The Curvature of a Single Operator on a Hilbert Space

2000/06/29 by Stephen Parrott, Parrott, Stephen · 1 citation
Mathematics · #47A13 (Primary) #47A20 (Secondary) #Algebraic and Geometric Analysis #FOS: Mathematics #Holomorphic and Operator Theory #Operator Algebras (math.OA) #Spectral Theory in Mathematical Physics #math.OA #msc:47A13 #msc:47A20

paper · pdf · doi:10.48550/arxiv.math/0006224

LaTeX, 15 pages

arxiv created 2000/06/29 · openalex publication_date 2000/06/29 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This note studies Arveson's curvature invariant for d-contractions specialized to the case d=1 of a single contraction operator on a Hilbert space. It establishes a formula which gives an easy-to-understand meaning for the curvature of a single contraction. The formula is applied to give an example of an operator with nonintegral curvature. Under the additional hypothesis that the contraction T be "pure", we show that its curvature K(T) is given by K(T) = - index(T) := -(dim ker T - dim coker T).

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