On intertwining operators
Let B(H) denote the algebra of operators on the Hilbert space H into itself. Given \(A,B\in B(H)\), define C(A,B) and R(A,B):B(H)\(\to B(H)\) by \(C(A,B)X=AX-XB\) and \(R(A,B)X=AXB-X\). Our purpose in this note is a two fold one. We show firstly that if A and \(B^*\in B(H)\) are dominant operators such that the pure part of B has non-trivial kernel, then \(C^ n(A,B)X=0\), n some natural number, implies that \(C(A,B)X=C(A^*,B^*)X=0\). Secondly it is shown that if A and \(B^*\) are contractions with \(C_{.0}\) completely non-unitary parts, then \(R^ n(A,B)X=0\) for some natural number n implies that \(R(A,B)X=R(A^*,B^*)X=C(A,B^*)X=C(A^*,B)X=0\). In the particular case in which X is compact, it is shown that this result extends to all contractions A and B.
contractions with \(C_{.0}\) completely non-unitary parts
scientific article; zbMATH DE number 4063852