Composition operators on <Emphasis Type="Italic">f</Emphasis>-algebras

By a composition operator between two f-algebras we mean a positive algebra homomorphism. This paper intends to give a systematic study of such operators. A particular attention is paid to their connection with separating regular operators as well as to their global behavior in the module of regular operators. The paper ends with some open problems.     

On a class of Szász-Mirakyan type operators

The actual construction of the Szász-Mirakyan operators and its various modifications require estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. Thus the question arises whether the Szász-Mirakyan operators and their generalizations cannot be replaced by a finite sum. In connection with this question we propose a new family of linear positive operators.     

Singular integrals and commutators on homogeneous groups

Let G be a homogeneous group. In this paper, the authors establish several general theorems for the boundedness of sublinear operators and commutators generated by linear operators and BMO(G)functions on the weighted Lebesgue space on G. The conditions of these theorems are satisfied by many important operators in analysis and these operators satisfy only some weak conditions on the size of operators and are known to be bounded in the unweighted case. Some of these theorems are best possible even when G is the Euclidean space. The authors also give some applications of their theorems to the boundedness on weighted spaces of rough singular integrals, oscillatory integrals, parabolic singular integrals, their commutators and the maximal operators associated with them.     

Unbounded pseudodifferential calculus on Lie groupoids

We develop an abstract theory of unbounded longitudinal pseudodifferential calculus on smooth groupoids (also called Lie groupoids) with compact basis. We analyze these operators as unbounded operators acting on Hilbert modules over C^∗(G), and we show in particular that elliptic operators are regular. We construct a scale of Sobolev modules which are the abstract analogues of the ordinary Sobolev spaces, and analyze their properties. Furthermore, we show that complex powers of positive elliptic pseudodifferential operators are still pseudodifferential operators in a generalized sense.     

Volterra type integral operators with homogeneous kernels in weighted <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-spaces

We consider multidimensional integral Volterra type operators with kernels homogeneous of degree (?n); the operators act in L _p -spaces with a submultiplicative weight. For these operators we obtain necessary and sufficient conditions of their invertibility. Besides, we describe the Banach algebra generated by the operators. For this algebra we construct the symbolic calculus, in terms of which we obtain an invertibility criterion of the operators.     

Conditional expectations on Riesz spaces

Conditional expectations operators acting on Riesz spaces are shown to commute with a class of principal band projections. Using the above commutation property, conditional expectation operators on Riesz spaces are shown to be averaging operators. Here the theory of f-algebras is used when defining multiplication on the Riesz spaces. This leads to the extension of these conditional expectation operators to their so-called natural domains, i.e., maximal domains for which the operators are both averaging operators and conditional expectations. The natural domain is in many aspects analogous to L¹.     

Rough oscillatory singular integral operators of nonconvolution type

In this paper, we study a classes of oscillatory singular integral operators of nonconvolution type with phases more general than polynomials. We prove that such operators are bounded on Lp provided their kernels satisfy a very weak condition. In addition, we also study the related truncated oscillatory singular integral operators. Moreover, we present a class of unbounded oscillatory singular integral operators.     

Isometries, Fock spaces, and spectral analysis of Schrödinger operators on trees

We construct conjugate operators for the real part of a completely non-unitary isometry and we give applications to the spectral and scattering theory of a class of operators on (complete) Fock spaces, natural generalizations of the Schrödinger operators on trees. We consider C^*-algebras generated by such Hamiltonians with certain types of anisotropy at infinity, we compute their quotient with respect to the ideal of compact operators, and give formulas for the essential spectrum of these Hamiltonians.     

Mean2-Bounded Operators on Hilbert Space and Weight Sequences of Positive Operators

We introduce and study the class of mean₂-bounded operators on Hilbert space. These operators, which are characterized by a uniform boundedness condition on the 'quadratic averages' of their powers, are intimately related to operator-valued weight sequences and weighted norm inequalities. Mean₂-bounded operators exhibit a lively interplay with the discrete Hilbert transform in an array of settings. Under appropriate circumstances, mean₂-bounded operators have a rich spectral theory leading to powerful transference properties.     

Reflection symmetry and symmetrizability of Hilbert space operators

A general factorization theorem for symmetrizable operators relating their spectra to spectra of selfadjoint operators induced by minimal factorizations is established. Its modified version essentially improves and completes a theorem of Jorgensen, which concerns diagonalizing operators with reflection symmetry.

