, H] and Ξ ⊆ [0, K], it is immediate to see that Φ(X) and Φ¯(X) are written as Φ(X) = ⊕ [0,H]×[0,K] φ(s, t)X(s, t) d(λ × µ)(s, t), Φ¯(X) = ⊕ [0,H]×[0,K] φ(s, t)X(s, t) d(λ × µ)(s, t) 2.3 Norm estimates 23 respectively. The measurable cross-section theorem guarantees that one can select measurable fields {J(s) : s ∈ [0, H]} and {J˜(s) : s ∈ [0, K]} of (conjugate linear) involutions J(s) : H(s) → H(s), J˜(s) : H˜(s) → H˜(s), and they give rise to the global involutions J = ⊕ [0,H] J(s) dλ(s) and J˜ = ⊕ [0,K] J˜(t) dµ(t) on the Hilbert space H. Then we observe JΦ˜ (X)J = ⊕ [0,H]×[0,K] φ(s, t)J˜(t)X(s, t)J(s) d(λ × µ)(s, t) = Φ¯(JXJ ˜ ). Since the map X → JXJ ˜ is obviously isometric on C1(H), the equality Φ(1,1) = Φ¯(1,1) is now obvious and the proposition has been proved. For each unitarily invariant norm ||| · |||, let I|||·||| and I(0) |||·||| be the associated symmetrically normed ideals, that is, I|||·||| = {X ∈ B(H) : |||X||| < ∞}, I(0) |||·||| = the ||| · |||-closure of Ifin in I|||·|||, where Ifin is the ideal of finite-rank operators (see [29, 37, 77] for details). For a Schur multiplier φ(t, s) we have shown Φ(X)1 ≤ kX1 (X ∈ C1(H)) and Φ(X) ≤ kX (X ∈ B(H)) (2.22) with k = Φ(1,1) = Φ(∞,∞) (Proposition 2.6). The next result says that φ(s, t) is automatically a “Schur multiplier for all operator ideals I|||·|||, I(0) |||·|||” with the same bound for Φ(|||·|||,|||·|||) = sup{|||Φ(X)||| : |||X||| ≤ 1}. Proposition 2.7. Let φ(s, t) be a Schur multiplier with κ = Φ(1,1) = Φ(∞,∞) (< ∞). For any unitarily invariant norm ||| · ||| we have |||Φ(X)||| ≤ κ|||X||| (≤ ∞) for all X ∈ B(H) so that Φ maps I|||·||| into itself. Moreover, Φ also maps the separable operator ideal I(0) |||·||| into itself. In particular, Φ(X) is a compact operator as long as X is.Proof. Recall the following expression for the Ky Fan norm as a K-functional: |||X|||(n) = n k=1 µk(X) = inf{nX0 + X11 : X = X0 + X1} (n = 1, 2, ···), where {µk(·)}k=1,2,··· denotes the singular numbers (see [26, p. 289] for example). This expression together with (2.22) clearly shows |||Φ(X)|||(n) ≤ κ|||X|||(n) for each n, which is known to be equivalent to the validity of |||Φ(X)||| ≤ κ|||X||| for each unitarily invariant norm (see [37, Proposition 2.10]). It remains to show Φ I(0) |||·||| ⊆ I(0) |||·|||. When X is a finite-rank operator, Φ(X) is of trace class and can be approximated by a sequence {Yn}n=1,2,··· of finite-rank operators in the ·1-norm. Notice |||Φ(X) − Yn||| ≤ Φ(X) − Yn1 −→ 0, showing Φ(X) ∈ I(0) |||·|||. For a general X ∈ I(0) |||·|||, one chooses a sequence {Xn}n=1,2,··· of finite-rank operators satisfying limn→∞ |||X −Xn||| = 0. Since Φ(Xn) ∈ I(0) |||·||| is already shown, the estimate |||Φ(X) − Φ(Xn)||| ≤ κ|||X − Xn||| → 0 (as n → ∞) guarantees Φ(X) ∈ I(0) |||·|||. 2.4 Technical results Here we collect technical results. When we deal with integral expressions of means of operators in later chapters, a careful handling for supports of relevant operators will be required and some lemmas are prepared for this purpose. In the sequel we will denote the support projection of H by sH. Lemma 2.8. Let φ, ψ be Schur multipliers (relative to (H, K)) with the corresponding double integral transformations Φ, Ψ respectively. Then, the pointwise product φ(s, t)ψ(s, t) is also a Schur multiplier, and the corresponding double integral transformation is the composition Φ ◦ Ψ (= Ψ ◦ Φ). Proof. As in Theorem 2.2, (iv) we can write φ(s, t) = Ω α(s, x)β(t, x) dσ(x), ψ(s, t) = Ω α (s, y)β (t, y) dσ (y). We consider the product space Ω × Ω equipped with the product measure σ × σ , and set 2.4 Technical results 25 a : (s, x, y) ∈ [0, H] × Ω × Ω → α(s, x)α (s, y), b : (t, x, y) ∈ [0, K] × Ω × Ω → β(t, x)β (t, y). At first we note Ω×Ω |a(s, x, y)| 2d(σ × σ )(x, y) = Ω |α(s, x)| 2dσ(x) × Ω |α (s, y)| 2dσ (y) ≤ Ω |α(·, x)| 2dσ(x) L∞(λ) × Ω |α (·, y)| 2dσ (y) L∞(λ) (and the similar estimate for b). Secondly, the Cauchy-Schwarz inequality implies Ω×Ω |a(s, x, y)b(t, x, y)| d(σ × σ )(x, y) ≤ Ω×Ω |a(s, x, y)| 2d(σ × σ )(x, y) 1/2 × Ω×Ω |b(t, x, y)| 2d(σ × σ )(x, y) 1/2 . From the two estimates we see the σ × σ -integrability of a(s, x, y)b(t, x, y), and the Fubini theorem clearly shows Ω×Ω a(s, x, y)b(t, x, y) d(σ × σ )(x, y) = Ω α(s, x)β(t, x) dσ(x) × Ω α (s, y)β (t, y) dσ (y) = φ(s, t)ψ(s, t). Therefore, the conditions stated in Theorem 2.2, (iv) have been checked for the product φ(s, t)ψ(s, t), and it is indeed a Schur multiplier. Let Π be the double integral transformation corresponding to φ(s, t)ψ(s, t). Then, it is straight-forward to see Π(X) = Φ(Ψ(X)) for each rank-one (hence finite-rank) operator X. Let {pn}n=1,2,··· be a sequence of finite-rank projections tending to 1 in the strong operator topology. Then, for each X ∈ B(H) the sequence {pnXpn} tends to X strongly and hence in the σ(B(H), C1(H))- topology (because of pnXpn≤X). Since Π(pnXpn) = Φ(Ψ(pnXpn)) as remarked above, by letting n → ∞ here, we conclude Π(X) = Φ(Ψ(X)) due to the continuity stated in Remark 2.5, (i). The additive version (which is much easier) is also valid. Namely, when φ, ψ are Schur multipliers, then so is the sum φ(s, t)+ψ(s, t) and the corresponding double integral transformation sends X to Φ(X) + Ψ(X). Lemma 2.9. Let φ(s, t) be a Schur multiplier (relative to (H, K)) with the corresponding double integral transformation Φ. With the support projections sH, sK of H, K we have sH(Φ(X))sK = Φ(sHXsK) and 26 2 Double integral transformations Φ(X) = sHΦ(X)sK + φ(H, 0)sHX(1 − sK) + (1 − sH)XsKφ(0, K) +φ(0, 0)(1 − sH)X(1 − sK). Proof. The equation sH(Φ(X))sK = Φ(sH XsK) is seen from Remark 2.5, (iii). Recall the following expression mentioned in Remark 2.5, (ii): Φ(X) = Ω α(H, x)Xβ(K, x) dσ(x) in the weak sense. Since α(H, x) = α(H, x)sH + α(0, x)(1 − sH), β(K, x) = β(K, x)sK + β(0, x)(1 − sK), we have α(H, x)Xβ(K, x) = α(H, x)sH XsKβ(K, x) +β(0, x)α(H, x)sH X(1 − sK) + α(0, x)(1 − sH)XsKβ(K, x) +α(0, x)β(0, x)(1 − sH)X(1 − sK). The integration of the first term over Ω is Φ(sHXsK). The second term gives us Ω β(0, x)α(H, x)sH X(1 − sK) dσ(x) = Ω H 0 α(s, x)β(0, x) dEs dσ(x) sHX(1 − sK) = H 0 Ω α(s, x)β(0, x) dσ(x) dEs sHX(1 − sK) = H 0 φ(s, 0) dEs sHX(1 − sK) = φ(H, 0)sHX(1 − sK). Of course the third term admits a similar integration. The last term gives us Ω α(0, x)β(0, x)(1 − sH)X(1 − sK) dσ(x) = Ω α(0, x)β(0, x) dσ(x) (1 − sH)X(1 − sK) = φ(0, 0)(1 − sH)X(1 − sK). The above estimates altogether yield the desired expression for Φ(X). 