Showing papers on "Multiplication operator published in 1999"
06 Jul 1999
137 citations
TL;DR: The Pick-Schwarz Lemma as mentioned in this paper states that B0 is a closed subspace of B. Good sources for results and references about Bloch functions are the papers of Anderson-Clunie-Pommerenke [ACP], Fernández [Fe], Pommererke [Po], and the book of Zhu [Zh, Chapter 5].
Abstract: and that B0 is a closed subspace of B. Good sources for results and references about Bloch functions are the papers of Anderson-Clunie-Pommerenke [ACP], Fernández [Fe], Pommerenke [Po], and the book of Zhu [Zh, Chapter 5]. If φ is an analytic function on D with φ(D) ⊂ D, then the equation Cφf = f ◦ φ defines a composition operator Cφ on the space of all holomorphic functions on D. The Pick-Schwarz Lemma (see [CM, p. 47], for instance) asserts that
92 citations
TL;DR: In this article, a photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase.
Abstract: A photon position operator with commuting components is constructed, and it is proved that it equals the Pryce operator plus a term that compensates for the adiabatic phase. Its eigenkets are transverse and longitudinal vectors, and thus states can be selected that have definite polarization or helicity. For angular momentum and boost operators defined in the usual way, all of the commutation relations of the Poincar\'e group are satisfied. This new position operator is unitarily equivalent to the Newton-Wigner-Pryce position operator for massive particles.
59 citations
TL;DR: In this article, the concept of a spectral shift operator was introduced and used to derive Krein's spectral shift function for pairs of self-adjoint operators and their logarithms.
Abstract: We introduce the concept of a spectral shift operator and use it to derive Krein’s spectral shift function for pairs of self-adjoint operators. Our principal tools are operator-valued Herglotz functions and their logarithms. Applications to Krein’s trace formula and to the Birman-Solomyak spectral averaging formula are discussed.
48 citations
15 Apr 1999
TL;DR: A function theoretic characterization of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation, is given in this paper.
Abstract: A function theoretic characterization is given of when a composition operator is compact on BMOA, the space of analytic functions on the unit disk having radial limits that are of bounded mean oscillation on the unit circle. When the symbol of the composition operator is univalent, compactness on BMOA is shown to be equivalent to compactness on the Bloch space, and a characterization in terms of the geometry of the image of the disk under the symbol of the operator results. ?
47 citations
TL;DR: In this paper, it was shown that the transfer operator for a general class of rational maps converges exponentially fast in the supremum norm and in Holder norms, for small enough Holder exponents, to its principal eigendirection.
Abstract: We prove that the transfer operator for a general class of rational maps converges exponentially fast in the supremum norm and in Holder norms, for small enough Holder exponents, to its principal eigendirection.
46 citations
TL;DR: A new computationally viable approach is derived for computing the induced norm of a state space compression operator, an integral operator defined on the finite length space L2[0,h], which may have application to a wider variety of integral operators.
37 citations
08 Feb 1999
TL;DR: New bounds for the generalization error of feature space machines, such as support vector machines and related regularization networks, are derived by obtaining new bounds on their covering numbers by virtue of the eigenvalues of an integral operator induced by the kernel function used by the machine.
Abstract: We derive new bounds for the generalization error of feature space machines, such as support vector machines and related regularization networks by obtaining new bounds on their covering numbers. The proofs are based on a viewpoint that is apparently novel in the field of statistical learning theory. The hypothesis class is described in terms of a linear operator mapping from a possibly infinite dimensional unit ball in feature space into a finite dimensional space. The covering numbers of the class are then determined via the entropy numbers of the operator. These numbers, which characterize the degree of compactness of the operator, can be bounded in terms of the eigenvalues of an integral operator induced by the kernel function used by the machine. As a consequence we are able to theoretically explain the effect of the choice of kernel functions on the generalization performance of support vector machines.
36 citations
Patent•
15 Nov 1999TL;DR: In this article, a correlation operator Θ provides a way for the results of sub-queries to be correlated by using an algebra to resolve into flags attached to appropriate operators in the sub-query.
