## 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**

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