Reducing Subspaces of de Branges-Rovnyak Spaces

2018/10/02 by Cheng Chu, Chu, Cheng
Mathematics · #Advanced Harmonic Analysis Research #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1810.01058

openalex publication_date 2018/10/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For b∈ H^∞₁, the closed unit ball of H^∞, the de Branges-Rovnyak spaces 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 Toepltiz 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.

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