An asymptotic formula for the commutators
type: Item
instance of: scholarly article
title: An asymptotic formula for the commutators
zbMATH Open document ID: 0914.47036
published in: Journal of Functional Analysis
publication date: 1999-06-23T00:00:00Z
review text: The main aim of this paper is to prove two asymptotic formulas which unify several ``Taylor type formulas'', defined in the context of commutators and analytic functional calculi. For the sake of simplicity, we shall present only the particular case of bounded linear operators acting in the same Banach space, although the authors' framework is slightly more general, involving two Banach spaces. Let \(Q= (Q_1,\dots, Q_n)\), \(Q'= (Q_1',\dots, Q_n')\) be commuting \(n\)-tuples. Let also \(\sigma(Q)\) (resp. \(\sigma(Q_j)\)) be the joint spectrum (resp. the spectrum) of \(Q\) (resp. of \(Q_j\), \(j= 1,\dots, n\)). One defines the commutators \(C(Q_j, Q_j')(T)= C_j(T)= Q_jT- TQ_j'\), and \(C^\alpha(Q, Q')(T)= C^{\alpha_1}_1\cdots C^{\alpha_n}_n(T)\), for every multi-index \(\alpha= (\alpha_1,\dots, \alpha_n)\). For a nonnegative \(r\), we denote by \(B(\sigma(Q_j'),r)\) the closed ball of center \(\sigma(Q_j')\) and radius \(r\). Let \(K'= B(\sigma(Q_1'),r)\times\cdots \times B(\sigma(Q_n'),r)\). Assume that \(\sigma(Q)\subset K'\) for a fixed \(r\geq \limsup_{|\alpha|\to \infty}\| C^\alpha(Q,Q')(T)\|^{1/| \alpha|}\), and let \(f\) be a holomorphic function in a neighborhood of \(K'\). With these conditions, the authors prove the formula \[ C(f(Q), f(Q'))(T)= \sum_{\alpha\in \mathbb{N}^n\setminus\{0\}} {1\over\alpha!} C^\alpha(Q, Q')(T)(\partial^\alpha f)(Q'), \] where \((\partial^\alpha f)(Q')\) is given by the holomorphic functional calculus of \(Q'\). A ``dual'' formula is also obtained. These formulas are then applied to get information about the quasinilpotent equivalence and the spectral semi-distance of some operators. Finally, a result about topologically irreducible quasisolvable Lie algebras of operators is also proved.
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 47A60
Mathematics Subject Classification ID: 47A13
zbMATH DE Number: 1215685
zbMATH Keywords: asymptotic formulas
zbMATH Keywords: commutators
zbMATH Keywords: analytic functional calculi
zbMATH Keywords: holomorphic functional calculus
zbMATH Keywords: topologically irreducible quasisolvable Lie algebras of operators
author: Daniel Beltiţă
author: Mihai Şabac
reviewed by: Florian-Horia Vasilescu
MaRDI profile type: Publication
full work available at URL: jfan.1997.3212
OpenAlex ID: W2071557760
cites work: Remarks on the perturbation and a topology for operators
cites work: Q5568730
cites work: Q5584224
cites work: Q3254057
cites work: The Taylor spectrum and spectral decompositions
cites work: Long-range many-body scattering. Asymptotic clustering for Coulomb-type potentials
cites work: Irreducible representations of infinite-dimensional Lie algebras
cites work: Nilpotent elements and solvable actions
cites work: Q4941272
cites work: The analytic-functional calculus for several commuting operators
cites work: Q3958979
DOI: 10.1006/JFAN.1997.3212
description: scientific article; zbMATH DE number 1215685