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

2015/06/01 by Olavi Nevanlinna, Nevanlinna, Olavi · 1 citation
Computer Science · Mathematics · #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Matrix Theory and Algorithms #math.FA

paper · pdf · doi:10.48550/arxiv.1506.00634

arxiv created 2015/06/01 · openalex publication_date 2015/06/01 · arxiv updated 2015/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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