Decompositions

Carlos S. Kubrusly
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Carlos S. Kubrusly: Catholic University of Rio de Janeiro

Chapter 6 in Hilbert Space Operators, 2003, pp 51-64 from Springer

Abstract: Abstract A contraction is an operator T on a formed space χ such that ‖T‖≤1. Equivalently, such that ‖T x ‖≤‖x‖for every x in χ. If T is a contraction on a Hilbert space Η, then $$\left\{ {{{T}^{{*n}}}{{T}^{{*n}}}} \right\}$$ is a decreasing sequence of nonnegative contractions. In fact, take an arbitrary positive integer n. Since $$ {T^{*n}} = {T^{n*}}{\text{ we get }}{T^{*n}}{T^n}{\text{ }}\underline > {\text{ O and}}\left\| {{T^{*n}}{T^n}} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{

Keywords: Hilbert Space; Unitary Operator; Invariant Subspace; Direct Summand; Strong Limit (search for similar items in EconPapers)
Date: 2003
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DOI: 10.1007/978-1-4612-2064-0_6

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