Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Time-Frequency Analysis of Pseudodifferential Operators

 In this paper we apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn–Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product of two symbols σ and τ, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of from the time-frequency content of σ and τ. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol . (Received 27 December 1999; in final form 9 November 2000)     

Time-Frequency Analysis of Pseudodifferential Operators

 In this paper we apply a time-frequency approach to the study of pseudodifferential operators. Both the Weyl and the Kohn–Nirenberg correspondences are considered. In order to quantify the time-frequency content of a function or distribution, we use certain function spaces called modulation spaces. We deduce a time-frequency characterization of the twisted product of two symbols σ and τ, and we show that modulation spaces provide the natural setting to exactly control the time-frequency content of from the time-frequency content of σ and τ. As a consequence, we discuss some boundedness and spectral properties of the corresponding operator with symbol .     

Locally stationary stochastic processes and Weyl symbols of positive operators

The paper treats locally stationary stochastic processes. A connection with the Weyl symbols of positive operators is observed and explored. We derive necessary conditions on the two functions that constitute the covariance function of a locally stationary stochastic process, some of which use this connection to time-frequency analysis and pseudodifferential operators. Finally, we discuss briefly the subclass of Cohen’s class of time–frequency representations having separable kernels, which is related to locally stationary stochastic processes.     

Bilinear oscillatory integrals and boundedness for new bilinear multipliers

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.     

Time-frequency partitions and characterizations of modulation spaces with localization operators

We study families of time-frequency localization operators and derive a new characterization of modulation spaces. This characterization relates the size of the localization operators to the global time-frequency distribution. As a by-product, we obtain a new proof for the existence of multi-window Gabor frames and extend the structure theory of Gabor frames.     

Time-Frequency analysis of localization operators

We study a class of pseudodifferential operators known as time-frequency localization operators, Anti-Wick operators, Gabor-Toeplitz operators or wave packets. Given a symbol a and two windows ?₁,?₂, we investigate the multilinear mapping from to the localization operator A_a^?₁,?₂ and we give sufficient and necessary conditions for A_a^?₁,?₂ to be bounded or to belong to a Schatten class. Our results are formulated in terms of time-frequency analysis, in particular we use modulation spaces as appropriate classes for symbols and windows.     

On Toeplitz localization operators

We present a unified approach to study properties of Toeplitz localization operators based on the Calderón and Gabor reproducing formula. We show that these operators with functional symbols on a plane domain may be viewed as certain pseudo-differential operators (with symbols on a line, or certain compound symbols).     

单位多圆柱上Bergman空间中的分别准齐次ToePlitz算子

本文研究了单位多圆柱上Bergman空间中以分别准齐次函数为记号的Toeplitz算子的代数性质.我们首先得到了两个以分别准齐次函数为记号的Toeplitz算子可以写成一个Toeplitz算子的充分必要条件,然后利用L2(Dn,dV)的一个极分解式证明了,只要其中有一个Toeplitz算子是分别准齐次的,则其零乘积问题...     

Toeplitz Operators and Group Representations

Toeplitz operators on the Bergman space of the unit disc can be written as integrals of the symbol against an invariant operator field of rank-one projections. We consider a generalization of the Toeplitz calculus obtained upon taking more general invariant operator fields, and also allowing more general domains than the disc; such calculi were recently introduced and studied by Arazy and Upmeier, but also turn up as localization operators in time-frequency analysis (witnessed by recent articles by Wong and others) and in representation theory and mathematical physics. In particular, we establish basic properties like boundedness or Schatten class membership of the resulting operators. A further generalization to the setting when there is no group action present is also discussed, and the various settings in which similar operator calculi appear are briefly surveyed.     

Commuting dual Toeplitz operators on the orthogonal complement of the Dirichlet space

In this paper we characterize commuting dual Toeplitz operators with harmonic symbols on the orthogonal complement of the Dirichlet space in the Sobolev space. We also obtain the sufficient and necessary conditions for the product of two dual Toeplitz operators with harmonic symbols to be a finite rank perturbation of a dual Toeplitz operator.     

