Non-compact λ-Hankel operators

A λ-Hankel operator X is a bounded operator on Hilbert space satisfying the operator equation S*X - XS = λX, where S is the (unilateral) forward shift and S* is its adjoint. We prove that there are non-compact λ-Hankel operators for λ 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 λ-Hankel operator associated with the Carleson measure given by the interpolating sequence is non-compact.