Simplifying operators by polynomials

TL;DR

This paper explores whether applying polynomial transformations to operators in Hilbert or Banach spaces can simplify their properties, enabling extended functional calculus applications through multicentric calculus.

Contribution

It organizes known results and introduces new findings on polynomial transformations that simplify operators, expanding the applicability of functional calculus methods.

Findings

Polynomial transformations can make certain operators 'simpler' or 'nicer'

Multicentric calculus enables extending functional calculus to original operators

Results cover various classes of operators like compact, Riesz, and quasitriangular

Abstract

We collect and organise known results and add some new ones of the following nature: if A is a bounded operator in a Hilbert or Banach space, does there exist a nonconstant polynomial p(z) such that p(A) is "simpler", "nicer" than A. The motivation for organising these is the following. Suppose a particular functional calculus is applicable to p(A) but not directly to A. Using "multicentric calculus" one can represent functions using p(z) as a new variable allowing the functional calculus to be extended to apply to A. Classes of operators considered are increasing chains like finite rank, compact , Riesz, almost algebraic, quasialgebraic, biquasitriangular, quasitriangular, bounded.