Shifts

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 Η_k isometrically onto Η_k+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 Η_k+1 = dimΗ_k,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 U_k+1 is any unitary transformation of Η_k onto Η_k+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.