Reducing subspaces of de Branges-Rovnyak spaces

Abstract

For b∈H1∞, the closed unit ball of H∞, the de Branges-Rovnyak space H(b) is a Hilbert space contractively contained in the Hardy space H2 that is invariant by the backward shift operator S⁎. We consider the reducing subspaces of the operator S⁎2|H(b). When b is an inner function, S⁎2|H(b) is a truncated Toeplitz operator and its reducibility was characterized by Douglas and Foias using model theory. We use another approach to extend their result to the case where b is extreme. We prove that if b is extreme but not inner, then S⁎2|H(b) is reducible if and only if b is even or odd, and describe the structure of reducing subspaces.