Polynomial as a new variable - a Banach algebra with a functional calculus

TL;DR

This paper introduces a new Banach algebra framework based on polynomials that allows a simple functional calculus for operators, effectively handling nontrivial Jordan blocks without requiring differentiability.

Contribution

It constructs a polynomial-dependent Banach algebra with a functional calculus applicable to operators with complex spectral properties, simplifying analysis of Jordan blocks.

Findings

Provides a functional calculus for operators with polynomial spectra

Handles nontrivial Jordan blocks without differentiability

Creates a polynomial-based Banach algebra for spectral analysis

Abstract

Given any square matrix or a bounded operator $A$ in a Hilbert space such that $p(A)$ is normal (or similar to normal), we construct a Banach algebra, depending on the polynomial $p$, for which a simple functional calculus holds. When the polynomial is of degree $d$, then the algebra deals with continuous $\mathbb C^d$-valued functions, defined on the spectrum of $p(A)$. In particular, the calculus provides a natural approach to deal with nontrivial Jordan blocks and one does not need differentiability at such eigenvalues.