Abstract: A correlation operator Θ provides a way for the results of sub-queries to be correlated. The correlation operator Θ has an implied existential quantifier property (i.e., a “for some” property) and is satisfied if any record matches its sub-query. If no record is found that matches the correlation operator Θ's sub-query, then the correlation operator Θ query fails. The implicit existential quantifier property of the correlation operator Θ can be converted into a universal quantifier property (i.e., a “for all” property) by transformation of the query. The correlation operator Θ uses an algebra to resolve into flags attached to appropriate operators in the sub-query. After the correlation operator Θ is eliminated from the query, the query can be performed. The operator flags perform the correlation as part of the sub-query. The correlation operator Θ can be inserted explicitly by the user if she knows what data she desires to be correlated. Alternatively, the correlation operator can be inserted by the schema automatically to improve the expected results of the query.
31 citations
TL;DR: In this paper, the spectral properties of the transition operator Ta were investigated and applied to the study of the smoothness properties of a normalized solution of the refinement equation for finitely supported sequences.
30 citations
TL;DR: In this paper, the authors considered continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling and provided necessary and sufficient conditions on a compactly supported function g(t) for its minimal dual (Wexler-Razdual) γ0 (t) to be compactly support.
Abstract: In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L2 (ℝ) for its minimal dual (Wexler-Razdual) γ0 (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L2 (ℝ) to have a band-limited minimal dual γ0 (t). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactly supported (band-limited) minimal dual γ0(t) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.
TL;DR: In this article, it was shown that one can use only four large vectors at the cost of executing the core conjugate algorithm twice, which is less than by a factor of 2, depending on the architecture of the computer one uses.
Abstract: The overlap lattice-Dirac operator contains the sign function ∊(H). Recent practical implementations replace ∊(H) by a ratio of polynomials, H Pn(H2)/Qn(H2), and require storage of 2n+2 large vectors. Here I show that one can use only four large vectors at the cost of executing the core conjugate algorithm twice. The slow-down might be less than by a factor of 2, depending on the architecture of the computer one uses.
TL;DR: For a closed densely defined sectorial operator using its Friedrichs and von Neumann-Krein m-sectorial extensions, the function of two complex variablesWF(λ,z) which determines the simple part of the operator up to unitary equivalence, is defined and studied in this article.
Abstract: For a closed densely defined sectorial operator using its Friedrichs and von Neumann-Krein m-sectorial extensions, the function of two complex variablesWF(λ,z) which determines the simple part of the operator up to unitary equivalence, is defined and studied The strong limitQF(λ)=−WF(λ, −∞) is an analog of the Q-function of positive symmetric operator These functions are used for the description of the resolvents of m-sectorial extensions
TL;DR: In this paper, the authors studied the local lifting property for operator spaces and showed that an operator space has local lifting if and only if it is an LΓ1, λ space.
TL;DR: The CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations of Ta = f* in Hilbert spaces, and two a posteriori stopping rules are introduced.
Abstract: We consider an ill-posed problem Ta = f* in Hilbert spaces and suppose that the linear bounded operator T is approximately available, with a known estimate for the operator perturbation at the solution. As a numerical scheme the CGNR-method is considered, that is, the classical method of conjugate gradients by Hestenes and Stiefel applied to the associated normal equations. Two a posteriori stopping rules are introduced, and convergence results are provided for the corresponding approximations, respectively. As a specific application, a parameter estimation problem is considered.
01 Jan 1999
TL;DR: In this article, the authors considered continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling and provided necessary and sufficient conditions on a compactly supported function g(t ) for its minimal dual (Wexler-Raz dual) yO (t) to be compactly support.
Abstract: In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t ) generating a Weyl-Heisenbergframe for L 2 (~,) for its minimal dual (Wexler-Raz dual) yO (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L 2 (~ ) to have a band-limited minimal dual yO ( t ). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t ) has a compactly supported (band-limited) minimal dual yO (t ) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.
TL;DR: In this article, a Banach space and a unitary operator are defined as isometrically equivalent if there exists an invertible isometry such that TV =VS.
Abstract: Let $X$ be a Banach space and let $B(X)$ denote the space of bounded operators on $X$. Two elements $S,T\inB(X)$ are isometrically equivalent if there exists an invertible isometry $V$ such that $TV=VS$. If $X$ is a Hilbert space, then $V$ is a unitary operator and $S$ and $T$ are said to be unitarily equivalent .
TL;DR: In this paper, the authors considered a complex third-order differential operator on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems and proved that two spectra are sufficient to determine the operator.
Abstract: We consider a complex third-order differential operator on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. It is proved that two spectra are sufficient to determine the operator. This result is valid under applying readily verifiable hypotheses simultaneously to the two spectra.
