The present monograph

 The present monograph is devoted to a thorough study of means for Hilbert
space operators, especially comparison of (unitarily invariant) norms of operator means and their convergence properties in various aspects.
The Hadamard product (or Schur product) A ◦ B of two matrices A =
[aij ], B = [bij ] means their entry-wise product [aij bij ]. This notion is a common and powerful technique in investigation of general matrix (and/or operator) norm inequalities, and particularly so in that of perturbation inequalities
and commutator estimates. Assume that n × n matrices H, K, X ∈ Mn(C)
are given with H, K ≥ 0 and diagonalizations
H = Udiag(s1, s2,...,sn)U∗ and K = V diag(t1, t2,...,tn)V ∗.
In our previous work [39], to a given scalar mean M(s, t) (for s, t ∈ R+), we
associated the corresponding matrix mean M(H, K)X by
M(H, K)X = U ([M(si, tj)] ◦ (U∗XV )) V ∗. (1.1)
For a scalar mean M(s, t) of the form
n
i=1 fi(s)gi(t) one easily observes
M(H, K)X =
n
i=1 fi(H)Xgi(K), and we note that this expression makes
a perfect sense even for Hilbert space operators H, K, X with H, K ≥ 0.
However, for the definition of general matrix means M(H, K)X (such as AL-G interpolation means Mα(H, K)X and binomial means Bα(H, K)X to
be explained later) the use of Hadamard products or something alike seems
unavoidable.
The first main purpose of the present monograph is to develop a reasonable theory of means for Hilbert space operators, which works equally well
for general scalar means (including Mα, Bα and so on). Here two difficulties have to be resolved: (i) Given (infinite-dimensional) diagonal operators
H, K ≥ 0, the definition (1.1) remains legitimate for X ∈ C2(H), the HilbertSchmidt class operators on a Hilbert space H, as long as entries M(si, tj )
stay bounded (and M(H, K)X ∈ C2(H)). However, what we want is a mean
M(H, K)X (∈ B(H)) for each bounded operator X ∈ B(H). (ii) General