Matrix convexity and unitary power dilations of Toeplitz-contractive operator tuples

Abstract

Using works of T.~Ando and L.~Gurvits, the well-known theorem of P.R.~Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces recast as a result for $d$-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator $d$-tuples satisfying this matrix-positivity condition are called, herein, Toeplitz-contractive, and a characterisation of the Toeplitz-contractivity condition is presented. The matrix-positivity condition leads to definitions of new distance-measures in several variable operator theory, generalising the notions of norm, numerical radius, and spectral radius to $d$-tuples of operators (commuting, for the spectral radius) in what appears to be a novel, asymmetric way. Toeplitz contractive operators form a noncommutative convex set, and a scaling constant $c_d$ for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle $S^1$ across $d$ complex dimensions is shown to exist.