Dual Piecewise Analytic Bundle Shift Models of Linear Operators

Yakubovich, Dmitry V.

doi:10.1006/jfan.1996.0032

Cited by 14 publications

(4 citation statements)

References 20 publications

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“…given by Remark. We observe that the idea of constructing model representations for A and for A* in dual functional spaces so that the duality formula (1.17) holds has already been exploited by the second author in [18] for Toeplitz operators with smooth symbols and related classes of operators. However, the model spaces constructed there were quite different.…”

Section: Verduyn Lunel and Yakubovichmentioning

confidence: 97%

A functional model approach to linear neutral functional differential equations

Lunel

Yakubovich

1997

Integr equ oper theory

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Section: Verduyn Lunel and Yakubovichmentioning

confidence: 97%

A functional model approach to linear neutral functional differential equations

Lunel

Yakubovich

1997

Integr equ oper theory

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“…Any answer involving the intrinsic complex geometry will be very interesting. In spite of the substantial work in [61] and [12], this question remains elusive. We discuss this a little more in Section 2.2.1.…”

Section: Some Future Directions and Further Thoughtsmentioning

confidence: 99%

Operators in the Cowen-Douglas Class and Related Topics

Misra¹

2019

Handbook of Analytic Operator Theory

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Linear spaces with an Euclidean metric are ubiquitous in mathematics, arising both from quadratic forms and inner products. Operators on such spaces also occur naturally. In recent years, the study of multivariate operator theory has made substantial progress. Although, the study of self adjoint operators goes back a few decades, the non-self adjoint theory has developed at a slower pace. While several approaches to this topic has been developed, the one that has been most fruitful is clearly the study of Hilbert spaces that are modules over natural function algebras like A (Ω), where Ω ⊆ C m is a bounded domain, consisting of complex valued functions which are holomorphic on some open set U containing Ω, the closure of Ω. The book [29] showed how to recast many of the familiar theorems of operator theory in the language of Hilbert modules. The books [31] and [14] provide an account of the achievements from the recent past. The impetus for much of what is described below comes from the interplay of operator theory with other areas of mathematics like complex geometry and representation theory of locally compact groups.such that the vectors {γ 1 (w ), . . . , γ n (w )} are linearly independent, w ∈ U . They also show that such an operator T defines a holomorphic Hermitian vector bundle E T :This means, for any fixed but arbitrary point w 0 ∈ Ω, there exists a holomorphic map γ T of the form γ T (w ) = (γ 1 (w ), . . ., γ n (w )), (T −w )γ i (w ) = 0 in some open neighbourhood U of w 0 . It is called a holomorphic frame for the operator T . Finally, Cowen and Douglas also assume that the linear span of {γ 1 (w ), . . ., γ n (w ) : w ∈ Ω} is dense in H . Let B n (Ω) denote this class of operators.One of the striking results of Cowen and Douglas says that there is a one to one correspondence between the unitary equivalence class of the operators T and the (local) equivalence classes of the holomorphic Hermitian vector bundles E T determined by them. As a result of this correspondence set up by the Cowen-Douglas theorem, the invariants the vector bundle E T like the curvature, the second fundamental form, etc.

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“…Completeness of almost eigenvectors is important in the construction of bundle shift models of left-invertible operators (see [22]). …”

Section: Preliminaries In This Paper We Discuss the Following Fundamentioning

confidence: 99%

On operators close to isometries

Chavan

2008

Studia Math.

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Abstract. We introduce and discuss a class of operators, to be referred to as operators close to isometries. The Bergman-type operators, 2-hyperexpansions, expansive pisometries, and certain alternating hyperexpansions are main examples of such operators. We establish a few decomposition theorems for operators close to isometries. Applications are given to the theory of p-isometries and of hyperexpansive operators. Preliminaries. In this paper, we discuss the following fundamental problems from single-variable operator theory. If S in B(H) is a completely non-unitary left-invertible operator, under what conditions does• S * admit a complete set of eigenvectors?• S possess the wandering subspace property?• S admit a complete set of almost eigenvectors?Recall that an operator S in B(H) admits a complete set of eigenvectors if H = {f ∈ H : (S − µI)f = 0 for some µ ∈ C} and that S possesses the wandering subspace property ifA left-invertible operator S in B(H) admits a complete set of almost eigenvectors ifOne of the main results of this paper asserts that if a completely nonunitary S in B(H) is close to isometry, then S * admits a complete set of eigenvectors. Moreover, under some additional hypotheses, such an S possesses the wandering subspace property as well as a complete set of almost 2000 Mathematics Subject Classification: Primary 47A16, 47B20.

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Dual Piecewise Analytic Bundle Shift Models of Linear Operators

Cited by 14 publications

References 20 publications

A functional model approach to linear neutral functional differential equations

A functional model approach to linear neutral functional differential equations

Operators in the Cowen-Douglas Class and Related Topics

On operators close to isometries

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