2.4 Technical results 27 We can consider sH(ΦH,K(X))sK as an operator from sKH to sHH, and denote it by ΦHsH ,KsK (sH XsK). It is possible to justify this (symbolic) notation by making use of double integral transformation for operators between two different spaces. The above lemma actually shows sH(ΦH,K(X))sK = ΦHsH ,KsK (sH XsK), sH(ΦH,K(X))(1 − sK) = sHφ(H, 0)X(1 − sK), (1 − sH)(ΦH,K(X))sK = (1 − sH)Xφ(0, K)sK, (1 − sH)(ΦH,K(X))(1 − sK) = φ(0, 0)(1 − sH)X(1 − sK). When dealing with means in later chapters we will mainly use Schur multipliers satisfying φ(s, 0) = φ(0, s) = bs (s ≥ 0) for some constant b ≥ 0. Then, the expression in Lemma 2.9 becomes ΦH,K(X) = sH(ΦH,K(X))sK + b (HX(1 − sK) + (1 − sH)XK) (2.23) thanks to φ(H, 0)sH = bHsH = bH, φ(0, K)sK = bKsK = bK and φ(0, 0) = 0. We fix signed measures νk (k = 1, 2, 3) on the real line R with finite total variation and also a scalar a. With the Fourier transforms of these measures we set a bounded function π on [0, ∞) × [0, ∞) as π(s, t) =    νˆ1(log s − log t) if s, t > 0, νˆ2(log s) if s > 0 and t = 0, νˆ3(− log t) if s = 0 and t > 0, a if s = t = 0. Lemma 2.10. The above π(s, t) is a Schur multiplier for any pair (H, K) of positive operators, and the corresponding double integral transformation Π is given by Π(X) = ∞ −∞ (HsH) ixX(KsK) −ixdν1(x) + ∞ −∞ (HsH) ixX(1 − sK) dν2(x) + ∞ −∞ (1 − sH)X(KsK) −ixdν3(x) +a(1 − sH)X(1 − sK). We give a few remarks before proving the lemma. In the above expression, (HsH)ix for instance denotes a unitary operator on sHH and it is zero on the orthogonal complement (1 − sH)H, i.e., (HsH)ix = (HsH)ixsH. We will mainly use this lemma (as well as the next Proposition 2.11) in the following special circumstances: 28 2 Double integral transformations π(s, 0) = π(0, t) = c (s > 0, t> 0) for some constant c and π(0, 0) = 0. This means ν2 = ν3 = cδ0 and a = 0, and hence in this case the expression in the lemma simply becomes Π(X) = ∞ −∞ (HsH) ixX(KsK) −ixdν1(x) +c(sHX(1 − sK) + (1 − sH)XsK). Proof. We decompose the domain {(s, t) : s, t ≥ 0} into the four regions {(s, t) : s, t > 0}, {(s, t) : s > 0, t = 0}, {(s, t) : s = 0,t> 0}, {(s, t) : s = t = 0}. We accordingly set π1(s, t) = π(s, t) if s, t > 0, 0 otherwise, π2(s, t) = π(s, 0) if s > 0 and t = 0, 0 otherwise, π3(s, t) = π(0, t) if s = 0 and t > 0, 0 otherwise, π4(s, t) = π(0, 0) if s = t = 0, 0 otherwise. So π(s, t) = 4 k=1 πk(s, t) is valid. We consider the following functions on R+ × R: α1(s, x) = six dν1 d|ν1|(x) if s > 0, 0 if s = 0, β1(t, x) = t−ix if t > 0, 0 if t = 0, α2(s, x) = six dν2 d|ν2|(x) if s > 0, 0 if s = 0, β2(t, x) = 0 if t > 0, 1 if t = 0, α3(s, x) = 0 if s > 0, 1 if s = 0, β3(t, x) = t−ix dν3 d|ν3|(x) if t > 0, 0 if t = 0, α4(s) = 0 if s > 0, a if s = 0, β4(t) = 0 if t > 0, 1 if t = 0. Here, dνk d|νk|(x) denotes the Radon-Nikodym derivative relative to the absolute value |νk|. It is plain to observe πk(s, t) = ∞ −∞ αk(s, x)βk(t, x) d|νk|(x) (for k = 1, 2, 3) and also π4(s, t) = α4(s)β4(t). The finiteness condition in Theorem 2.2, (iv) is obviously satisfied (since dνk d|νk| ’s are bounded functions and |νk|’s are finite measures) so that all πk’s are Schur multipliers. Thus, so is the sum π as was mentioned in the paragraph right after Lemma 2.8. We begin with π1 (with the corresponding double integral transformation Π1). Since π1(s, t) = 0 for either s = 0 or t = 0, we note Π1(X) = sH(Π1(X))sK by Lemma 2.9. For a rank-one operator X = ξ ⊗ ηc, (2.4) shows 2.4 Technical results 29 Π1(X) = ∞ −∞ (sHH) ixξ ⊗ (sKK) ixη c dν1 d|ν1| (x) d|ν1|(x) = ∞ −∞ (sHH) ix(ξ ⊗ ηc )(sKK) −ixdν1(x) = ∞ −∞ (sHH) ixX(sKK) −ixdν1(x), which remains of course valid for finite-rank operators. Actually this integral expression for Π1(X) is also valid for an arbitrary operator X ∈ B(H). In fact, as in the proof of Lemma 2.8 we approximate X by the sequence {pnXpn}n=1,2,···. At first Π1(pnXpn) tends to Π1(X) in the weak operator topology as remarked there. Therefore, it suffices to show the weak convergence ∞ −∞ (HsH) ixpnXpn(KsK) −ixdν1(x) −→ ∞ −∞ (HsH) ixX(KsK) −ixdν1(x). However, it simply follows from the Lebesgue dominated convergence theorem. We next consider π2 (with the double integral transformation Π2). By Lemma 2.9 (and Remark 2.5) we have Π2(X) = sH(Π2(X))(1 − sK) = π2(H, 0)sHX(1 − sK). Recall π2(s, 0) = ˆν2(log s) (s > 0) so that π2(H, 0)sH = (0,H] νˆ2(log s) dEs = (0,H] ∞ −∞ sixdν2(x) dEs = ∞ −∞ (HsH) ixdν2(x) due to the Fubini theorem. Therefore, we have Π2(X) = ∞ −∞ (HsH) ixX(1 − sK) dν2(x). Symmetric arguments also show Π3(X) = (1 − sH)XsKπ3(0, K) = ∞ −∞ (1 − sH)X(KsK) −ixdν3(x) while Π4(X) = (1 − sH)(Π4(X))(1 − sK) = a(1 − sH)X(1 − sK) is just trivial. By summing up all the Πk’s computed so far, we get the desired expression for Π(X). 30 2 Double integral transformations Proposition 2.11. Let π(s, t) be the Schur multiplier in the previous lemma. If φ(s, t) is a Schur multiplier relative to a pair (H, K), then so is the pointwise product ψ(s, t) = π(s, t)φ(s, t). Furthermore, for each X ∈ B(H) the corresponding double integral transformations Φ(X) and Ψ(X) are related by Ψ(X) = ∞ −∞ (HsH) ix(Φ(X))(KsK) −ixdν1(x) + ∞ −∞ (HsH) ix(Φ(X))(1 − sK) dν2(x) + ∞ −∞ (1 − sH)(Φ(X))(KsK) −ixdν3(x) +a(1 − sH)(Φ(X))(1 − sK). Proof. The first statement follows from Lemmas 2.8 and 2.10. To get the expression for Ψ(X), in the formula appearing in Lemma 2.10 we should just replace X by Φ(X). We end the chapter with the following remark on the standard 2×2-matrix trick, that will be sometimes useful in later chapters: Remark 2.12. We set H˜ = H 0 0 K , and assume that φ is a Schur multiplier relative to (H, ˜ H˜ ) (or equivalently, so is φ relative to (H, H), (H, K) and (K, K)). Then, φ (on [0, H˜ ] × [0, H˜ ]) admits an integral expression as (2.3) relative to (H, ˜ H˜ ). For X˜ = 0 X 0 0 we compute α(H,x ˜ )Xβ˜ (H,x ˜ ) = α(H, x) 0 0 α(K, x) 0 X 0 0 β(H, x) 0 0 β(K, x) = 0 α(H, x)Xβ(K, x) 0 0 . Therefore, the (1, 2)-component of ΦH, ˜ H˜ X˜ = Ω α(H,x ˜ )Xβ˜ (H,x ˜ ) dσ(x) is exactly Ω α(H, x)Xβ(K, x) dσ(x) = ΦH,K(X). The support projection of H˜ is of the form sH˜ = sH 0 0 sK , and sH(ΦH,K(X))sK is the (1, 2)-component of sH˜ ΦH, ˜ H˜ X˜sH˜ . 2.5 Notes and references 31 2.5 Notes and references Motivated from perturbations of a continuous spectrum, scattering theory and triangular representations of Volterra operators (see [30]) as well as study of Hankel operators (see [71] for recent progress of the subject matter), in [14, 15, 16] M. Sh. Birman and M. Z. Solomyak systematically developed theory of double integral transformations formally written as Y = φ(s, t) dFtXdEs Besides the definition given at the beginning of this chapter (first defined on C2(H)), another definition by repeated integration Y (s) = φ(s, t) dFt X, Y = Y (s) dEs (2.24) was also taken by Birman and Solomyak, where the latter integration is understood as the limit of Riemann-Stieltjes sums. Indeed, the articles [15, 16] were largely devoted to the well-definedness of the repeated integration in certain symmetric operator ideals in cases when φ is a function in some classes of Lipschitz type or of Sobolev type. For example, the following criterion was obtained: Theorem Let φ(s, t) be a bounded Borel function on [a, b] × [c, d] satisfying Lip α with respect to variable s with a constant (of H¨older continuity of order α) independent of t. Assume that Es and Ft are supported in [a, b] and [c, d] respectively. If α > 1 2 , then φ is a Schur multiplier and for any X ∈ B(H) the repeated integral (2.24) exists and coincides with Φ(X) (defined in §2.1). If α ≤ 1 2 , then for any X ∈ Cp(H) with 1 p > 1 2 − α the repeated integral (2.24) exists as a compact operator. But this type of results are not so useful in the present monograph because we mostly treat means (introduced in Definition 3.1) which do not at all satisfy the Lipschitz type condition. As was shown in [69, 70] (also [15]), double integral transformations are closely related to problems of operator perturbations. For a