Noncommutative Schwarz lemma and Pick–Julia Theorems for Generalized Derivations in Norm Ideals of Compact Operators

Abstract

If A and B are strict contractions on a Hilbert space H and the derivation AX- XB is a trace class ([InlineEquation not available: see fulltext.]) operator for some bounded operator [InlineEquation not available: see fulltext.] acting on a Hilbert space H, then for all holomorphic function f, which maps the open unit disc D⊂ C into itself, we have shown by (3.13) in Theorem 3.5 that [InlineEquation not available: see fulltext.] and ||I-A∗A(f(A)X-Xf(B))I-BB∗||1⩽||I-f(A)∗f(A)(AX-XB)I-f(B)f(B)∗||1.If AX- XB is in a Hilbert-Schmidt class [InlineEquation not available: see fulltext.] then [InlineEquation not available: see fulltext.] as well, and it satisfies ||f(A)X-Xf(B)-A(f(A)X-Xf(B))B∗||2⩽||AX-XB-f(A)(AX-XB)f(B)∗||2.