2−n
n k=0 n k s k n t n−k n . We indeed show that all of Bα(s, t) are (by proving Bα
M∞). Thus, all of the above give rise to operator means. Note M1/2(s, t) = √st (the geometric mean), M2 = 1 2 (s + t) (the arithmetic mean) and M1(s, t)  = limα→1 Mα(s, t)  = s − t log s − log t =  1 0 sxt 1−xdx (the logarithmic mean). Because of these reasons {Mα(s, t)}−∞≤α≤∞ will be referred to as the A-L-G interpolation means. The convergence (1.10) (see also (5.1)) means lim m→∞ |||M m m+1 (H, K)X − L||| = limn→∞ |||M n n−1 (H, K)X − L||| = 0 with the logarithmic mean L = M1(H, K)X =  1 0 HxXK1−xdx, and the main result in Chapter 5 is the following generalization: lim α→α0 |||Mα(H, K)X − Mα0 (H, K)X||| = 0 under the assumption |||Mβ(H, K)X||| < ∞ for some β>α0. This is a “dominated convergence theorem” for the A-L-G means, the proof of which is indeed based on Lebesgue’s theorem applied to the relevant integral expression (1.11) with the concrete form of the density dν(x)/dx. Similar dominated convergence theorems for the Heinz-type means Aα(H, K)X = 1 2 (HαXK1−α + H1−αXKα) (or rather the single components HαXK1−α) and the binomial means Bα(H, K)X are also obtained together with other related results in Chapters 6 and 7. A slightly different subject is covered in Chapter 8, that might be of independent interest. The homogeneous alternating sums    A(n) = n k=1 (−1)k−1H k n+1 XK n+1−k n+1 (with n = 1, 2, ···), B(m) = m −1 k=0 (−1)kH k m−1 XK m−1−k m−1 (with m = 2, 3, ···) are not necessarily symmetric (depending upon parities of n, m), but our method works and integral expressions akin to (1.11) (sometimes with signed measures ν) are available. This enables us to determine behavior of unitarily invariant norms of these alternating sums of operators such as mutual comparison, uniform bounds, monotonicity and so on. Some technical results used in the monograph are collected in Appendices, and §A.1 is concerned with extension of our arguments to certain nonsymmetric means