Operators That Commute with Slant Toeplitz Operators

Abstract

Let H be a separable Hilbert space and {en : n ∈ Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if 〈T ej, ei〉 = c2i− j, where cn is the nth Fourier coefficient of a bounded Lebesgue measurable function φ on the unit circle T = {z ∈ C : |z| = 1}. It has been shown [9], with some assumption on the smoothness and the zeros of φ, that T ∗ is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. These results, together with the theory of shifts (e.g., in [11]), allows us to identify all bounded operators on H commuting with such T .