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.