C1-function ϕ on an interval I (⊆ R) and self-adjoint operators A = s dEs, B = t dFt with spectra contained in I we formally have ϕ(A) − ϕ(B) = I I ϕ[1](s, t) dEs(A − B)dFt (2.25) with the divided difference ϕ[1](s, t) =    ϕ(s) − ϕ(t) s − t (if s = t), ϕ (s) (if s = t).If ϕ[1](s, t) is known to be a Schur multiplier relative to say some p-Schatten ideal Cp(H), then (2.25) for A − B sitting in the ideal is justified and hence one gets the perturbation norm inequality ϕ(A) − ϕ(B)p ≤ const. A − Bp, (2.26) showing ϕ(A)−ϕ(B) ∈ Cp(H), i.e., the stability of perturbation. The following is a folk result (whose proof is an easy but amusing exercise): If ϕ(s) is of the form ϕ(s) = ∞ −∞ eistdν(t) with a signed measure ν satisfying ∞ −∞(1 + |t|) d|ν|(t) < ∞, then ϕ[1](s, t) is a Schur multiplier relative to C1(H) (and hence relative to any Cp(H)). On the other hand, in [27] Yu. B. Farforovskaya obtained an example of ϕ ∈ C1(I) for which (2.26) fails to hold for ·1. The next result due to E. B. Davies is very powerful: Theorem ([24, Theorem 17]) Let ϕ be a function of the form ϕ(s) = as + b + s −∞ (s − t) dν(t) with a, b ∈ R and a signed measure ν of compact support. Then, the estimate (2.26) is valid for any p ∈ (1, ∞). The following “unitary version” of (2.25) is also useful: If ϕ is a C1-function on the unit circle T (with a Schur multiplier ϕ[1](s, t)), then we have ϕ(U) − ϕ(V ) = T T ϕ[1](ζ,η) dEζ (U − V ) dFη for unitary operators U = T ζ dEζ , V = T η dFη. This technique was often used in M. G. Krein’s works and is closely related to his famous spectral shift function. Peller’s characterization theorem (Theorem 2.2) was given in [69] ([70] is an announcement) while general results such as Propositions 2.6 and 2.7 were shown in [15, 16] by M. Sh. Birman and M. Z. Solomyak. Unfortunately these articles [15, 16, 69] (especially [69]) were not widely circulated. Our arguments here are basically taken from their articles, but we have tried to present more details. In fact, for the reader’s convenience we have supplied some arguments that were omitted in the original articles. 3 Means of operators and their comparison From now on we will study means M(H, K)X of operators H, K, X with H, K ≥ 0 (for certain scalar means M(s, t)). In fact, our operator means M(H, K)X are defined as double integral transformations studied in Chapter 2 so that corresponding scalar means M(s, t) are required to be Schur multipliers. In this chapter general properties of such operator means are clarified while some special series of concrete means will be exemplified in later chapters. Here we are mostly concerned with integral expressions (Theorem 3.4), comparison of norms (Theorem 3.7), norm estimate (Theorem 3.12) and the determination of the kernel and the closure of the range of the “mean transform” M(H, K) (Theorem 3.16). 3.1 Symmetric homogeneous means We begin by introducing a class of means for positive scalars and a partial order among them. This order will be quite essential in the sequel of the monograph. We confine ourselves to that class of means for convenience sake while all the results in the next §3.2 remain valid (with obvious modification) for more general means (as will be briefly discussed in §A.1). Definition 3.1. A continuous positive real function M(s, t) for s, t > 0 is called a symmetric homogeneous mean (or simply a mean) if M satisfies the following properties: (a) M(s, t) = M(t, s), (b) M(rs, rt) = rM(s, t) for r > 0, (c) M(s, t) is non-decreasing in s, t, (d) min{s, t} ≤ M(s, t) ≤ max{s, t}. We denote by M the set of all such symmetric homogeneous means. Definition 3.2. We assume M,N ∈ M. We write M N when the ratio M(ex, 1)/N(ex, 1) is a positive definite function on R, or equivalently, the F. Hiai and H. Kosaki: LNM 1820, pp. 33–55, 2003. c Springer-Verlag Berlin Heidelberg 2003 34 3 Means of operators and their comparison matrix M(si, sj ) N(si, sj ) i,j=1,··· ,n is positive semi-definite for any s1,...,sn > 0 with any size n. By the Bochner theorem it is also equivalent to the existence of a symmetric probability measure ν on R satisfying M(ex, 1) = ˆν(x)N(ex, 1) (x ∈ R), that is, M(s, t)=ˆν(log s − log t)N(s, t) (s, t > 0). (3.1) Here, ˆν(x) means the Fourier transform ˆν(x) = ∞ −∞ eixydν(y) (x ∈ R). The reason why a symmetric probability ν comes out is that the real function M(ex, 1)/N(ex, 1) takes value 1 at the origin. (See [39, Theorem 1.1] for details.) Also, note that the order M N is strictly stronger than the usual (point-wise) order M(s, t) ≤ N(s, t) (s, t > 0) (see [39, Example 3.5]). The domain of M ∈ M naturally extends to [0, ∞) × [0, ∞) in the following way: M(s, 0) = limt0 M(s, t) (s > 0), M(0, t) = lims0 M(s, t) (t > 0), M(0, 0) = lims0 M(s, 0) = limt0 M(0, t), and M(s, t) remains continuous on the extended domain. It is easy to check M(s, 0) = M(0, s) = sM(1, 0) (s > 0) (3.2) and hence M(0, 0) = 0. (3.3) The most familiar means in M are probably A(s, t) = s + t 2 (arithmetic mean), L(s, t) = s − t log s − log t = 1 0 sxt 1−xdx (logarithmic mean), G(s, t) = √ st (geometric mean), Mhar(s, t) = 2 s−1 + t−1 (harmonic mean). The largest and smallest means in M M∞(s, t) = max{s, t} and M−∞(s, t) = min{s, t} will play an important role in our discussions below. We have the following order relation among the above means: M−∞ Mhar G L A M∞. (3.4) The proof is found in the more general [39, Theorem 2.1] (i.e., (5.2) right before Theorem 5.1 in Chapter 5; see also [38, Proposition 1 or more generally 3.1 Symmetric homogeneous means 35 Theorem 5]). However, here for the reader’s convenience we prove this special case by bare-handed computations. Firstly Example 3.6, (c) below shows A M∞. For L A we just note L(ex, 1) A(ex, 1) = ex − 1 x × 2 ex + 1 = 2 sinh(x/2) x cosh(x/2) = 1 0 cosh(ax/2) cosh(x/2) da. Since cosh(ax/2)/ cosh(x/2) is positive definite for each a ∈ [0, 1] (see §6.3, 1), so is the above integral. (The Fourier transform can be also explicitly determined; see (6.8) or the computations in [38, p. 305].) For G L we observe G(ex, 1) L(ex, 1) = ex/2 × x ex − 1 = x 2 sinh(x/2). The well-known formula ∞ −∞ x 2 sinh(x/2) eixydx = 1 4 cosh2(πy) (3.5) and its inverse transform guarantee the positive definiteness of the ratio. Finally, both of Mhar(ex, 1) G(ex, 1) = 2 e−x + 1 × e−x/2 = 1 cosh(x/2), M−∞(ex, 1) Mhar(ex, 1) = min{ex, 1} × e−x + 1 2 = e−|x| + 1 2 are obviously positive definite (see (5.8) and (7.3)), and we are done. Now let H, K be positive operators in B(H) with the spectral decompositions H = H 0 s dEs and K = K 0 t dFt. For a mean M ∈ M we would like to define the corresponding double integral transformation relative to the pair (H, K): M(H, K)X = MH,K(X) = H 0 K 0 M(s, t) dEsXdFt for X ∈ B(H), and we consider this transformation acting on operators on H as a “mean of H and K”. The transformation M(H, K) always makes sense if restricted on the Hilbert-Schmidt class C2(H) (in particular, on the ideal Ifin); it is the function calculus on C2(H) via M(s, t) of the left multiplication by H and the right multiplication by K. But, to define M(H, K) = MH,K on the whole B(H), we have to verify that M is a Schur multiplier relative to (H, K). For instance, if H, K have finite spectra so that they