positive operators

 

positive operators H, K are no longer diagonal so that continuous spectral
decomposition has to be used. The requirement in (i) says that the concept
of a Schur multiplier ([31, 32, 66]) has to enter our picture, and hence what
we need is a continuous analogue of the operation (1.1) with this concept
built in. The theory of (Stieltjes) double integral transformations ([14]) due
to M. Sh. Birman, M. Z. Solomyak and others is suited for this purpose. With
this apparatus the operator mean M(H, K)X is defined (in Chapter 3) as
M(H, K)X =


H

0


K

0
M(s, t) dEsXdFt (1.2)
with the spectral decompositions
H =


H

0
s dEs and K =


K

0
t dFt.
Double integral transformations as above were actually considered with
general functions M(s, t) (which are not necessarily means). This subject has
important applications to theories of perturbation, Volterra operators, Hankel
operators and so on (see §2.5 for more information including references), and
one of central problems here (besides the justification of the double integral
(1.2)) is to determine for which unitarily invariant norm the transformation
X
→ M(H, K)X is bounded. Extensive study has been made in this direction, and V. V. Peller’s work ([69, 70]) deserves special mentioning. Namely,
he completely characterized (C1-)Schur multipliers in this setting (i.e., boundedness criterion relative to the trace norm ·1, or equivalently, the operator
norm · by the duality), which is a continuous counterpart of U. Haagerup’s
characterization ([31, 32]) in the matrix setting. Our theory of operator means
is built upon V. V. Peller’s characterization (Theorem 2.2) although just an
easy part is needed. Unfortunately, his work [69] with a proof (while [70] is
an announcement) was not widely circulated, and details of some parts were
omitted. Moreover, quite a few references there are not easily accessible. For
these reasons and to make the monograph as self-contained as possible, we
present details of his proof in Chapter 2 (see §2.1).
As emphasized above, the notions of Hadamard products and double integral transformations play important roles in perturbation theory and commutator estimates. In this monograph we restrict ourselves mainly to symmetric
homogeneous means (except in Chapter 8 and §A.1) so that these important
topics will not be touched. However, most of the arguments in Chapters 2 and
3 are quite general and our technique can be applicable to these topics (which
will be actually carried out in our forthcoming article [55]). It is needless to
say that there are large numbers of literature on matrix and/or operator norm
inequalities (not necessarily of perturbation and/or commutator-type) based
on closely related techniques. We also remark that the technique here is useful
for dealing with certain operator equations such as Lyapunov-type equations
(see §3.7 and [39, §4]). These related topics as well as relationship to other