Construction of a tensor product algebra with a multicentric functional calculus

TL;DR

This paper develops a tensor product algebra framework for multicentric functional calculus, enabling a functional calculus for pairs of commuting matrices by extending the algebraic structure to matrix-valued functions.

Contribution

It introduces a tensor product algebra for C^2-valued functions into matrix spaces, extending multicentric calculus to pairs of matrices.

Findings

Constructed a Banach algebra for C^2 to C^{d_1 x d_2} functions.

Extended multicentric calculus to commuting matrix pairs.

Provided a framework for functional calculus in this new algebraic setting.

Abstract

In the multicentric calculus one takes a polynomial with simple roots as a new global variable and replaces scalar functions {\varphi} by functions f taking values in C^d with d the degree of the polynomial leading to an efficient holomorphic functional calculus for bounded operators. This calculus was extended for non-holomorphic functions, so that if A is not diagonalizable one can find a p such that p(A) is diagonalizable and apply the calculus to all matrices. This was done in [7] by creating a Banach algebra for C^d-valued continuous functions in such a way that the original functions {\varphi} appear as Gelfand transforms of f. In this paper we consider constructing a Banach algebra for functions from C^2 into C^(d_1 x d_2) which then likewise leads to a functional calculus for commuting pairs of matrices.