have the discrete spectral decompositions H = m i=1 siPi and K = n j=1 tjQj 36 3 Means of operators and their comparison with projections Pi, Qj such that m i=1 Pi = n j=1 Qj = 1, then each M ∈ M is a Schur multiplier relative to (H, K) and M(H, K)X = m i=1 n j=1 M(si, tj)PiXQj (this is the case even for any Borel function on [0, ∞) × [0, ∞)). In what follows we simply say that M ∈ M is a Schur multiplier if it is so relative to any pair (H, K) of positive operators. As for the means A, L and G, the corresponding double integral transformations have the concrete forms A(H, K)X = 1 2 (HX + XK), L(H, K)X = 1 0 HxXK1−xdx, G(H, K)X = H 1 2 XK 1 2 , showing that they are indeed Schur multipliers. But it is not so obvious to determine whether a given M ∈ M is a Schur multiplier. The next proposition provides a handy sufficient condition. Proposition 3.3. Let M,N ∈ M and H, K be positive operators. (a) If M N and N is a Schur multiplier relative to (H, K), then so is M. (b) If M M∞, then M is a Schur multiplier (relative to any (H, K)). Proof. (a) By Definition 3.2 there exists a symmetric probability measure ν satisfying (3.1). Noting M(1, 0) ≤ N(1, 0) (following from M(s, t) ≤ N(s, t) when s, t > 0) we set c = M(1, 0)/N(1, 0) if N(1, 0) > 0, otherwise c = 0. Then, thanks to (3.2) and (3.3) we have M(s, t) = π(s, t)N(s, t) for all s, t ≥ 0 with π(s, t) =    νˆ(log s − log t) if s, t > 0, c if s > 0 and t = 0, c if s = 0 and t > 0, 0 if s = t = 0. (3.6) Hence the assertion is a consequence of Proposition 2.11 (based on Lemma 2.8). (b) By virtue of (a) it suffices to show that M∞ is a Schur multiplier. Since A is obviously a Schur multiplier (as was mentioned above), (a) implies by (3.4) that M−∞ is a Schur multiplier. Hence so is M∞ because of the simple formula M∞(s, t)=2A(s, t) − M−∞(s, t) (3.7) (see the remark after Lemma 2.8). 3.2 Integral expression and comparison of norms 37 The fact that M±∞ are Schur multipliers can be also seen from the discrete decompositions explained in §A.3 (see (A.4) and Theorem A.6). All the concrete means treated in later chapters satisfy M M∞ so that they are all Schur multipliers. It is easy to write down examples of M ∈ M not satisfying M M∞; nevertheless we have so far no explicit example of M ∈ M which is not a Schur multiplier. 3.2 Integral expression and comparison of norms We begin with the integral expression (Theorem 3.4) for operator means, which is an adaptation of the integral expression in Proposition 2.11 (also Lemma 2.9) in the present setting of means in M. (Similar integral expressions for wider classes of means will be worked out in §8.1 and §A.1.) Then, comparison of norms of means will be an easy consequence. In [49, p. 138] the following formula appears as an exercise: t∈R |µ({t})| 2 = lim T→∞ 1 2T T −T |µˆ(t)| 2dt for a complex measure µ on R. A related fact will be needed in the proof of the theorem, and the proofs for this fact as well as the above formula will be presented in §A.4 for the reader’s convenience. Theorem 3.4. Let M,N ∈ M and H, K be positive operators. If M N with the representing measure ν for M(ex, 1)/N(ex, 1) (see Definition 3.2) and if N is a Schur multiplier relative to (H, K), then so is M and M(H, K)X = ∞ −∞ (HsH) ix(N(H, K)X)(KsK) −ixdν(x) +M(1, 0)(HX(1 − sK) + (1 − sH)XK) (3.8) for all X ∈ B(H). In this case we also have M(H, K)X = {x=0} (HsH) ix(N(H, K)X)(KsK) −ixdν(x) +ν({0})N(H, K)X. (3.9) Proof. We use the same notations as in the proof of Proposition 3.3, (a). Use of Lemma 2.9 (see (2.23)) to N with (3.2) and (3.3) yields N(H, K)X = sH(N(H, K)X)sK +N(1, 0)(HX(1 − sK) + (1 − sH)XK). (3.10) Since M(s, t) = π(s, t)N(s, t) for all s, t ≥ 0 with π defined by (3.6), Proposition 2.11 implies 38 3 Means of operators and their comparison M(H, K)X = ∞ −∞ (HsH) ix(N(H, K)X)(KsK) −ixdν(x) +c sH(N(H, K)X)(1 − sK) + (1 − sH)(N(H, K)X)sK . Since M(1, 0) = cN(1, 0), the expression (3.8) is obtained by substituting (3.10) into the above integral expression. To show (3.9), we begin with the claim M(1, 0) = ν({0})N(1, 0). When N(1, 0) = 0, we must have M(1, 0) = 0 due to M(1, 0) ≤ N(1, 0) and there is nothing to prove. Thus we may and do assume N(1, 0) > 0. In this case we note lim x→−∞ νˆ(x) = lim x→−∞ M(ex, 1) N(ex, 1) = M(0, 1) N(0, 1) = M(1, 0) N(1, 0) , limx→∞ νˆ(x) = limx→∞ M(ex, 1) N(ex, 1) = limx→∞ M(1, e−x) N(1, e−x) = M(1, 0) N(1, 0). Therefore, we conclude lim x→±∞ νˆ(x) = M(1, 0) N(1, 0), and the claim follows from Corollary A.8 in §A.4. The claim and (3.8) yield M(H, K)X = {x=0} (HsH) ix(N(H, K)X)(KsK) −ixdν(x) +ν({0}) sH(N(H, K)X)sK + N(1, 0)(HX(1 − sK) + (1 − sH)XK) = {x=0} (HsH) ix(N(H, K)X)(KsK) −ixdν(x) + ν({0})N(H, K)X. Here, the second equality is due to (3.10). From the expression (3.9) in the preceding theorem and Theorem A.5 we have Corollary 3.5. Let M,N ∈ M (M N) and H, K be as in the theorem. Then for any unitarily invariant norm ||| · ||| we have |||M(H, K)X||| ≤ |||N(H, K)X||| for all X ∈ B(H). In particular, M(H, K)(|||·|||,|||·|||) ≤ N(H, K)(|||·|||,|||·|||). Example 3.6. The following examples are applications of the integral expression in the above theorem to means in (3.4). 3.2 Integral expression and comparison of norms 39 (a) Since the ratio G(ex, 1)/A(ex, 1) = cosh x 2 −1 is the Fourier transform of cosh(πx) −1 , H 1 2 XK 1 2 = ∞ −∞ (HsH) ix(HX + XK)(KsK) −ix dx 2 cosh(πx) . Actually, the observation of this expression is the starting point of our works on means of operators in a series of recent articles ([54, 38, 39]). We also point out that the use of this integral transformation was crucial in [22, 23]. (b) Since M−∞(ex, 1)/G(ex, 1) = e−|x|/2 is the Fourier transform of 1 2π x2 + 1 4 −1 (see (5.8) and (7.3)), M−∞(H, K)X = ∞ −∞ (HsH) 1 2 +ixX(KsK) 1 2 −ix dx 2π x2 + 1 4 . (c) Since A(ex, 1)/M∞(ex, 1) = 1 2 (1 + e−|x| ) is the Fourier transform of the measure 1 2 δ0 + 1 2π (x2 + 1)−1 dx, HX + XK = M∞(H, K)X + ∞ −∞ (HsH) ix(M∞(H, K)X)(KsK) −ix dx π(x2 + 1). The opposite direction of this is also possible. Since M−∞(ex, 1)/A(ex, 1) = 2e−|x| /(1 + e−|x| ) = e−|x|/2/ cosh x 2 is the Fourier transform of the convolution product f(x) = 1 cosh(πx) ∗ 1 2π x2 + 1 4 , one obtains thanks to (3.7) M∞(H, K)X = HX + XK − 1 2 ∞ −∞ (HsH) ix(HX + XK)(KsK) −ixf(x) dx. (3.11) The general comparison theorem for means in M was summarized in [39, Theorem 1.1] in the setting of matrices, and its extension to the operator setting was stated at the end of [39]. However, the statement there is quite rough and its sketch for the proof contains some inaccurate arguments. So, for completeness let us prove the next theorem in a precise form. Theorem 3.7. For M,N ∈ M the following conditions are all equivalent : (i) there exists a symmetric probability measure ν on R with the following property : if N is a Schur multiplier relative to (H, K) of non-singular positive operators, then so is M and 40 3 Means of operators and their comparison M(H, K)X = ∞ −∞ Hix(N(H, K)X)K−ixdν(x) for all X ∈ B(H); (ii) if N is a Schur multiplier relative to a pair (H, K) of positive operators, then so is M and |||M(H, K)X||| ≤ |||N(H, K)X||| for all unitarily invariant norms and all X ∈ B(H); (iii) M(H, H)X≤N(H, H)X for all H ≥ 0 and all X ∈ Ifin; (iv) M