Shifts

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.