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W. Auzinger:
"Sectorial operators and normalized numerical range";
Applied Numerical Mathematics, 45 (2003), 367 - 388.

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 S_σ = { z in C: |Arg z| <= σ }, and this is equivalent to a certain inverse estimate valid outside S_σ. 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 R_N(A). We also show how R_N(A) can be computed numerically.

http://dx.doi.org/10.1016/S0168-9274(02)00254-4

http://aleph.ub.tuwien.ac.at/F?base=tuw01&func=find-c&ccl_term=AC04405559

http://www.math.tuwien.ac.at/~winfried/papers/article_APNUM45_2003_ERSCHIENEN.pdf