Localizable Spectrum and Bounded Local Resolvent Functions
نویسندگان
چکیده
Given a Banach space operator with interior points in the localizable spectrum and without non-trivial divisible subspaces, this article centers around the construction of an infinite-dimensional linear subspace of vectors at which the local resolvent function of the operator is bounded and even admits a continuous extension to the closure of its natural domain. As a consequence, it is shown that, for any measure with natural spectrum on a locally compact abelian group, the corresponding operator of convolution on the group algebra admits a non-zero bounded local resolvent function precisely when its spectrum has non-empty interior. Mathematics Subject Classification (2000). Primary 47A11; Secondary 43A25,
منابع مشابه
Local Spectrum of a Family of Operators
Starting from the classic definitions of resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the local resolvent set and local spectrum, the local spectral space and the single-valued extension property of a family of linear bounded operators on a Banach space. Keeping the analogy with the classic case, we extend some of the known results from the case of a l...
متن کاملOn smooth local resolvents
We exhibit an example of a bounded linear operator on a Banach space which admits an everywhere defined local resolvent with continuous derivatives of all orders. Let T be a bounded linear operator acting on a complex Banach space X. It is well known that the resolvent z 7→ (T − z)−1 defined on the complement of the spectrum σ(T ) is unbounded. More precisely, ‖(T − z)−1‖ → ∞ whenever z approac...
متن کاملOn the Resolvent of a Linear Operator Associated with a Well-Posed Cauchy Problem
We show how local estimates may be obtained for holomorphic functions of a class of linear operators on a finite-dimensional linear vector space. This is accomplished by classifying the spectrum of each operator and then estimating its resolvent on certain contours in the left half-plane. We apply these methods to prove some known theorems, and in addition we obtain new estimates for the invers...
متن کاملSpectrum of a Family of Operators
Having as start point the classic definitions of resolvent set and spectrum of a linear bounded operator on a Banach space, we introduce the resolvent set and spectrum of a family of linear bounded operators on a Banach space. In addition, we present some results which adapt to asymptotic case the classic results.
متن کاملCompact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions
We characterize compact composition operators on real Banach spaces of complex-valued bounded Lipschitz functions on metric spaces, not necessarily compact, with Lipschitz involutions and determine their spectra.
متن کامل