We extend the Fuglede-Putnam theorem modulo the Hilbert-Schmidt class to almost normal operators with finite Hilbert-Schmidt modulus of quasi-triangularity.
The question whether a hyponormal weighted shift with trace class self-commutator is the compression modulo the Hilbert-Schmidt class of a normal operator, remains open. It is natural to ask whether Putinar's construction through which he proved that hyponormal operators are subscalar operators provides the answer to the above question. We show that the normal extension provided by Putinar's theory does not lead to the extension that would provide a positive answer to the question.
We prove that for normal operators the generalized commutator approaches zero when tends to zero in the norm of the Schatten-von Neumann class with and varies in a bounded set of such a class.
Page 1