An asymptotic formula for the commutators
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.
topologically irreducible quasisolvable Lie algebras of operators
scientific article; zbMATH DE number 1215685