N. Proof. (iv) ⇒ (i) is contained in Theorem 3.4, and (iv) ⇒ (ii) follows from Corollary 3.5. When H, K and X are of finite-rank, (ii) and (iii) reduce to the same condition in the matrix case (of any size). So (ii) ⇒ (iv) and (iii) ⇒ (iv) are seen from [39, Theorem 1.1]. (Necessary arguments under a slightly weaker assumption will be actually presented in the proof of Theorem A.3 in §A.1.) For (i) ⇒ (iv) put H = s1 (s > 0) and K = X = 1; then the integral expression in (i) reduces to M(s, 1) = ˆν(log s)N(s, 1), i.e., M N. It remains to show (iv) ⇒ (iii), which is not quite trivial because N in (iii) is not a priori a Schur multiplier relative to (H, H). At first, when H is also of finite-rank, the inequality in (iii) follows from (iv) by [39, Theorem 1.1] (or from (ii) since we have already had (iv) ⇒ (ii)). For a general H choose a sequence {Hn} of finite-rank positive operators such that Hn≤H and Hn → H in the strong operator topology. Then π(Hn) → π(H) strongly on C2(H) because for a rank-one operator ξ ⊗ ηc we get π(Hn)(ξ ⊗ ηc) − π(H)(ξ ⊗ ηc)2 = (Hnξ − Hξ) ⊗ ηc2 = Hnξ − Hξ×η −→ 0. Similarly πr(Hn) → πr(H) strongly on C2(H). Since M(s, t) is uniformly approximated on [0, H] × [0, H] by polynomials in two variables s and t, it follows that M(Hn, Hn) → M(H, H) strongly on C2(H). For every X ∈ Ifin (⊆ C2(H)) we thus get M(Hn, Hn)X − M(H, H)X≤M(Hn, Hn)X − M(H, H)X2 → 0 and the same is true for N too. Hence the required inequality is obtained by taking the limit from M(Hn, Hn)X≤N(Hn, Hn)X. 3.3 Schur multipliers for matrices In estimating the norm of a double integral transformation, it is sometimes useful to reduce the problem to the matrix case by approximation (thoughcomputing the Schur multiplication norm is usually difficult even in the matrix case). Such an approximation technique is developed here, which will be indispensable in §3.5. We begin with basics on Schur multiplication on matrices. Let A = [aij ]i,j=1,2,··· be an infinite complex matrix such that supi,j |aij | < ∞. Then one can formally define a Schur multiplication operator SA on the space of infinite matrices as SA(X) = A ◦ X = [aijxij ] for X = [xij ], where ◦ is the Schur product or the Hadamard product (i.e., the entry-wise product). Consider the Hilbert space 2 with the canonical basis {ei}i=1,2,··· and identify an operator X ∈ B(2) as the matrix (Xej, ei) i,j=1,2,···. We then say that A is a Schur multiplier if SA gives rise to a bounded transformation of C1(2) into itself (or equivalently, of B(2) into itself). A Schur multiplication operator SA as above is realized as a double integral transformation of discrete type. In fact, assume that H, K ≥ 0 are diagonalizable with H = ∞ i=1 siξi ⊗ ξc i and K = ∞ i=1 tiηi ⊗ ηc i for some orthonormal bases {ξi} and {ηi}. For any Borel function φ on [0, ∞) × [0, ∞) the corresponding double integral transformation ΦH,K can be represented as ΦH,K(UXV ∗) = USA(X)V ∗ for X = [xij ] ∈ B(2), (3.12) where A = [φ(si, tj )]i,j=1,2,··· and U, V are unitary operators given by Uei = ξi, V ei = ηi. In this way, φ is a Schur multiplier relative to (H, K) if and only if A = [φ(si, tj )] is a Schur multiplier, and in this case ΦH,K(1,1) = SA(1,1). (3.13) Moreover, the characterization (iv) of Theorem 2.2 reads as follows: there exist a Hilbert space K (= L2(Ω, σ) there) and bounded sequences {ui} and {vj} of vectors in K such that aij (= φ(si, tj )) = (ui, vj )K (i, j = 1, 2,...). This criterion (known as Haagerup’s criterion) was independently obtained by U. Haagerup (see 4 in §3.7). In particular, when A = [aij ]i,j=1,··· ,n is an n × n matrix, the Schur multiplication operator SA is defined on Mn(C), the algebra of n × n matrices, and furthermore the following is known (see 4 in §3.7): SA(1,1) = SA(∞,∞) = min{κ ≥ 0 : there are ξ1,...,ξn, η1,...,ηn ∈ Cn such that ξi ≤ κ1/2, ηj ≤ κ1/2, aij = (ξi, ηj ) for i, j = 1,...,n}. (3.14) 42 3 Means of operators and their comparison If A is a positive semi-definite matrix, then SA(∞,∞) = max i aii. (3.15) In fact, this is immediately seen from (3.14); if ξ1,...,ξn are the row vectors of A1/2, then aij = (ξi, ξj ) for all i, j. (A different proof without using (3.14) can be found in [4, 42].) Lemma 3.8. An infinite matrix A = [aij ]i,j=1,2,··· is a Schur multiplier if and only if sup n≥1 S[aij ]i,j=1,··· ,n (1,1) < ∞. In this case, SA(1,1) is equal to the above supremum. Proof. If A is a Schur multiplier, then it is obvious that S[aij]i,j=1,··· ,n (1,1) ≤ SA(1,1) (for n = 1, 2,...). Conversely, assume that κ = sup n≥1 S[aij]i,j=1,··· ,n (1,1) < ∞. Let pn = n i=1 ei⊗ec i with the canonical basis {ei} for 2. For every X ∈ C1(2) and n = 1, 2,... we get SA(pnXpn)1 = [aijxij ]i,j=1,··· ,n1 ≤ κpnXpn1 ≤ κX1, and SA(pmXpm) − SA(pnXpn)1 ≤ κpmXpm − pnXpn1. By approximating X by finite-rank operators in the norm ·1, one observes limm,n→∞ pmXpm −pnXpn1 = 0 so that {SA(pnXpn)}n=1,2,··· is Cauchy in C1(2) from the second inequality and SA(pnXpn) − Y 1 → 0 for some Y ∈ C1(2). Since the convergence also takes place in the weak operator topology, this limit Y must be equal to SA(X) and consequently SA(X)1 = limn→∞ SA(pnXpn)1 ≤ κX1 from the above first estimate. The next lemma will play a key role in §3.5. The assumption of φ here may not be best possible, however it is enough for our purpose. Lemma 3.9. Let φ(s, t) be a function on [0, α] × [0, α] where 0 <α< ∞, and assume that φ is bounded and continuous at any point possibly except at (0, 0). Then the following conditions are equivalent : (i) φ is a Schur multiplier relative to every pair (H, K) of positive operators with H, K ≤ α; 3.3 Schur multipliers for matrices 43 (ii) sup S[φ(si,sj )]i,j=1,··· ,n (1,1) : 0 ≤ s1,...,sn ≤ α, n ≥ 1 < ∞, where repetition is allowed for s1,...,sn. Furthermore, if (ii) holds with finite supremum κ, then ΦH,K(1,1) ≤ κ for any (H, K) with H, K ≤ α. Proof. (i) ⇒ (ii). By assuming (i) and the failure of (ii), we will obtain a contradiction. Since (ii) fails to hold, for each n one can choose s (n) 1 ,...,s(n) n from [0, α] in such a way that sup n≥1 S φ(s(n) i ,s(n) j ) i,j=1,··· ,n (1,1) = ∞. Let {si}i=1,2,··· be the sequence s (1) 1 , s(2) 1 , s(2) 2 , s(3) 1 , s(3) 2 , s(3) 3 , ··· , s(n) 1 ,...,s(n) n , ··· obtained so far. We set A = [φ(si, sj )]i,j=1,2,··· and H = ∞ i=1 siξi ⊗ ξc i where {ξi} is an orthonormal basis. Then (i) implies that φ is a Schur multiplier relative to (H, H), so A must be a Schur multiplier as remarked just after (3.12). But, since φ(s (n) i , s(n) j ) i,j=1,··· ,n is a principal submatrix of A, it is obvious that S φ(s(n) i ,s(n) j ) i,j=1,··· ,n (1,1) ≤ SA(1,1) for all n. The supremum of the above left-hand side is ∞, a contradiction. (ii) ⇒ (i). Assume that the supremum κ in (ii) is finite. Let H be a positive operator with H ≤ α and the spectral decomposition H = α 0 s dEs. For each n = 1, 2,... we divide [0, α] into subintervals Λ(n) i = i − 1 n α, i n α (i = 1,...,n − 1) and Λ(n) n = n − 1 n α, α , and let t (n) i = i−1 n α (i = 1,...,n). Define φn(s, t) = n i,j=1 φ(t (n) i , t(n) j )χΛ(n) i ×Λ(n) j (s, t) for (s, t) ∈ [0, α] × [0, α] and Hn = n i=1 t (n) i EΛ(n) i . Then the double integral transformation Φn = ΦHn,Hn corresponding to φn is given by Φn(X) = n i,j=1 φ(t (n) i , t(n) j )EΛ(n) i XEΛ(n) j . 