Spectrum of the Compression of a Slant Toeplitz Operator

Abstract

A slant Toeplitz operator A_φ with symbol φ in L^∞(T) where T is the unit circle on the complex plane, is an operator where the representing matrix M=(a_ij) is given by a_ij=<φ,z^2i-j> where is the usual inner product in L²(T). The operator B_φ denotes the compression of A_φ, to H²(T) (Hardy space). In this paper, we prove that the spectrum of B_φ contains a closed disc and the interior of this disc consists of eigenvalues with infinite multiplicity, if T_φ is invertible, where T_φ is the Toeplitz operator on H²(T).