Sectorial operators and normalized numerical range

Authors:
W. Auzinger
Affiliations:
Institut für Angewandte und Numerische Mathematik, Technische Universität Wien, Wiedner Hauptstraße 8-10/115, A-1040 Wien, Austria
Venue:
Applied Numerical Mathematics
Year:
2003

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Spectral theory and differential operators

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Abstract

We study the notion of sectorial operator in a Hilbert space. According to the classical definition, the numerical range R(A) of a sectorial operator A is contained in a sector Sσ = {z ∈ C: |Arg z| ≤ σ}, and this is equivalent to a certain inverse estimate valid outside Sσ. In this paper we show that the validity of the same estimate, but with a factor 1, is equivalent to the validity of a certain strengthened Cauchy-Schwarz inequality for all pairs w, Aw. This extends the original characterization in terms of R(A) by a more general characterization based on a normalized numerical range RN(A). We also show how RN(A) can be computed numerically.