A characteristic operator function for the class of n-hypercontractions

Olofsson, Anders

doi:10.1016/j.jfa.2006.03.004

Cited by 39 publications

(33 citation statements)

References 21 publications

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“…We note that the last statement in the theorem is a consequence of part (2b) of Lemma 6.1. We also note that realization formula (7.40) appears in [27,29]) in slightly different terms. 7.5.…”

Section: Beurling-lax Theorem For a Nmentioning

confidence: 89%

“…In the quest for a parallel system-theoretic interpretation for Bergman inner functions on the unit disk, Olofsson [27,29] has shown that realization formulas of the type Θ(z) = D + zCR n (zA)B involving higher-order resolvents R n (zA) arise naturally and he also associated a certain higher-order time-invariant linear system having such a Θ(z) as transfer function. We shall see below that functions of the form Θ(z) = D + zCR n,k (zA)B involving shifted higher-order resolvents R n,k (zA) also arise naturally.…”

Section: System Theory Motivationmentioning

confidence: 99%

“…Following [27,29] we say that the function Θ is a wandering-subspace Bergmaninner function whenever (1) M Θ : U 0 → A n (Y) is isometric, and (2) Θ · U 0 is orthogonal to S ℓ n Θ · U 0 for ℓ ≥ 1. In case M Θ U 0 = E is the wandering subspace for the S n -invariant subspace M, then we have the Beurling-Lax-type representation of M as the closure of Θ · C[z] ⊗ U 0 , where C[z] denotes the algebra of polynomials with coefficients in C. Construction of a Beurling inner function for the wandering subspace E of M amounts to focusing on the first element Θ 0 in the Bergman-inner family associated with M. Specifying Lemma 7.7 for the case k = 0 then leads to the following.…”

Section: Beurling-lax Theorem For a Nmentioning

confidence: 99%

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Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems

Ball

Bolotnikov

2013

Integr. Equ. Oper. Theory

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Abstract. It is well known that subspaces of the Hardy space over the unit disk which are invariant under the backward shift occur as the image of an observability operator associated with a discrete-time linear system with stable state-dynamics, as well as the functional-model space for a Hilbert space contraction operator, while forward shift-invariant subspaces have a representation in terms of an inner function. We discuss several variants of these statements in the context of weighted Bergman spaces on the unit disk. IntroductionLet H 2 be the standard Hardy space over the open unit diskand we let S : f (z) → zf (z) be the shift operator on H 2 . The classical theorem of Beurling [11] asserts that any S-invariant subspace M of H 2 has the form M = θ · H 2 where θ is an inner function (analytic on the disk with boundary-value function on the unit circle having modulus 1 almost everywhere). The result was extended to the vector-valued case by Lax [22] (for the finite-dimensional case) and by Halmos [18] (for the general case). The main thrust of Beurling's paper was the development of a theory of inner-outer factorization for H 2 and H ∞ -functions; this theory was then used to arrive at the famous characterization of invariant subspacesThere are now a number of more operator-theoretic and/or system-theoretic proofs which better handle the vector-valued case. For Y a Hilbert space, we write H 2 (Y) for the Hardy space H 2 (Y) = H 2 ⊗Y of Y-valued functions and we let M be a shift-invariant subspace of H 2 (Y). We single out four distinct approaches toward what we shall call simply the Beurling-Lax theorem for the vector-valued case.(1) Find a Hilbert space X (playing the role of a state space from the system theory point of view) and construct operators C : X → Y and A : X → X so that [ A C ] is isometric (A * A + C * C = I X ) and so that M ⊥ has the representation

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Section: Beurling-lax Theorem For a Nmentioning

confidence: 89%

Section: System Theory Motivationmentioning

confidence: 99%

Section: Beurling-lax Theorem For a Nmentioning

confidence: 99%

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Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems

Ball

Bolotnikov

2013

Integr. Equ. Oper. Theory

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“…We mention that for n = α + 2 positive integer the result of Theorem 1.4 originates from [18,Section 1] where it was used in a calculation of operator-valued Bergman inner functions.…”

Section: The Operator Identitymentioning

confidence: 99%

Operator identities for standard weighted Bergman shift and Toeplitz operators

Olofsson¹,

Wennman²

2013

J. Operator Theory

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Abstract. We prove an operator identity for the shift operator in the scale of standard weighted Bergman spaces in the unit disc. This operator identity is then applied in the context of functional calculus for the shift operator and a characterization of harmonic symbol Bergman space Toeplitz operators is obtained generalizing an earlier result by Louhichi and Olofsson. Duality arguments lead to operator inequalities and structure formulas for reproducing kernel functions which make contact with work of Richter, Shimorin, and others.

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“…Recently, biharmonic Bergman space potential theory has been applied by Hedenmalm, Shimorin and others in the study of Hele-Shaw flows and related problems of differential geometric nature (see [8,9,10]). In recent work the author has indicated how related methods can be used to develop a generalized systems theory for weighted Bergman space norms (see [16,17,18]). …”

Section: Introductionmentioning

confidence: 99%

A computation of Poisson kernels for some standard weighted biharmonic operators in the unit disc

Olofsson¹

2008

Complex Variables and Elliptic Equations

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Abstract. We compute Poisson kernels for integer weight parameter standard weighted biharmonic operators in the unit disc with Dirichlet boundary conditions. The computations performed extend the supply of explicit examples of such kernels and suggest similar formulas for these Poisson kernels to hold true in more generality. Computations have been carried out using the open source computer algebra package Maxima. IntroductionWe address in this paper the problem of finding explicit formulas for Poisson kernels for weighted biharmonic operators of the form ∆w −1 ∆ in the unit disc D with Dirichlet boundary conditions, where ∆ = ∂ 2 /∂z∂z, z = x + iy, is the Laplacian in the complex plane and w = w γ is a weight function of the formfor some real parameter γ > −1. Such a weight function w γ is commonly referred to as a standard weight.Let us first describe the context of these Poisson kernels. Let w : D → (0, ∞) be a smooth radial weight function and consider the weighted biharmonic Dirichlet problemHere T = ∂D is the unit circle and ∂ n denotes differentiation in the inward normal direction. The first equation in (0.1), the biharmonic equation ∆w −1 ∆u = 0, is evaluated in the distributional sense and defines a class of functions which we call w-biharmonic. In full generality the boundary datas f j ∈ D ′ (T) (j = 0, 1) are distributions on T and the boundary conditions in (0.1) are interpreted in a distributional sense as follows: Let u be a smooth function in D and let f 0 ∈ D ′ (T). We say that u = f 0 on T in the distributional sense if lim r→1 u r = f 0 in D ′ (T), where (0.2) u r (e iθ ) = u(re iθ ), e iθ ∈ T,

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A characteristic operator function for the class of n-hypercontractions

Cited by 39 publications

References 21 publications

Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems

Weighted Bergman Spaces: Shift-Invariant Subspaces and Input/State/Output Linear Systems

Operator identities for standard weighted Bergman shift and Toeplitz operators

A computation of Poisson kernels for some standard weighted biharmonic operators in the unit disc

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