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Based on lectures given at an instructional course, this volume enables readers with a basic knowledge of functional analysis to access key research in the field. The authors survey several areas of current interest, making this volume ideal preparatory reading for students embarking on graduate work as well as for mathematicians working in related areas.
Based on lectures given at an instructional course, this volume enables readers with a basic knowledge of functional analysis to access key research in the field. The authors survey several areas of current interest, making this volume ideal preparatory reading for students embarking on graduate work as well as for mathematicians working in related areas.
Modern local spectral theory is built on the classical spectral
theorem, a fundamental result in single-operator theory and Hilbert
spaces. This book provides an in-depth introduction to the natural
expansion of this fascinating topic of Banach space operator
theory, whose pioneers include Dunford, Bishop, Foias, and others.
Assuming only modest prerequisites of its readership, it gives
complete coverage of the field, including the fundamental recent
work by Albrecht and Eschmeier which provides the full duality
theory for Banach space operators. It is highlighted by many
characterizations of decomposable operators, and of other related,
important classes of operators, as well as an in-depth study of
their spectral properties, including identifications of
distinguished parts, and results on permanence properties of
spectra with respect to several types of similarity. Also found is
a thorough and quite elementary treatment of the modern single-
operator duality theory; this theory has many applications, both to
general issues of classification and to such celebrated problems as
the invariant subspace problems. A long chapter - almost a book in
itself - is devoted to the use of local spectral theory in the
study of spectral properties of multipliers and convolution
operators. Another one describes its connections to automatic
continuity theory. Written in a careful and detailed style, it
contains numerous examples, many simplified proofs of classical
results, and extensive references. It concludes with a list of
interesting open problems, suitable for continued research.
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