Advertisement

Shifts

  • Carlos S. Kubrusly

Abstract

An operator S+ on a Hilbert space Η is a unilateral shift if there exists an infinite sequence \(\left\{ {{{H}_{k}}} \right\}_{{k = 0}}^{\infty }\)of nonzero pairwise orthogonal subspaces of Η such that\(H = \oplus _{{k = 0}}^{\infty }{{H}_{k}}\)(i.e., the orthogonal family \(\left\{ {{{H}_{k}}} \right\}_{{k = 0}}^{\infty }\)spans Ηand S+ maps each Ηk isometrically onto Ηk+1.Two Hilbert spaces are unitarily equivalent if and only if they have the same dimension (see e.g., [32, p. 365]). Since \({{S}_{ + }}{{|}_{{{{H}_{k}}}}}:{{H}_{k}} \to {{H}_{{k + 1}}}\)is unitary (a surjective isometry), it follows that dim Ηk+1 = dimΗk,for everyk≥0.This constant dimension is the multiplicity of S+.The adjoint \(S_{ + }^{*} \in \mathcal{B}\left[ \mathcal{H} \right]{\mkern 1mu} of{\mkern 1mu} {{S}_{ + }} \in \mathcal{B}\left[ \mathcal{H} \right]\)is referred to as a backward unilateral shift, also denoted by S_. Writing \(\oplus _{{k = 0}}^{\infty }{{x}_{k}}{\mkern 1mu} for{\mkern 1mu} \left\{ {{{x}_{k}}} \right\}_{{k = 0}}^{\infty }{\mkern 1mu} in{\mkern 1mu} \oplus _{{k = 0}}^{\infty }{{H}_{k}}\),it follows that S+ and S + * are given by the formulas \({{S}_{ + }}x = 0 \oplus \mathop{ \oplus }\limits_{{k = 1}}^{\infty } {{U}_{k}}{{x}_{{k - 1}}}\;\;\;and\;\;\;\;\;\;{{S}_{ + }}x = \mathop{ \oplus }\limits_{{k = 0}}^{\infty } U_{k}^{*}{{x}_{{k + 1}}}\) for every \(x = \oplus _{{k = 0}}^{\infty }{{x}_{k}}{\mkern 1mu} in{\mkern 1mu} H = \oplus _{{k = 0}}^{\infty }{{H}_{k}}\),where 0 is the origin of Η0and Uk+1 is any unitary transformation of Ηk onto Ηk+1 so that\({{S}_{ + }}{{|}_{{{{H}_{k}}}}} = {{U}_{{k + 1}}}\),for each k≥0.These are identified with the infinite matrices
$${S_ + }\left( {\begin{array}{*{20}{c}} O&{}&{}&{}&{} \\ {{U_1}}&O&{}&{}&{} \\ {}&{{U_2}}&O&{}&{} \\ {}&{}&{{U_3}}&O&{} \\ {}&{}&{}&{}& \ddots \end{array}} \right){\text{ }}and{\text{ }}S_ + ^*\left( {\begin{array}{*{20}{c}} O&{U_1^*}&{}&{}&{} \\ {}&O&{U_2^*}&{}&{} \\ {}&{}&O&{U_3^*}&{} \\ {}&{}&{}&O&{} \\ {}&{}&{}&{}& \ddots \end{array}} \right)$$
of transformations where every entry below (above) the main block diagonal in the matrix of S+ (S + * )is unitary and the remaining entries are all null.

Keywords

Hilbert Space Orthonormal Basis Direct Summand Unitary Transformation Weight Sequence 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Birkhäuser Boston 2003

Authors and Affiliations

  • Carlos S. Kubrusly
    • 1
  1. 1.Catholic University of Rio de JaneiroRio de JaneiroBrazil

Personalised recommendations