By Fumio Hiai

The monograph is dedicated to a scientific research of technique of Hilbert house operators by way of a unified approach in keeping with the speculation of double vital adjustments and Peller's characterization of Schur multipliers. common homes on technique of operators similar to comparability effects, norm estimates and convergence standards are tested. After a few basic concept, unique investigations are fascinated about 3 one-parameter households of A-L-G (arithmetic-logarithmic-geometric) interpolation ability, Heinz-type potential and binomial ability. particularly, norm continuity within the parameter is tested for such potential. a few precious technical effects are accumulated as appendices.

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**Example text**

In this section the optimal bound will computed for the mean M = M∞ . 11. For every s1 , . . , sn ≥ 0, 2 2 ≤ √ max si − min si + min si ≤ √ max si , i i i 3 3 i √ where si ∨ tj = max{si , tj }. Moreover, 2/ 3 is the optimal bound in the above estimate. S[si ∨sj ]i,j=1,··· ,n (1,1) Proof. 14) and it indeed says S¾ 11 10 ¿ (∞,∞) 2 = √ . ) 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 −1 XΓ ) Γ −1 .

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). 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(sH XsK ) and 26 2 Double integral transformations Φ(X) = sH Φ(X)sK + φ(H, 0)sH X(1 − sK ) + (1 − sH )XsK φ(0, K) +φ(0, 0)(1 − sH )X(1 − sK ).

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˜ P1 B(H ⊕ H)P2 sH˜ +(1 − sH˜ )P1 B(H ⊕ H)P2 sH˜ + sH˜ P1 B(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. 7 Notes and references 1. 1). 1]), and many norm inequalities were obtained.