A new resolvent equation for the S-functional calculus

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 $s=(s_0,s_1,\dots ,s_n)\in {\mathbb{R}}^{n+1}$ and $T=(T_0,T_1,\dots ,T_n)$ is an $(n+1)$-tuple of non commuting operators. These two S-resolvent operators satisfy the S-resolvent equations $S_L^{-1}(s,T)s-T{S}_L^{-1}(s,T)=\mathcal{I}$, and $s{S}_R^{-1}(s,T)-S_R^{-1}(s,T)T=\mathcal{I}$, 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.

References in corpus (2)

• A new functional calculus for non-commuting operators
• Realizations of slice hyperholomorphic generalized contractive and positive functions

Cited by in corpus (2)

• S-spectrum and Associated Continuous Frames on Quaternionic Hilbert Spaces
• A new approach to slice analysis via slice topology