Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2112.05204

Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators

Fabrizio Colombo·Jonathan Gantner·David P. Kimsey·Irene Sabadini
View original source ↗Read source document ↗Export metadata ↓

Abstract

The spectral theory on the $S$-spectrum was born out of the need to give quaternionic quantum mechanics (formulated by Birkhoff and von Neumann) a precise mathematical foundation. Then it turned out that this theory has important applications in several fields such as fractional diffusion problems and, moreover, it allows one to define several functional calculi for $n$-tuples of noncommuting operators. With this paper we show that the spectral theory on the $S$-spectrum is much more general and it contains, just as particular cases, the complex, the quaternionic and the Clifford settings. More precisely, we show that the $S$-spectrum is well defined for objects in an algebra that has a complex structure and for operators in general Banach modules. We show that the abstract formulation of the $S$-functional calculus goes beyond quaternionic and Clifford analysis. Indeed we show that the $S$-functional calculus has a certain {\em universality property}. This fact makes the spectral theory on the $S$-spectrum applicable to several fields of operator theory and allows one to define functions of noncommuting matrix variables, and operator variables, as a particular case.

Explore related subjects

math.FA

Keep this discovery

BibTeXRIS

Fabrizio Colombo, Jonathan Gantner, David P. Kimsey, Irene Sabadini. 2021-12-09. Universality property of the $S$-functional calculus, noncommuting matrix variables and Clifford operators. https://doi.org/10.1016/j.aim.2022.108719

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Typical dynamical properties of operators on $\ell_p$

We investigate the typical dynamical properties of hypercyclic operators in $\mathcal{L}_M(X)$, the set of all bounded linear operators on $X$ whose norms are at most $M$, when $X=\ell_p$, $1< p<\infty$. We show that, with respect to SOT$^*$, a typical operator $T\in \mathcal{L}_M(X)$ is weakly mixing, is weakly disjoint from a given hypercyclic operator $S$, is not topologically ergodic, and satisfies $(T,T^2,\dotsc,T^k)$ is disjoint hypercyclic for any $k\geq 2$. We also study the typical dynamical properties for the concrete family $\mathcal{M}=\{I+B_w\in \mathcal{L}(X)\colon w\in c_0(\mathbb{Z})\}$, endowed with the norm topology, where $B_w$ is a bilateral weighted backward shift.

math.FA↗
arXiv2026

A bi-Lipschitz characterization of strong minimum-attainment for Lipschitz maps

We completely characterize the denseness of strongly minimum-attaining Lipschitz functions, a minimum analogue for strongly norm-attaining Lipschitz functions, in terms of bi-Lipschitz embeddings. More precisely, our main result shows that the set of strongly minimum-attaining Lipschitz functions defined on a complete metric space $M$ fails the denseness if and only if $M$ is bi-Lipschitz equivalent to a subset of $\mathbb{R}$ with positive Lebesgue measure, or equivalently, if $M$ admits a bi-Lipschitz embedding into $\mathbb{R}$ and $M$ has positive 1-dimensional Hausdorff measure. As a consequence, we provide an isometric characterization of the pure 1-unrectifiability of $M$ in terms of strongly minimum-attaining Lipschitz maps defined on bi-Lipschitz copies of closed subsets of $M$. Several counterexamples showing that the main result cannot be naturally extended to the vector-valued setting are also presented.

math.FA↗
arXiv2026

On weak dominance of t-conorms over t-norms

The weak dominance of aggregation operators, particularly between triangular norms (t-norms) and triangular conorms (t-conorms), has attracted considerable attention in aggregation operator theory. While several characterizations have been obtained for Archimedean and continuous cases, a general criterion for continuous t-conorms over continuous t-norms remains to be fully clarified. In this paper, we provide a complete characterization of a continuous t-conorm weakly dominating a continuous t-norm. We first reduce the problem for ordinal sum operators to that for their single Archimedean components, and then express the weak dominance condition entirely in terms of the additive generators of these components. Our approach covers both strict and nilpotent cases uniformly, and recovers the known results for Archimedean operators as a special case.

math.FA↗