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An Asymptotic Formula for the Commutators

โœ Scribed by Daniel Beltita; Mihai Sabac

Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
245 KB
Volume
153
Category
Article
ISSN
0022-1236

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โœฆ Synopsis

In what follows we shall describe some asymptotic formulas for the commutators defined by the values of the analytic functional calculus of some commuting n-tuple of bounded operators in a complex Banach space.

1998 Academic Press Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. For Q and A in B(X) we denote (ad Q) A=[Q, A]= QA&AQ. Related to the invariant subspace problem for certain Lie algebras of operators, in [8,9] occurred the problem of describing properties of the operator ad f (Q) (where f # O(_(Q)) i.e. f is analytic on a neighborhood of _(Q); f (Q) denotes the corresponding value of the analytic functional calculus) in terms of the corresponding properties of ad Q. The complexity of this problem is suggested by the following Taylor type formula announced in [9]:

where \ is a certain positive number depending on Q and T (see Theorem 1 below) and f (n) is the n th derivative of f. Connected to the study of the perturbation topology defined by the spectral semi-distance p in B(X), in [1,2] are proved other Taylor type formulas article no.

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