Measure Schur Complements and Spectral Functions of Unitary Operators with Respect to Different Scales

Boiko, S. S.; Dubovoy, V. K.; Kheifets, Alexander

doi:10.1007/978-3-0348-8247-7_5

Cited by 8 publications

(24 citation statements)

References 7 publications

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“…The incoming subspace W * := ⊕ −1 n=−∞ U n F * is orthogonal to the outgoing subspace W := ⊕ ∞ n=0 U n F . (For more general notions of the scattering system in wandering subspace formulation, where the pair of wandering subspaces F and F * is replaced by a general mapping ρ : N → K, called a scale or a control/observation operator, see BoikoDubovoy-Kheifets [14]). The scattering operator S : W → W * associated with the scattering system S is defined to be simply…”

Section: Introductionmentioning

confidence: 99%

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

Ball

Sadosky

Vinnikov

2005

Integr. equ. oper. theory

111

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The one-to-one correspondence between one-dimensional linear (stationary, causal) input/state/output systems and scattering systems with one evolution operator, in which the scattering function of the scattering system coincides with the transfer function of the linear system, is well understood, and has significant applications in H ∞ control theory. Here we consider this correspondence in the d-dimensional setting in which the transfer and scattering functions are defined on the polydisk. Unlike in the onedimensional case, the multidimensional state space realizations and the corresponding multi-evolution scattering systems are not necessarily equivalent, and the cases d = 2 and d > 2 differ substantially. A new proof of Andô's dilation theorem for a pair of commuting contraction operators and a new statespace realization theorem for a matrix-valued inner function on the bidisk are obtained as corollaries of the analysis. (2000). Primary 47A20; Secondary 47A40, 93C55, 93C35. Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 99%

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

Ball

Sadosky

Vinnikov

2005

Integr. equ. oper. theory

111

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“…Proof. This is essentially Theorem 4.1 in [15]. For the reader's convenience, we recall the proof here.…”

Section: Unitary Scattering Systemsmentioning

confidence: 99%

“…Suppose that w is a vector measure such that (4.12) holds, where σ and σ are the positive measures given by (4.13). Consider the Hellinger space L σ (see [15]) with operator U σ being multiplication by the independent variable t. We define the embeddings i K : K → L σ and i K : K → L σ as follows: first we map K onto L σ and K onto L σ by Fourier representations…”

Section: J a Ball And A Kheifets Ieotmentioning

confidence: 99%

“…The lift Y is uniquely determined from its moments w Y (n) = i * * Y n i where i * and i are certain isometric embedding operators (or scale operators in the sense of [15]). Calculation of such moments (the collection of which we call the symbol of the lift Y ) requires the computation of powers of U * = F (U 0 , U 1 ) in terms of the coefficients of the universal unitary colligation U 0 determined by the problem data and the coefficients of the free-parameter unitary colligation U 1 (or in terms of its characteristic function ω(ζ)).…”

Section: Introductionmentioning

confidence: 99%

“…2 we present the general principle for computation of powers of U = F (U 0 , U 1 ) via the trajectory-level feedback connection of the augmented input-output operator of U 0 with that of U 1 . Section 3 reviews results from [15] concerning general unitary scattering systems which will be needed in the sequel. Section 4 reviews basic ideas from [1] concerning the correspondence between contractive intertwiners Y of two unitary operators U and U on the one hand and unitary couplings U of U and U on the other.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

Ball

Kheifets

2011

Integr. Equ. Oper. Theory

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It is known that the set of all solutions of a commutant lifting and other interpolation problems admits a Redheffer linear-fractional parametrization. The method of unitary coupling identifies solutions of the lifting problem with minimal unitary extensions of a partially defined isometry constructed explicitly from the problem data. A special role is played by a particular unitary extension, called the central or universal unitary extension. The coefficient matrix for the Redheffer linear-fractional map has a simple expression in terms of the universal unitary extension. The universal unitary extension can be seen as a unitary coupling of four unitary operators (two bilateral shift operators together with two unitary operators coming from the problem data) which has special geometric structure. We use this special geometric structure to obtain an inverse theorem (Theorem 8.4) which characterizes the coefficient matrices for a Redheffer linear-fractional map arising in this way from a lifting problem. The main tool is the formalism of unitary scattering systems developed in Boiko et al. (Operator theory, system theory and related topics (Beer-Sheva/Rehovot 1997), pp. 89-138, 2001) and Kheifets (Interpolation theory, systems theory and related topics, pp. 287-317, 2002) Mathematics Subject Classification (2010). Primary 47A20; Secondary 47A57.

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Formal Reproducing Kernel Hilbert Spaces: The Commutative and Noncommutative Settings

Ball

Vinnikov

2003

Reproducing Kernel Spaces and Applications

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Measure Schur Complements and Spectral Functions of Unitary Operators with Respect to Different Scales

Cited by 8 publications

References 7 publications

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

Scattering Systems with Several Evolutions and Multidimensional Input/State/Output Systems

The Inverse Commutant Lifting Problem. I: Coordinate-Free Formalism

Formal Reproducing Kernel Hilbert Spaces: The Commutative and Noncommutative Settings

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