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

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 is recast as a result for <i>d</i>-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator <i>d</i>-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 <i>d</i>-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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>c</mml:mi> <mml:mi>d</mml:mi></mml:msub> </mml:math> for inclusions of the minimal and maximal matrix convex sets determined by a stretching of the unit circle <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>S</mml:mi> <mml:mn>1</mml:mn></mml:msup> </mml:math> across <i>d</i> complex dimensions is shown to exist.