44 3 Means of operators and their comparison Since Hn is obviously diagonalizable, we write Hn = ∞ i=1 s (n) i ξ (n) i ⊗ ξ (n)c i with an orthonormal basis {ξ (n) i }i=1,2,··· and set An = φ(s (n) i , s(n) j ) i,j=1,2,···. Then, thanks to (3.13) we get Φn(1,1) = SAn (1,1). By assumption (ii) we apply Lemma 3.8 to conclude SAn (1,1) ≤ κ so that Φn(1,1) ≤ κ for all n. Now let ξ, η, ξ , η ∈ H be arbitrary. For Φ = ΦH,H we get (Φ(ξ ⊗ ηc )ξ , η ) = Φ(ξ ⊗ ηc ), η ⊗ ξc C2(H) = α 0 α 0 φ(s, t) d Es(ξ ⊗ ηc )Et, η ⊗ ξc C2(H) = α 0 α 0 φ(s, t) d(Esξ,η ) d(ξ , Etη), and similarly (Φn(ξ ⊗ ηc )ξ , η ) = α 0 α 0 φn(s, t) d(Esξ,η ) d(ξ , Etη). Here, the complex-valued measures d(Esξ,η ), d(ξ , Etη) are denoted by λ, µ respectively with their absolute values |λ|, |µ|. By assumption, |φ(s, t)| ≤ m (so |φn(s, t)| ≤ m as well) on [0, α] × [0, α] for some m < ∞. For each 0 <δ<α, since φn(0, 0) = φ(0, 0), we estimate |(Φn(ξ ⊗ ηc )ξ , η ) − (Φ(ξ ⊗ ηc )ξ , η )| ≤ ([0,α]×[0,α])\([0,δ)×[0,δ)) (φn(s, t) − φ(s, t)) d(λ × µ)(s, t) + ([0,δ)×[0,δ))\{(0,0)} φn(s, t) d(λ × µ)(s, t) + ([0,δ)×[0,δ))\{(0,0)} φ(s, t) d(λ × µ)(s, t) ≤ ([0,α]×[0,α])\([0,δ)×[0,δ)) |φn(s, t) − φ(s, t)| d(|λ|×|µ|)(s, t) +2m(|λ|×|µ|) ([0, δ) × [0, δ)) \ {(0, 0)} . For any δ > 0 the first term of the latter expression tends to 0 as n → ∞ because φ is continuous (hence uniformly continuous) on [0, α] × [0, α] \ [0, δ) × [0, δ) so that φn → φ uniformly there. But the second term can be arbitrarily small when δ > 0 is small enough. Therefore, we arrive at limn→∞(Φn(ξ ⊗ ηc)ξ , η )=(Φ(ξ ⊗ ηc)ξ , η ). 3.4 Positive definite kernels 45 This implies that Φn(X) → Φ(X) in the weak operator topology for all X ∈ Ifin. Since Φn(1,1) ≤ κ for all n as stated above, the lower semi-continuity of ·1 in the weak operator topology (see [37, Proposition 2.11]) yields Φ(X)1 ≤ lim inf n→∞ Φn(X)1 ≤ κX1 for all X ∈ Ifin. For each X ∈ C1(H) we approximate X by pnXpn with finiterank projections pn 1. Then {Φ(pnXpn)} is ·1-Cauchy and Φ(pnXpn) → Y ∈ C1(H) in the norm ·1 as in the proof of Lemma 3.8. However, we claim Y = Φ(X). In fact, since Φ is a bounded operator on C2(H), we have Φ(pnXpn) − Φ(X)2 → 0 (as well as Φ(pnXpn) − Y 2 → 0 thanks to ·2 ≤ ·1). Since Y = Φ(X), from the above estimate for operators in ∈ Ifin we have Φ(X)1 = limn→∞ Φ(pnXpn)1 ≤ κX1 for all X ∈ C1(H). Finally, the standard 2×2-matrix trick can be conveniently used to extend this inequality to a pair (H, K) with H, K ≤ α. In fact, with H˜ and X˜ as in Remark 2.12 we notice ΦH, ˜ H˜ (X˜ ) = 0 ΦH,K(X) 0 0 , which implies ΦH,K(X)1 = ΦH, ˜ H˜ (X˜ )1 ≤ κX˜1 = κX1 for X ∈ C1(H). Thus, φ is a Schur multiplier relative to (H, K) and ΦH,K(1,1) ≤ κ. 3.4 Positive definite kernels We say that M ∈ M is a positive definite kernel if [M(si, sj )]i,j=1,··· ,n is positive semi-definite for any s1,...,sn > 0 with any n. If N ∈ M is a positive definite kernel, then so is M ∈ M with M N. This is an immediate consequence of the famous Schur theorem on the Schur product of two positive semi-definite matrices. The next proposition says that the geometric mean G is the largest in the order among means in M that are positive definite kernels. When H is a matrix with eigenvalues s1,...,sn ≥ 0, M(H, H) is essentially equal to the Schur multiplication by [M(si, sj )]i,j=1,··· ,n (up to unitary conjugation, see (3.12)). So one may consider the property (i) below as a generalization of the Schur theorem. Proposition 3.10. The following conditions are equivalent for M ∈ M: 46 3 Means of operators and their comparison (i) M is a Schur multiplier and M(H, H)X is positive if so are H, X ∈ B(H); (ii) M is a positive definite kernel ; (iii) M G. If this is the case, then M(H, K)(1,1) ≤ H×K for all H, K ≥ 0. Proof. (i) ⇒ (ii). Choose an orthonormal basis {ξi}. For each n, by setting X = n i,j=1 ξi ⊗ ξc j and H = n i=1 siξi ⊗ ξc i with s1,...,sn ≥ 0, we get M(H, H)X = n i,j=1 M(si, sj )ξi ⊗ ξc j . Hence (i) implies the positive definiteness of [M(si, sj )]i,j=1,··· ,n. (ii) ⇒ (iii). This is immediate because of M(si, sj ) G(si, sj ) = diag(s −1/2 1 ,...,s−1/2 n ) M(si, sj ) diag(s −1/2 1 ,...,s−1/2 n ) for any s1,...,sn > 0. (iii) ⇒ (i). Assume (iii) with the representing measure ν for the ratio M(ex, 1)/G(ex, 1). Then Theorem 3.4 implies that M is a Schur multiplier and M(H, H)X = ∞ −∞ (HsH) ix(H1/2XH1/2)(HsH ) −ixdν(x), (because of M(1, 0) = 0), which is positive if so is X. Furthermore, by Corollary 3.5 we get |||M(H, K)X||| ≤ |||H1/2XK1/2||| ≤ H×K |||X||| for any unitarily invariant norm. Therefore, M(H, K)(1,1) ≤ H×K. 3.5 Norm estimates for means When M is one of A, L and G, it is straight-forward to see M(H, K)(1,1) ≤ M(H, K). In fact, this was noticed for G in the proof of Proposition 3.10, and for L we have |||L(H, K)X||| ≤ 1 0 |||HxXK1−x||| dx ≤ 1 0 HxK1−x dx × |||X||| = L(H, K)|||X||| for any unitarily invariant norm. As long as M M∞ we also get the estimate 3.5 Norm estimates for means 47 |||M(H, K)X||| ≤ |||M∞(H, K)X||| ≤ 3 2 |||HX + XK||| ≤ 3 2 (H + K)|||X||| (3.16) which is a consequence of Corollary 3.5 and (3.11). The problem to compute the best possible bound of M(H, K)(1,1) (in terms of H and K) is not easy in general. In this section the optimal bound will computed for the mean M = M∞. Lemma 3.11. For every s1,...,sn ≥ 0, S[si∨sj ]i,j=1,··· ,n (1,1) ≤ 2 √3 max i si − min i si + min i si ≤ 2 √3 max i si, where si∨tj = max{si, tj}. Moreover, 2/ √ 3 is the optimal bound in the above estimate. Proof. The explicit formula of SA(∞,∞) for a real 2 × 2 matrix A was obtained in [21] by using Haagerup’s criterion (3.14) and it indeed says S 1 1 1 0 (∞,∞) = 2 √3 . (3.17) (In fact, a direct computation of (3.17) with Haagerup’s criterion is also easy.) Next, let s1,...,sn ≥ 0. For a permutation γ on {1, 2,...,n} with the corresponding permutation matrix Γ we obviously have S[sγ(i)∨sγ(j)](X) = Γ S[si∨sj ](Γ −1XΓ) Γ −1. Thus, we may and do assume s1 ≥ s2 ≥···≥ sn ≥ 0, and the matrix [si ∨ sj ] can be written as [si ∨ sj ]=(s1 − s2)J(n) 1 + (s2 − s3)J(n) 2 + ··· + (sn−1 − sn)J(n) n−1 + snJ(n) n , where J(n) k =           1 ··· 1 1 ··· 1 . . . ... . . . . . . ... . . . 1 ··· 1 1 ··· 1 1 ··· 1 0 ··· 0 . . . ... . . . . . . ... . . . 