2022
DOI: 10.48550/arxiv.2201.08045
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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 obtai… Show more

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“…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.…”
mentioning
confidence: 99%
“…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.…”
mentioning
confidence: 99%