Dilation to Unilateral Shifts

Abstract

Which bounded linear operator on a complex separable Hilbert space can be dilated to a unilateral shift? This is the problem we are going to address here. Recall that an operator A on space H is said to dilate (resp. power dilate) to operator B on K if there is an isometry V from H to K such that A = V ^* BV (resp. A ⁿ =V ^* B ⁿ V for all n > 1) or, equivalently, if B is unitarily equivalent to a 2-by-2 operator matrix $\left[ {\begin{array}{*{20}{c}} A & * \\ * & * \\ \end{array} } \right]$ with A in its upper left corner (resp.B ⁿ is unitarily equivalent to $\left[ {\begin{array}{*{20}{c}} {{{A}^{n}}} & * \\ * & * \\ \end{array} } \right]$ under the same unitary operator for all n>1). Theunilateral shift S _k of multiplicity $k (1 \leqslant k \leqslant \infty )$ is the operator ${S_k}(x0,x1,x2,...) = (0,{x_0}{x_1}, \ldots )on\sum\nolimits_{n = 0}^\infty { \oplus H with dim H = k.}$