Multicentric calculus and the Riesz projection

Diana Apetrei; Olavi Nevanlinna

doi:10.33993/jnaat442-1064

Authors

Diana Apetrei Aalto University, Finland
Olavi Nevanlinna Aalto University, Finland

DOI:

https://doi.org/10.33993/jnaat442-1064

Keywords:

multicentric calculus, Riesz projection, spectral projections, sign function of an operator, lemniscates

Abstract views: 356

Abstract

In multicentric holomorphic calculus one represents the function ? using a new polynomial variable $w = p (z)$ in such a way that when evaluated at the operator $p (A)$ is small in norm. Here it is assumed that $p$ has distinct roots. In this paper we discuss two related problems, separating a compact set, such as the spectrum, into different components by a polynomial lemniscate, and then applying the calculus for computation and estimation of the Riesz spectral projection. It may then be desirable to move to using $p (z)^{n}$ as a new variable and we develop the necessary modifications to incorporate the multiplicities in the roots.

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References

D. Apetrei, O. Nevanlinna,Multicentric calculus and the Riesz projection, http://arxiv.org/abs/1602.08337, Feb 29, 2016, https://doi.org/10.48550/arXiv.1602.08337

J.B. Conway, A Course in Functional Analysis, Springer, New York, 1990.

M. Marden,Geometry of Polynomials, AMS, Providence, Rhode Island, 1989.

O. Nevanlinna,Computing the spectrum and representing the resolvent, Numer. Funct.Anal. Optimiz.30(2009) 9-10, 1025-1047, https://doi.org/10.1080/01630560903393162 DOI: https://doi.org/10.1080/01630560903393162

O. Nevanlinna,Convergence of Iterations for Linear Equations, Birkhauser, Basel,1993. DOI: https://doi.org/10.1007/978-3-0348-8547-8

O. Nevanlinna,Hessenberg matrices in Krylov subspaces and the computation of thespectrum, Numer. Funct. Anal. Optimiz.,16(1995) 3-4, pp. 443-473, https://doi.org/10.1080/01630569508816627 DOI: https://doi.org/10.1080/01630569508816627

O. Nevanlinna,Lemniscates and K-spectral sets, J. Funct. Anal.,262(2012), pp.1728-1741, https://doi.org/10.1016/j.jfa.2011.11.019 DOI: https://doi.org/10.1016/j.jfa.2011.11.019

O. Nevanlinna,Meromorphic Functions and Linear Algebra, AMS Fields Institute Monograph 18, 2003. DOI: https://doi.org/10.1090/fim/018

O. Nevanlinna,Multicentric holomorphic calculus, Comput. Methods Funct. Theory,12(2012) 1, pp. 45-65. DOI: https://doi.org/10.1007/BF03321812

O. Nevanlinna,Polynomial as a new variable - a Banach algebra with a functionalcalculus, , July 3, 2015, http://arxiv.org/abs/1506.00634 DOI: https://doi.org/10.7153/oam-10-33

V. Paulsen,Completely Bounded Maps and Operator Algebras, Cambridge UniversityPress, 2002, https://www.researchgate.net/profile/Vern-Paulsen/publication/200524409, DOI: https://doi.org/10.1017/CBO9780511546631

Th. Ransford,Potential Theory in Complex Plane, London Math. Soc. Student Texts28, Cambridge University Press, 1995.Received by the editors: November 9, 2015. DOI: https://doi.org/10.1017/CBO9780511623776