1 ··· 1 0 ··· 0           (the zero block is (n − k) × (n − k)). According to (3.17) and Haagerup’s criterion, there are u1, u2, v1, v2 ∈ C2 such that ui2, vj2 ≤ 2/ √ 3 an 48 3 Means of operators and their comparison (u1, v1)=(u1, v2)=(u2, v1)=1, (u2, v2)=0. For k = 1,...,n − 1 we get J(n) k = (ξi, ηj ) when ξ1 = ··· = ξk = u1, ξk+1 = ··· = ξn = u2, η1 = ··· = ηk = v1 and ηk+1 = ··· = ηn = v2. This implies SJ(n) k (∞,∞) ≤ 2 √3 (for k = 1,...,n − 1), and obviously SJ(n) n (∞,∞) = 1. Since S[si∨sj ] = (s1 − s2)SJ(n) 1 + (s2 − s3)SJ(n) 2 + ··· + (sn−1 − sn)SJ(n) n−1 + snSJ(n) n with positive coefficients, we get S[si∨sj ](∞,∞) ≤ 2 √3 (s1 − sn) + sn as desired. Finally the optimality of 2/ √3 is clear from (3.17). The next theorem is a consequence of Lemmas 3.11 and 3.9 (for φ = M∞) together with Corollary 3.5 (or Theorem 3.7). Theorem 3.12. If M ∈ M satisfies M M∞, then M(H, K)(1,1) ≤ 2 √3 max{H, K} for all H, K ≥ 0. Consequently, for any unitarily invariant norm ||| · ||| we have |||M(H, K)X||| ≤ 2 √3 max{H, K} |||X||| for all X ∈ B(H). For each mean M ∈ M one can define the mean M(−) ∈ M dual to M by M(−) (s, t) = M(s−1, t−1) −1 for s, t > 0 (3.18) (see [39, §1]). For M,N ∈ M note that M N is equivalent to N(−) M(−) . For example, G(−) = G, A(−) = Mhar and M(−) ∞ = M−∞ concerning means in (3.4). It is easy to see that if H, K are invertible positive operators, then M(−) (H−1, K−1)(M(H, K)X) = M(H, K)(M(−) (H−1, K−1)X) = X (3.19) for all X ∈ C2(H). Indeed, this is the application of function calculus to the equality M(−) (s−1, t−1)M(s, t) = 1. Whenever both M and M(−) are Schur multipliers, (3.19) remains valid for all X ∈ B(H) so that M(−) (H−1, K−1) is the inverse of M(H, K) on B(H). Hence Theorem 3.12 implies 3.6 Kernel and range of M(H,K) 49 Proposition 3.13. If M ∈ M satisfies M−∞ M M∞ and H, K are invertible positive operators, then |||M(H, K)X||| ≥ √3 2 min{H−1−1, K−1−1} |||X||| for all unitarily invariant norms and all X ∈ B(H). Remark 3.14. The “mean transform” M(H, K) (when M M∞ for example) sends I|||·||| (and I(0) |||·|||) into itself (see Propositions 2.7 and 3.3). However, if H, K are positive compact operators in some Schatten class, then one can do better. For example, let us assume H, K ∈ Cp0 (H) (1 ≤ p0 ≤ ∞) and M = A, the arithmetic mean. Then, thanks to the (generalized) H¨older inequality XY p2 ≤ Xp1Y p0 with p−1 1 + p−1 0 = p−1 2 , (3.20) M(H, K) sends the Schatten class Cp1 (H) into the smaller one Cp2 (H) with the norm bound A(H, K)Xp2 ≤ 1 2 (HXp2 + XKp2) ≤ 1 2 (Hp0 + Kp0) Xp1 ≤ max{Hp0, Kp0 } Xp1. We point out that this is a general phenomenon. Namely, let us assume M M∞ and p−1 1 + p−1 0 = p−1 2 (1 ≤ p0, p1, p2 ≤ ∞). If positive operators H, K belong to Cp0 (H), then M(H, K) is a bounded linear operator from Cp1 (H) into Cp2 (H) satisfying M(H, K)Xp2 ≤ 3 max{Hp0, Kp0} Xp1. In fact, the general estimate (3.16) gives M(H, K)Xp2 ≤ 3 2 HX + XKp2 ≤ 3 2 (HXp2 + XKp2) so that the assertion follows from (3.20) as before. 3.6 Kernel and range of M(H, K) Assume M−∞ M M∞. When both of H, K ≥ 0 are invertible, the mean transform M(H, K) : B(H) → B(H) is bijective due to (3.19) (for each X ∈ B(H)). In this section we determine the kernel and the closure of the range for general positive H, K. Lemma 3.15. Assume that M ∈ M satisfies M−∞ M M∞, and let H be a non-singular positive operator. 50 3 Means of operators and their comparison (i) If X ∈ B(H) and M(H, H)X = 0, then X = 0. (ii) The range of M(H, H) is dense in B(H) in the strong operator topology. Proof. (i) For δ > 0 we note 0 = E(δ,∞)(M(H, H)X)E(δ,∞) = M(HE(δ,∞),HE(δ,∞))(E(δ,∞)XE(δ,∞)) with the spectral projection E(δ,∞) of H. Here, the second equality easily follows from the integral expression pointed out in Remark 2.5, (ii). By restricting everything to the subspace E(δ,∞)H (where HE(δ,∞) is an invertible operator), from Proposition 3.13 (and (3.19)) we get E(δ,∞)XE(δ,∞) = 0. We then see X = 0 because the non-singularity of H yields the strong convergence E(δ,∞) 1 (as δ 0). (ii) Choose and fix X ∈ B(H) and δ > 0 at first. As above we regard E(δ,∞)XE(δ,∞) and HE(δ,∞) (≥ δ) as operators on E(δ,∞)H. Then, the operator equation M(HE(δ,∞),HE(δ,∞))Y = E(δ,∞)XE(δ,∞) for an unknown operator Y ∈ B(E(δ,∞)H) possesses a solution, i.e., Y = M(−) ((HE(δ,∞)) −1,(HE(δ,∞)) −1)(E(δ,∞)XE(δ,∞)) (see (3.19)). However, since Y ∈ B(E(δ,∞)H) (⊆ B(H)), we observe M(HE(δ,∞),HE(δ,∞))Y = M(H, H)Y once again based on the expression in Remark 2.5, (ii). Consequently we have M(H, H)Y = E(δ,∞)XE(δ,∞), meaning that E(δ,∞)XE(δ,∞) sits in the range of M(H, H). We thus get the conclusion by letting δ 0. Theorem 3.16. Assume that M ∈ M satisfies M−∞ M M∞, and let H, K be positive operators. I. Case M(1, 0) = 0. (i) For X ∈ B(H) we have M(H, K)X = 0 if and only if sHXsK = 0. (ii) The closure of the range of M(H, K) in the strong operator topology is sHB(H)sK. II. Case M(1, 0) > 0. (iii) For X ∈ B(H) we have M(H, K)X = 0 if and only if sHXsK = sHX(1 − sK) = (1 − sH)XsK = 0. (iv) The closure of the range of M(H, K) in the strong operator topology is {X ∈ B(H) : (1 − sH)X(1 − sK)=0}. 3.6 Kernel and range of M(H,K) 51 Proof. We begin with the special case H = K. We recall M(H, H)X = sH(M(H, H)X)sH + M(1, 0)(HX(1 − sH) + (1 − sH)XH) = M(HsH,HsH)(sHXsH) + M(1, 0)(HX(1 − sH) + (1 − sH)XH) (see Lemma 2.9 and (3.10)). By restricting everything to the subspace sHH (where HsH is non-singular) Lemma 3.15, (i) says M(HsH,HsH)(sHXsH) = 0 if and only if sHXsH = 0, showing (i). When M(1, 0) > 0, the additional requirement HX(1 − sH) = (1 − sH)XH = 0 is needed. However, this is obviously equivalent to sHX(1−sH) = (1−sH)XsH = 0, which corresponds to (iii). On the other hand, from Lemma 3.15, (ii) (and the above decomposition) we easily get (ii) and (iv). Note that to show (iv) we need the following obvious fact for instance: HB(H)(1 − sH) is strongly dense in {X ∈ B(H) : sHXsH = (1 − sH)XsH = (1 − sH)X(1 − sH)=0}, i.e., operators with only (non-zero) “(1, 2)-components”. In the rest of the proof we will deal with the general case. With H, ˜ X˜ in Remark 2.12 we have M(H, K)X = 0 ⇐⇒ M(H, ˜ H˜ )X˜ = 0, which is also equivalent to sH˜ Xs ˜ H˜ = 0 (with the additional requirement sH˜ X˜(1 − sH˜ ) = (1 − sH˜ )Xs ˜ H˜ = 0 when M(1, 0) > 0) from the first part of the proof. But, since sH˜ = sH 0 0 sK , we easily get (i) and (iii) (in the general setting). Indeed, we have sH˜ Xs ˜ H˜ = 0 ⇐⇒ sHXsK = 0, sH˜ X˜(1 − sH˜ )=0 ⇐⇒ sHX(1 − sK)=0, (1 − sH˜ )X˜(1 − sH˜ )=0 ⇐⇒ (1 − sH)X(1 − sK)=0. To investigate the range, we consider the projections P1 = 1 0 0 0 , P2 = 0 0 0 1 (in B(H⊕H)). The range M(H, K)(B(H)) is P1(M(H, ˜ H˜ )(B(H⊕H))P2 (see Remark 2.12) with the natural identification of the (1, 2)-corner of B(H⊕H) with B(H). We claim P1(M(H, ˜ H˜ )(B(H⊕H))P2 = P1(M(H, ˜ H˜ )(B(H⊕H))P2. At first, ⊇ is obvious. To see ⊆, we choose and fix Y from the left-hand side. We note Y = P1Y P2 and can choose Yλ = M(H, ˜ H˜ )Zλ (for some Zλ ∈ B(H⊕H)) such that P1YλP2 → Y strongly. But notice 52 3 Means of operators and their comparison P1YλP2 = P1(M(H, ˜ H˜ )Zλ)P2 = M(H, ˜ H˜ )(P1ZλP2) due to the fact that P1 and P2 commute with H˜ (recall the integral expression in Remark 2.12). Therefore, each P1YλP2 actually belongs to the range M(H, ˜ H˜ )(B(H⊕H)) so that the limit Y sits in the strong closure M(H, ˜ H˜ )(B(H⊕H)). Hence, we have Y = P1Y P2 ∈ P1M(H, ˜ H˜ )(B(H⊕H))P2, and the claim is established. From the discussions so far we have M(H, K)(B(H)) = P1(M(H, ˜ H˜ )(B(H⊕H))P2 = P1(M(H, ˜ H˜ )(B(H⊕H))P2. (3.21) When M(1, 0) = 0, we have M(H, K)(B(H)) = P1sH˜ B(H⊕H)sH˜ P2 = sH˜P1B(H⊕H)P2sH˜ = sH˜ B(H)sH˜ . Here, the first equality follows from (3.21) and the first part of the proof (i.e, (ii) in the special case H = K) while the second is a consequence of the commutativity of P1, P2 with H˜ . The last equality comes from the abovementioned natural identification. We note that the B(H) (appearing in the far right side) is the one sitting at the (1, 2)-corner so that sH˜ B(H)sH˜ actually means sHB(H)sK (sitting at the same place). Therefore, we have shown (ii). On the other hand, when M(1, 0) > 0, from (3.21) (and (iv) in the special case) we similarly get M(H, K)(B(H)) = P1 sH˜ B(H⊕H)sH˜ +(1 − sH˜ )B(H⊕H)sH˜ + sH˜ B(H⊕H)(1 − sH˜ ) P2 = sH˜ P1B(H⊕H)P2sH˜ +(1 − sH˜ )P1B(H⊕H)P2sH˜ + sH˜ P1B(H⊕H)P2(1 − sH˜ ) = sH˜ B(H)sH˜ + (1 − sH˜ )B(H)sH˜ + sH˜ B(H)(1 − sH˜ ). The B(H) appearing at the end is once again the one at the (1, 2)-corner, and the same reasoning as in the last part of the preceding paragraph yields (iv) in the general case. 3.7 Notes and references 53 3.7 Notes and references 1. Means of operators In [39] the class M (in Definition 3.1) of homogeneous symmetric means was introduced, and for matrices H, K, X (with H, K ≥ 0) and M ∈ M the matrix mean M(H, K)X was defined by (1.1). With this definition Theorem 3.7 was obtained for matrices (as [39, Theorem 1.1]), and many norm inequalities were obtained. We cannot determine if every M ∈ M is a Schur multiplier (probably not), and this problem seems to deserve further investigation. Anyway the criterion M M∞ obtained in Proposition 3.3, (b) is good enough in almost all circumstances. The implication (iv) ⇒ (ii) in Theorem 3.7 (at least in the matrix case, or equivalently (3.15)) has been known to many specialists ([42, p. 343] and [4, p. 363] for example) and indeed used as a standard tool for showing norm inequalities. We actually have the bi-implication here. Therefore, the theorem can be also used to check failure of certain norm inequalities, which will be carried out in our forthcoming article [55]. In §8.1 and §A.1 we will deal with “operator means” M(H, K)X for functions M in wider classes. This will make it possible to study norm inequalities for certain operators which are not operator means in the sense of the present chapter. Our theory of operator means is useful in study of certain operator equations. Let us assume the invertibility of H, K ≥ 0 for simplicity and regard M(H, K)X = Y as an operator equation with an unknown operator X. Then, (3.18) and (3.19) show that X = M(−) (H−1, K−1)Y gives rise to a solution. With this idea concrete integral expressions for solutions to many operator equations were obtained in [39, §4]. In [68] related analysis was also made by G. K. Pedersen from the viewpoint of “operator differentials” (see also [33, 67]). Theorem 3.16 in §3.6 provides us useful information on uniqueness of solutions to the above operator equation. Another important notion of operator means, quite different from those treated in the present monograph, is the one axiomatically introduced by F. Kubo and T. Ando in [57]. An operator mean in their sense is a binary operation B(H)+ × B(H)+ → B(H)+, and it bijectively corresponds to an operator monotone function on R+. For example, the geometric mean (formerly introduced by W. Pusz and L. Woronowicz in [73]) is given as H#K = H 1 2 (H− 1 2 KH− 1 2 ) 1 2 H 1 2 for positive invertible H, K ∈ B(H) while our geometric mean G(H, K)X = H 1 2 XK 1 2 is no longer positive even when X = 1. 2. Arithmetic-geometric mean inequality and related topics The arithmetic-geometric mean inequality (1.4) for unitarily invariant norms was first noticed by R. Bhatia and C. Davis in [10], and its alternative proofs (and/or some discussions) were worked out by many authors including 54 3 Means of operators and their comparison R. A. Horn ([41]), F. Kittaneh ([50, 51]), R. Mathias ([63]) and probably some others. Proofs presented in [41, 63] are indeed based on the method explained in 1. The article [13] by R. Bhatia and K. Parthasarathy is closely related to our previous works [38, 39, 54], and this method was systematically used to derive an abundance of known and new norm inequalities. The same method was used by X. Zhan ([83, Theorem 6 and Corollary 7]) to show the following generalizations of the arithmetic-geometric mean inequality (as well as the Heinz inequality (1.3)): (i) for x ∈ (−2, 2] and θ ∈ [1/4, 3/4], 2 + x 2 |||HθXK1−θ + H1−θXKθ||| ≤ |||HX + XK + xH1/2XK1/2|||; (ii) for x ∈ (−2, 2], (2 + x)|||H1/2XK1/2||| ≤ |||HX + XK + xH1/2XK1/2|||. Similar results (based on the similar method) were also obtained in [78]. The following inequality was obtained by D. Joci´c ([45, Theorem 3.1]) as an application of the arithmetic-geometric mean inequality: ||| |HX + XK| p ||| ≤ 2p−1Xp−1||| |H| p−1HX + XK|K| p−1||| for p ≥ 3 and self-adjoint operators H, K. It generalizes the earlier result |||(H − K) 2n+1||| ≤ 22n|||H2n+1 − K2n+1||| due to D. Joci´c and F. Kittaneh ([46], and also see [7]). In fact, when p = 2n+1 odd, by setting X = 1 and using −K instead one gets |H| 2nH = H2n+1 and (−K)|(−K)| 2n = −K2n+1. This perturbation estimate in particular shows H − K ∈ C(2n+1)p as long as H2n+1 − K2n+1 ∈ Cp and p ∈ [1, ∞), which improves L. S. Koplienko’s result in [52]. G. Corach, H. Porta and L. Recht studied the set of invertible self-adjoint operators (and some other sets) as a space equipped with a certain natural Finsler metric (see [60]). In [19] from the differential geometry viewpoint they arrived at the inequality X ≤ 1 2 HXH−1 + H−1XH for an invertible self-adjoint operator H. This corresponds to the normdecreasing property of a certain tangential map, and their proof actually uses Schur products. As noticed in [28, 51] for example (change X to HXH and use the standard 2 × 2-matrix trick in Remark 2.12), their inequality is nothing but the arithmetic-geometric mean inequality (in the operator norm). In [20] they also gave a geometric interpretation of the Segal inequality eH+K≤eH/2eKeH/2 ≤ eHeK 3.7 Notes and references 55 for self-adjoint operators H, K. 3. Arithmetic-logarithmic-geometric mean inequality The arithmetic-logarithmic-geometric mean inequality (1.8) (as well as some further extensions such as monotonicity of the norms (1.9) in m and n) was proved in [38]. In [9] R. Bhatia pointed out a close connection between the logarithmic-geometric mean inequality and the Golden-Thompson-type norm inequality (extending the Segal inequality) |||eH+K||| ≤ |||eHeK||| for self-adjoint operators H, K based on the differential geometry viewpoint (akin to [19, 20]). (See [8, 37, 77] for the Golden-Thompson-type inequality.) 4. Schur multipliers in the matrix case Haagerup’s criterion and (3.14) were presented in his unpublished notes [31, 32], and a proof is available in the literature. Namely, the formula was shown in the article [5] by T. Ando and K. Okubo as a consequence of its variant for the numerical radius norm. The Ando-Okubo theorem was recently extended to B(H) by T. Itoh and M. Nagisa in [44]. Materials in §3.3 are somewhat technical. But, we need them (especially Lemma 3.9) to reduce the proof of Theorem 3.12 in §3.5 to the matrix case. In fact, this technique enables us to make use of Lemma 3.11 (based on (3.14)). (Sub)majorization theory for eigenvalues and singular values of matrices provides a powerful tool in study of matrix (also operator) norm inequalities for unitarily invariant norms (see [34, 62] and also [1, 2, 8] for surveys on recent results). Among others, T. Ando, R. A. Horn and C. R. Johnson obtained in [4] a fundamental majorization for singular values of Hadamard (or Schur) products of matrices, which implies (3.15) as a corollary. Majorization method was implicitly used in the proof of Proposition 2.6; however it does not have much to do with the present monograph.

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