×

An asymptotic formula for the commutators. (English) Zbl 0914.47036

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.

MSC:

47B47 Commutators, derivations, elementary operators, etc.
47A60 Functional calculus for linear operators
47A13 Several-variable operator theory (spectral, Fredholm, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Apostol, C., Remarks on the perturbation and a topology for operators, J. Funct. Anal., 2, 395-408 (1968) · Zbl 0174.44405
[2] Apostol, C., Spectral decompositions and functional calculus, Rev. Roumaine Math. Pures Appl., XIII, 1481-1528 (1968) · Zbl 0176.43701
[3] Colojoara, I.; Foias, C., Theory of Generalized Spectral Operators (1968), Gordon and Breach: Gordon and Breach New York · Zbl 0189.44201
[4] Dunford, N.; Schwartz, J. T., Linear Operators. Part I (1958), Interscience: Interscience New York
[5] Frunza, St., The Taylor spectrum and spectral decompositions, J. Funct. Anal., 19, 390-421 (1975) · Zbl 0306.47013
[6] Sigal, I. M.; Soffer, A., Long range many-body scattering, Invent. Math., 99, 115-143 (1990) · Zbl 0702.35197
[7] Sabac, M., Irreducible representations of infinite dimensional Lie algebras, J. Funct. Anal., 52, 303-314 (1983) · Zbl 0528.17005
[8] Sabac, M., Nilpotent elements and solvable actions, Collect. Math., XLVII, 91-104 (1996) · Zbl 0856.46029
[9] M. Sabac, Analytic commutativity and nilpotence criteria; M. Sabac, Analytic commutativity and nilpotence criteria · Zbl 0942.47025
[10] Taylor, J. L., The analytic functional calculus for several commuting operators, Acta Math., 125, 1-38 (1970) · Zbl 0233.47025
[11] Tomescu, I., Introduction to Combinatorics (1975), Collet’s Ltd: Collet’s Ltd London and Wellingborough
[12] Vasilescu, F.-H., Analytic Functional Calculus and Spectral Decompositions (1982), Academiei-Reidel: Academiei-Reidel Bucharest-Dordrecht
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.