     

Idempotent Block Splitting on Partial Partitions,I: Isotone Operators

Image segmentation algorithms can be modelled as image-guided operators (maps) on the complete lattice of partitions of space, or on the one of partial partitions (i.e., partitions of subsets of the space). In particular region-splitting segmentation algorithms correspond to block splitting operators on the lattice of partial partitions, in other words anti-extensive operators that act by splitting each block independently. This first paper studies in detail block splitting operators and their lattice-theoretical and monoid properties; in particular we consider their idempotence (a requirement in image segmentation). We characterize block splitting openings (kernel operators) as operators splitting each block into its connected components according to a partial connection; furthermore, block splitting openings constitute a complete sublattice of the complete lattice of all openings on partial partitions. Our results underlie the connective approach to image segmentation introduced by Serra. The second paper will study two classes of non-isotone idempotent block splitting operators, that are also relevant to image segmentation.     

Wavelets from Laguerre Polynomials and Toeplitz-Type Operators

We study a parameterized family of Toeplitz-type operators with respect to specific wavelets whose Fourier transforms are related to Laguerre polynomials. On the one hand, this choice of wavelets underlines the fact that these operators acting on wavelet subspaces share many properties with the classical Toeplitz operators acting on the Bergman spaces. On the other hand, it enables to study poly-Bergman spaces and Toeplitz operators acting on them from a different perspective. Restricting to symbols depending only on vertical variable in the upper half-plane of the complex plane these operators are unitarily equivalent to a multiplication operator with a certain function. Since this function is responsible for many interesting features of these Toeplitz-type operators and their algebras, we investigate its behavior in more detail. As a by-product we obtain an interesting observation about the asymptotic behavior of true poly-analytic Bergman spaces. Isomorphisms between the Calderón-Toeplitz operator algebras and functional algebras are described and their consequences in time-frequency analysis and applications are discussed.     

Resonant Delocalization on the Bethe Strip

Recently, Aizenman and Warzel discovered a mechanism for the appearance of absolutely continuous spectrum for random Schrödinger operators on the Bethe lattice through rare resonances (resonant delocalization). We extend their analysis to operators with matrix-valued random potentials drawn from ensembles such as the Gaussian Orthogonal Ensemble. These operators can be viewed as random operators on the Bethe strip, a graph (lattice) with loops.     

Problèmes de petites valeurs propres sur les surfaces de courbure moyenne constante

This paper deals with the spectra of the Laplace and stability operators of a constant mean curvature surface in the hyperbolic space. In a preceding work, the author described the essential spectra of these operators, assuming that the surface is of finite total curvature. In this paper, we prove that these two operators have a finite number of eigenvalues below their essential spectra.

     

Extension of Some Generalized Hermite-Type Interpolation Operators to the Triangle with One Curved Side

Abstract

We construct some generalized Hermite-type interpolation operators for the case of a triangle with one curved side, and we consider their product and Boolean sum operators. We study the interpolation properties of these operators, the orders of accuracy and the remainders of the interpolation formulas. Finally, we give some numerical examples.     

r-ASYMPTOTICALLY QUASI-FINITE RANK OPERATORS AND THE SPECTRUM OF MEASURES

Abstract

The r-asymptotically quasi finite rank operators were introduced in [10]. For regular operators on Banach lattices, these operators are the order theoretic analogue of Riesz operators on Banach spaces. We establish their basic properties and apply these in the spectral analysis of convolution operators.     

Large time behavior for discontinuous dynamics on Hilbert spaces

This paper is concerned with the behavior in time for a certain class of dynamics which are discontinuous with respect to the time variable. We introduce the corresponding wave operators and we ensure their existence. Moreover, under suitable conditions this class of wave operators can be approximated in the strong sense by a sequence of ordinary wave operators. Our results can be applied to impulsive dynamical systems.

     

Hermitian Weighted Composition Operators and Bergman Extremal Functions

Weighted composition operators have been related to products of composition operators and their adjoints and to isometries of Hardy spaces. In this paper, Hermitian weighted composition operators on weighted Hardy spaces of the unit disk are studied. In particular, necessary conditions are provided for a weighted composition operator to be Hermitian on such spaces. On weighted Hardy spaces for which the kernel functions are ${(1 - \overline{w}z)^{-\kappa}}$ for κ ≥ 1, including the standard weight Bergman spaces, the Hermitian weighted composition operators are explicitly identified and their spectra and spectral decompositions are described. Some of these Hermitian operators are part of a family of closely related normal weighted composition operators. In addition, as a consequence of the properties of weighted composition operators, we compute the extremal functions for the subspaces associated with the usual atomic inner functions for these weighted Bergman spaces and we also get explicit formulas for the projections of the kernel functions on these subspaces.     

Dixmier classes on generalized Segal–Bargmann–Fock spaces

We give criteria for the membership of Toeplitz operators and products of Hankel operators, with symbols of a certain type, in the Dixmier class, and formulas for their Dixmier trace, on a variety of weighted Segal–Bargmann–Fock spaces on the complex plane.     

Complex symplectic geometry with applications to ordinary differential operators

Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems.