A new resolvent equation for the S-functional calculus

TL;DR

This paper introduces a novel resolvent equation for the S-functional calculus, unifying the left and right S-resolvent operators, which enhances the theoretical framework for analyzing noncommuting operator tuples.

Contribution

The paper presents a new resolvent equation for the S-functional calculus that involves both left and right S-resolvent operators simultaneously, extending classical resolvent theory.

Findings

Derived a new resolvent equation for S-functional calculus

Unified the treatment of left and right S-resolvent operators

Enhanced the theoretical tools for noncommuting operators

Abstract

The S-functional calculus is a functional calculus for $(n+1)$-tuples of non necessarily commuting operators that can be considered a higher dimensional version of the classical Riesz-Dunford functional calculus for a single operator. In this last calculus, the resolvent equation plays an important role in the proof of several results. Associated with the S-functional calculus there are two resolvent operators: the left $S_L^{-1}(s,T)$ and the right one $S_R^{-1}(s,T)$, where $s=(s_0,s_1,\ldots,s_n)\in \mathbb{R}^{n+1}$ and $T=(T_0,T_1,\ldots,T_n)$ is an $(n+1)$-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations $S_L^{-1}(s,T)s-TS_L^{-1}(s,T)=\mathcal{I}$, and $sS_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, respectively, where $\mathcal{I}$ denotes the identity operator. These equations allows to prove some properties of the S-functional calculus. In this paper we prove a new resolvent equation for the S-functional calculus which is the analogue of the classical resolvent equation. It is interesting to note that the equation involves both the left and the right S-resolvent operators simultaneously.