A semigroup approach to the numerical range of operators on Banach spaces

2015/07/06 by Adler, Martin, Dada, Waed, Radl, Agnes
#47A10 #47A12 #47D06 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1507.01418

Abstract

We introduce the numerical spectrum σ_n(A)⊂ ℂ of an (unbounded) linear operator A on a Banach space X and study its properties. Our definition is closely related to the numerical range W(A) of A and always yields a superset of W(A). In the case of bounded operators on Hilbert spaces, the two notions coincide. However, unlike the numerical range, σ_n(A) is always closed, convex and contains the spectrum of A. In the paper we strongly emphasise the connection of our approach to the theory of C₀-semigroups.

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