Conservative Structured Noncommutative Multidimensional Linear Systems
TL;DRAbstract
We introduce a class of conservative structured multidimensional linear systems with evolution along a free semigroup. The system matrix for such a system is unitary and the associated transfer function is a formal power series in noncommuting indeterminates. A formal power series T(z1, ⋯ , zd) in the noncommuting indeterminates z1,⋯, zd arising in this way satisfies a noncommutative von Neumann inequality, i.e., substitution of a d-tuple of noncommuting operators δ = (δ1,⋯, δd) on a fixed separable Hilbert space which is contractive in the appropriate sense yields a contraction operator T(δ) = T(δ1,⋯, δd). We also obtain the converse realization theorem: any formal power series satisfying such a von Neumann inequality can be realized as the transfer function of such a conservative structured multidimensional linear system.
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We introduce a class of conservative structured multidimensional linear systems with evolution along a free semigroup. The system matrix for such a system is unitary and the associated transfer function is a formal power series in noncommuting indeterminates. A formal power series T(z1, ⋯ , zd) in the noncommuting indeterminates z1,⋯, zd arising in this way satisfies a noncommutative von Neumann inequality, i.e., substitution of a d-tuple of noncommuting operators δ = (δ1,⋯, δd) on a fixed separable Hilbert space which is contractive in the appropriate sense yields a contraction operator T(δ) = T(δ1,⋯, δd). We also obtain the converse realization theorem: any formal power series satisfying such a von Neumann inequality can be realized as the transfer function of such a conservative structured multidimensional linear system.
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