M N if and only if M(

 

M
N if and only if M(ex, 1)/N(ex, 1) is positive definite. If this is the case, then for non-singular positive operators H, K we have the integral expression M(H, K)X =
∞ −∞ Hix(N(H, K)X)K−ixdν(x) (1.11) with a probability measure ν (see Theorems 3.4 and 3.7 for the precise statement), and of course the Bochner theorem is behind. Under such circumstances (thanks to the general fact explained in §A.2) we actually have |||M(H, K)X||| ≤ |||N(H, K)X||| (1.12) (even without the non-singularity of H, K ≥ 0). This inequality actually characterizes the order M
N, and is a source for a variety of concrete norm inequalities (as was demonstrated in [40]). The order
and (1.11), (1.12) were also used in [39] for matrices, but much more involved arguments are required for Hilbert space operators, which will be carried out in Chapter 3. It is sometimes not an easy task to determine if a given mean M(s, t) is a Schur multiplier. However, the mean M∞(s, t) = max{s, t} comes to the rescue: (i) The mean M∞ itself is a Schur multiplier. (ii) A mean majorized by M∞ (relative to
) is a Schur multiplier. These are consequences of (1.11), (1.12), and enable us to prove that all the means considered in [39] are indeed Schur multipliers. The observation (i) also follows from the discrete decomposition of max{s, t} worked out in §A.3, which might be of independent interest. Furthermore, a general norm estimate of the transformation X → M(H, K)X is established for means M
M∞. In Chapter 4 we study the convergence M(Hn, Kn)X → M(H, K)X (in ||| · ||| or in the strong operator topology) under the strong convergence Hn → H, Kn → K of the positive operators involved. The requirement for the convergence (1.10) in the appendix to [39] was the following finiteness condition: either |||H|||, |||K||| < ∞ or |||X||| < ∞. This requirement is somewhat artificial (and too restrictive), and the arguments presented there were ad hoc. The second main purpose of the monograph is to present systematic and thorough investigation on such convergence phenomena. In [39] we dealt with the following one-parameter families of scalar means: Mα(s, t) = α − 1 α × sα − tα sα−1 − tα−1 (−∞ ≤ α ≤ ∞), Aα(s, t) = 1 2 (sαt 1−α + s1−αt α) (0 ≤ α ≤ 1), Bα(s, t) = sα + tα 2 1/α (−∞ ≤ α ≤ ∞). It is straight-forward to see that Mα(s, t), Aα(s, t) are Schur multipliers, and also so is B1/n(s, t) thanks to the the binomial expansion B1/n(s, t)