Decompositions

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 \( {T^{*n}} = {T^{n*}}{\text{ we get }}{T^{*n}}{T^n}{\text{ }}\underline > {\text{ O and}}\left\| {{T^{*n}}{T^n}} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ < } \left\| {{T^{*n}}} \right\|\left\| {{T^n}} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ < } {\left\| T \right\|^{2n}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ < } 1.{\text{Moreover}},\left\langle {{T^{*n + 1}}{T^{n + 1}}x;x} \right\rangle = {\left\| {{T^{n + 1}}x} \right\|^2}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ < } \;{\left\| {{T^n}x} \right\|^2} = \left\langle {{T^{*n}}{T^n}x;x} \right\rangle \) for every x in Η Thus \(\left\{ {{{T}^{{*n}}}{{T}^{{*n}}}} \right\}\) is a bounded monotone sequence of self-adjoint operators, and therefore it converges strongly (Problem 3.5). Summing up: if T is a contraction on a Hilbert space Η, then

for some operator A on Η ;the strong limit of \(\left\{{{T^{*n}}{T^{*n}}}\right\}\).

Authors and Affiliations

1. Catholic University of Rio de Janeiro, R. Marques de S. Vicente 255, 22453-900, Rio de Janeiro, RJ, Brazil

   Carlos S. Kubrusly

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