A relative of the shift operator

Abstract

The spectrum of a unilateral shift has circular symmetry. Here an operator is studied which has elliptical symmetry. Let U be the unweighted shift on a separable Hilbert space H , and let α be a nonzero complex number. The spectrum of the operator U α = 1 2 (αU+α −1U ∗) is studied, and its resolvent is given in interesting form. The operator U α is a generalization of the real part of the shift U: U 1 = 1 2 (U + U ∗) . If D α denotes the diagonal operator with weights 1, α, α 2,…, then formally U α = U α U 1 D α −1, but the spectrum of U α equals the spectrum of U 1 if and only if α=1.