Abstract.
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.
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The first author was supported by grant No. 201/06/0128 of GA CR and by IRP AV 0Z 1019053.
Received: 15 November 2007
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Müller, V., Neumann, M.M. Localizable spectrum and bounded local resolvent functions. Arch. Math. 91, 155–165 (2008). https://doi.org/10.1007/s00013-008-2652-6
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DOI: https://doi.org/10.1007/s00013-008-2652-6