Spline Approximation Methods with Uniform Meshes in Algebras of Multiplication and Convolution Operators

The purpose of this paper is to obtain necessary and sufficient conditions for maximum defect spline approximation methods with uniform meshes to be stable. The methods are applied to operators belonging to the closed subalgebra of ℒ︁ (L² (ℝ)) generated by operators of multiplication by piecewise continuous functions on ℝ and convolution operators also with piecewise continuousgenerating functions. To that purpose, a C*-algebra of sequences is introduced, which contains the special sequences of approximating operators we are interested in. There is a direct relationship between the applicability of the approximation method to a given operator and invertibility of the corresponding sequence in this C*-algebra. Exploring this relationship, applicability criteria are derived by the use of C*-algebra and Banach algebra techniques (essentialization, localization andidentification of the local algebras by means of construction of locally equivalent representations). Finally, examples are presented, including explicit conditions for the applicability of spline Galerkin methods to Wiener-Hopf operators with piecewise continuous symbols.     

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.     

Commuting Toeplitz operators on the Dirichlet space

In this paper, we study the commutativity of Toeplitz operators with continuous symbols on the Dirichlet space. First, under a mild condition concerning absolute continuity we characterize (semi-)commuting Toeplitz operators. This is a generalization of the case of harmonic symbols. Also, if one of the symbol is radial or analytic, we get another characterization, which is different from the case on the Bergman space.     

On C*-Algebras of Toeplitz Operators on the Harmonic Bergman Space

Commutative algebras of Toeplitz operators acting on the Bergman space on the unit disk have been completely classified in terms of geometric properties of the symbol class. The question when two Toeplitz operators acting on the harmonic Bergman space commute is still open. In some papers, conditions on the symbols have been given in order to have commutativity of two Toeplitz operators. In this paper, we describe three different algebras of Toeplitz operators acting on the harmonic Bergman space: The C*-algebra generated by Toeplitz operators with radial symbols, in the elliptic case; the C*-algebra generated by Toeplitz operators with piecewise continuous symbols, in the parabolic and hyperbolic cases. We prove that the Calkin algebra of the first two algebras are commutative, like in the case of the Bergman space, while the last one is not.     

Fock空间上径向函数诱导的Toeplitz算子

本文讨论了Fock空间上以径向函数和拟齐次函数为符号的Toeplitz算子的代数性质,给出了两个以径向函数为符号的Toeplitz算子的积仍为Toeplitz算子的充分必要条件,并且研究了以拟齐次函数为符号的Toeplitz算子的交换性.     

Dirichlet空间上的Bergman型Toeplitz算子

本文给出了Dirichlet空间上以有界调和函数为符号的Bergman型Toeplitz算子是紧算子的充要条件.同时刻画了此类Bergman型Toeplitz算子在Dirichlet空间上的交换性.     

Convolution operators on a finite interval with periodic kernel-Fredholm property and invertibility

Finite interval convolution operators with periodic kernel-functions are studied from the point of view of Fredholm properties and invertibility. These operators are associated with Wiener-Hopf operators with matrix-valued symbols defined on a space of functions whose domain is a contour consisting of two parallel straight-lines. For the Fredholm study a Wiener-Hopf operator is considered on a space of functions defined on a contour composed of two closed curves having a common multiple point. Invertibility of the finite interval operator is studied for a subclass of symbols related to the problem of wave diffraction by a strip grating.The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

Finite-Dimensional Approximations of Operators in the Hilbert Spaces of Functions on Locally Compact Abelian Groups

A new approach to the approximation of operators in the Hilbert space of functions on a locally compact Abelian (LCA) group is developed. This approach is based on sampling the symbols of such operators. To choose the points for sampling, we use the approximations of LCA groups by finite groups, which were introduced and investigated by Gordon. In the case of the group R ⁿ , the constructed approximations include the finite-dimensional approximations of the coordinate and linear momentum operators, suggested by Schwinger. The finite-dimensional approximations of the Schrödinger operator based on Schwinger's approximations were considered by Digernes, Varadarajan, and Varadhan in Rev. Math. Phys. 6 (4) (1994), 621–648 where the convergence of eigenvectors and eigenvalues of the approximating operators to those of the Schrödinger operator was proved in the case of a positive potential increasing at infinity. Here this result is extended to the case of Schrödinger-type operators in the Hilbert space of functions on LCA groups. We consider the approximations of p-adic Schrödinger operators as an example. For the investigation of the constructed approximations, the methods of nonstandard analysis are used.     

Subnormality and composition operators on the Bergman space

Following the work of C. Cowen and T. Kriete on the Hardy space, we prove that under a regularity condition, all composition operators with a subnormal adjoint on A²(D) have linear fractional symbols of the form. Moreover, we show that all composition operators on the Bergman space having these symbols have a subnormal adjoint, with larger range for the parameterr than found in the Hardy space case.