Multicentric Calculus: A New Look at \(f(A)\)

Olavi Nevanlinna (Aalto University)

Frank Adams Room 1, Alan Turing Building,

We outline a path from polynomial numerical hulls to multicentric calculus for evaluating \(f(A)\). Consider \[V_p(A) = \{z\in\mathbb{C}:|p(z)|\le\|p(A)\|\}\] where \(p\) is a polynomial and \(A\) a bounded linear operator (or matrix). Intersecting these sets over polynomials of degree 1 gives the closure of the numerical range, while intersecting over all polynomials gives the spectrum of A, with possible holes filled in [1].

Outside any set \(V_p(A)\) one can write the resolvent down explicitly and this leads to multicentric holomorphic functional calculus [3], [4].

The spectrum, pseudospectrum or the polynomial numerical hulls can move rapidly in low rank perturbations. However, this happens in a very controlled way and when measured correctly one gets an identity which shows e.g. the following: if you have a low-rank homotopy between self-adjoint and quasinilpotent, then the identity forces the nonnormality to increase in exact compensation with the spectrum shrinking [2].

In this talk we shall mention how the multicentric calculus leads to a nontrivial extension of von Neumann theorem \[\|f(A)\|\le \sup_{|z|\le 1}\|f(z)\|\] where \(A\) is a contraction in a Hilbert space, [5], and conclude with some new results on (nonholomorphic) functional calculus for operators for which \(p(A)\) is (similar to ) normal at a nontrivial polynomial \(p\). The results are new even for matrices.

References

  1. Convergence of Iterations for Linear Equations, Birkhäuser, 1993
  2. Meromorphic Functions and Linear Algebra, Fields Institute Monographs, 18, AMS 2003
  3. Computing the Spectrum and Representing the Resolvent, Numer. Funct. Anal. Optimiz. Volume 30, Issue 9 and 10 (2009) 1025 - 1047
  4. Multicentric Holomorphic Calculus, Computational Methods and Function Theory, June 2012, Volume 12, Issue 1, pp 45-65
  5. Lemniscates and K-spectral sets, J. Funct. Anal. 262(2012), 1728-1741.
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