Non-commutative rational Clark measures

Abstract

We characterize the non-commutative Aleksandrov-Clark measures and the minimal realization formulas of contractive and, in particular, isometric non-commutative rational multipliers of the Fock space.Here, the full Fock space over C d is defined as the Hilbert space of square-summable power series in several non-commuting formal variables, and we interpret this space as the non-commutative and multi-variable analogue of the Hardy space of square-summable Taylor series in the complex unit disk. We further obtain analogues of several classical results in Aleksandrov-Clark measure theory for non-commutative and contractive rational multipliers.Non-commutative measures are defined as positive linear functionals on a certain self-adjoint subspace of the Cuntz-Toeplitz algebra, the unital C * −algebra generated by the left creation operators on the full Fock space. Our results demonstrate that there is a fundamental relationship between NC Hardy space theory, representation theory of the Cuntz-Toeplitz and Cuntz algebras and the emerging field of non-commutative rational functions.

Sub-Hardy Hilbert spaces in the non-commutative unit row-ball

“…H(Z) + I n H(0) − I n Re H(0) = H(Z) − iI n Im H(0), see[28,17]. Since H T and H are NC Herglotz functions which differ by an imaginary constant, µ T is the NC Clark measure of both b T and b, where b T is the inverse Cayley transform of H T and b is the inverse Cayley transform of H[34].Theorem 5 (NC rational Fejér-Riesz). Any NC rational, positive semi-definite left Toeplitz operator,T = Re H(R) ≥ 0, H ∈ H ∞ d , is factorizable.…”