Non-Compact $\lambda$-Hankel Operators

Abstract

A $\lambda$-Hankel operator $X$ is a bounded operator on Hilbert space satisfying the operator equation $S*X–XS = \lambda X$, where $S$ is the (unilateral) forward shift and $S*$ is its adjoint. We prove that there are non-compact $\lambda$-Hankel operators for $\lambda$ a complex number of modulus less than 2, by first exhibiting a way to obtain bounded solutions to the above equation by associating to it a Carleson measure. We then show that an interpolating sequence can be given such that the $\lambda$-Hankel operator associated with the Carleson measure given by the interpolating sequence is non-compact.