TL;DR: This theorem strengthens the result of Burosch, Demetrovics and Katona, who propose a characterization of the generating system of a closure operator defined by the quasi-closed sets of the closure operator.
Abstract: The purpose of this paper is to provide a characterization result for the canonical basis of an arbitrary closure operator. This theorem strengthens the result of Burosch, Demetrovics and Katona, who propose a characterization of the generating system of a closure operator defined by the quasi-closed sets of the closure operator.
Posted Content•
12 Jun 1999
TL;DR: In this article, the authors study operator spaces, operator algebras, and operator modules from the point of view of the noncommutative Shilov boundary, and find that Hilbert C$^*-$modules and their properties, which they studied earlier in the operator space framework, replace certain topological tools.
Abstract: We study operator spaces, operator algebras, and operator modules, from the point of view of the `noncommutative Shilov boundary'. In this attempt to utilize some `noncommutative Choquet theory', we find that Hilbert C$^*-$modules and their properties, which we studied earlier in the operator space framework, replace certain topological tools. We introduce certain multiplier operator algebras and C$^*-$algebras of an operator space, which generalize the algebras of adjointable operators on a C$^*-$module, and the `imprimitivity C$^*-$algebra'. It also generalizes a classical Banach space notion. This multiplier algebra plays a key role here. As applications of this perspective, we unify, and strengthen several theorems characterizing operator algebras and modules, in a way that seems to give more information than other current proofs. We also include some general notes on the `commutative case' of some of the topics we discuss, coming in part from joint work with Christian Le Merdy, about `function modules'.
TL;DR: In this paper, the Central Limit Theorem is interpreted as a fixed point theorem for a certain operator, and the problem of linearizing this operator is studied in both classical and free probability theory.
Abstract: We interpret the Central Limit Theorem as a fixed point theorem for a certain operator, and consider the problem of linearizing this operator In classical as well as in free probability theory [VDN92], we consider two methods giving such a linearization, and interpret the result as a weak form of the CLT In the classical case the analysis involves dilation operators; in the free case more general composition operators appear
TL;DR: In this article, sufficient and necessary conditions are obtained for the observation operator C to be infinite-time admissible for a control system Σ(A, B, −), due to duality argument, for the control operator B to be extended admissible.
Abstract: Let Σ(A, −, C) be an abstract dynamical system withA being the generator of aC
0-semigroup on a Hilbert spaceH, C:D(A)→Y a linear operator,Y another Hilbert space. In this paper, some sufficient and necessary conditions are obtained for the observation operatorC to be infinite-time admissible. For a control system Σ(A, B, −), due to duality argument, some sufficient and necessary conditions are also given for the control operatorB to be extended admissible. It is wellknown that observation operatorC is admissible if and only if the operator Lyapunov equation associated with the system has a nonnegative solution. In this paper, all nonnegative solutions to this equation are represented parametrically.
Metz1
TL;DR: In this paper, it was shown that the multiplication operator T_f: {g, g, mapsto f {g} is decomposable on H ∞ + C if and only if f is quasi-continuous.
Abstract: It is shown that the multiplication operator \(T_f: {g}\mapsto f {g}\) is decomposable on \(H^\infty +C\) if and only if f is quasi-continuous
01 Jan 1999
TL;DR: In this paper, the authors considered a complex third-order dierential operator on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems and proved that two spectra are sufficient to determine the operator.
Abstract: We consider a complex third-order dierential operator on a bounded interval with boundary conditions presenting a mixed aspect of the Dirichlet and the periodic problems. It is proved that two spectra are sucient to determine the operator. This result is valid under applying readily veriable hypotheses simultaneously to the two spectra.
TL;DR: In this article, the amplitude integral (AI) is used for solving inverse problems, and the convergence of the AI is discussed and the role of the diagonal in the problem of triangular factorization is elucidated.
Abstract: An operator integral, referred to as the amplitude integral (AI) and used in the BC-method (based on boundarycontrol theory) for solving inverse problems, is systematically studied. For a continuous operator and two families of increasing subspaces, the continual analog of the matrix diagonal in the form of an AI is introduced. The convergence of the AI is discussed. An example of an operator with no diagonal is provided. The role of the diagonal in the problem of triangular factorization is elucidated. The well-known result of matrix theory stating the uniqueness of triangular factorization with a prescribed diagonal is extended. It is shown that the corresponding factor can be represented in the AI form. The correspondence between the AI and the classical representation of the triangular factor of an operator that is a sum of the identity and a compact operator is established.
01 Sep 1999