Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some disintegration condition which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic leaves) between the endpoints of jumps.These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra generated by SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIO's from in terms of their symbols.     

Shift invariant spaces inL 2

If is a complex, separable Hilbert space, letL ² () denote theL ²-space of functions defined on the unit circle and having values in . The bilateral shift onL ²() is the operator (U f)()=f(). A Hilbert spaceH iscontractively contained in the Hilbert spaceK ifHK and the inclusion mapH–K is a contraction. We describe the structure of those Hilbert spaces, contractively contained inL ²(), that are carried into themselves contractively byU . We also do this for the subcase of those spaces which are carried into themselves unitarily byU .     

Atomic and molecular decompositions of anisotropic Besov spaces

In this work we develop the theory of weighted anisotropic Besov spaces associated with general expansive matrix dilations and doubling measures with the use of discrete wavelet transforms. This study extends the isotropic Littlewood- Paley methods of dyadic -transforms of Frazier and Jawerth [19, 21] to non-isotropic settings.Several results of isotropic theory of Besov spaces are recovered for weighted anisotropic Besov spaces. We show that these spaces are characterized by the magnitude of the -transforms in appropriate sequence spaces. We also prove boundedness of an anisotropic analogue of the class of almost diagonal operators and we obtain atomic and molecular decompositions of weighted anisotropic Besov spaces, thus extending isotropic results of Frazier and Jawerth [21].The author was partially supported by the NSF grant DMS-0441817.     

Generalization of the Newman-Shapiro isometry theorem and Toeplitz operators

We give a generalization of the Newman-Shapiro Isometry Theorem to the case of Hilbert space-valued entire functions, which are square-summable with respect to the Gaussian measure on ⁿ , together with some applications in the theory of Toeplitz operators with operator-valued symbols. The study of various properties (such as density of domains, cores, closedness and boundedness from below) of these operators in illustrated with many relevant examples.Research supported by KBN under grant no. 2 P03A 041 10.     

Multiplication operators on Hilbert spaces of analytic functions

Let be a Hilbert space of functions analytic on a plane domain such that for every in the functional of evaluation at is bounded. Assume further that contains the constants and admits multiplication by the independent variable z, M_z, as a bounded operator. We give sufficient conditions for M_z to be reflexive.Received: 17 February 2004     

Contractibility of the linear group in Banach spaces of measurable operators

We show contractibility to a point of the linear group for a wide class of symmetric spaces of measurable operators affiliated with several concrete non-atomic semifinite von Neumann algebras.Research supported by the Australian Research Council     

Smooth points in some spaces of bounded operators

We determine the smooth points of certain spaces of bounded operatorsL(X,Y), including the cases whereX andY arel ^p -orc ₀-direct sums of finite dimensional Banach spaces or subspaces of the latter enjoying the metric compact approximation property. We also remark that the operators not attaining their norm are nowhere dense inL(X,Y) wheneverK(X,Y) is anM-ideal inL(X,Y).     

Automorphisms on the Toeplitz Algebra with Piecewise Continuous Symbol on the Unit Ball

In this paper, we obtain a Fredholm index formula for Toeplitz operators whose symbols are certain piecewise continuous function matrices on the unit ball. Moreover, using this formula, we discuss the automorphisms on the corresponding Toeplitz algebra     

Interpolation problems,extensions of symmetric operators and reproducing kernel spaces II

By an oversight on the part of the authors this section was not included in the paper previously published in Integral Equations Operator Theory, volume 14/4 (1991), 466–500. Present address:Department of Mathematics Ben-Gurion University of the Negev Beersheva Israel     

Spectral measures,II: Characterization of scalar operators

We continue the development of part I. The Riesz representation theorem is proved without assuming local convexity. This theorem is applied to give sufficient conditions for an operator (continuous or otherwise) to be spectral. A uniqueness problem is pointed out and the function calculus is extended to the case of several variables. A Radon—Nikodym theorem is proved.     

Convergence of invariant measures for singular stochastic diffusion equations

It is proved that the solutions to the singular stochastic p$p$-Laplace equation, p∈(1,2)$p\in (1,2)$ and the solutions to the stochastic fast diffusion equation with nonlinearity parameter r∈(0,1)$r\in (0,1)$ on a bounded open domain Λ⊂R^d$\Lambda \subset {\mathbb{R}}^d$ with Dirichlet boundary conditions are continuous in mean, uniformly in time, with respect to the parameters p$p$ and r$r$ respectively (in the Hilbert spaces L²(Λ)$L^2\left(\Lambda \right)$, H⁻¹(Λ)$H^{-1}\left(\Lambda \right)$ respectively). The highly singular limit case p=1$p=1$ is treated with the help of stochastic evolution variational inequalities, where P$\mathbb{P}$-a.s. convergence, uniformly in time, is established.     

Lifting strong commutants of unbounded subnormal operators

Various theorems on lifting strong commutants of unbounded subnormal (as well as formally subnormal) operators are proved. It is shown that the strong symmetric commutant of a closed symmetric operatorS lifts to the strong commutant of some tight selfadjoint extension ofS. Strong symmetric commutants of orthogonal sums of subnormal operators are investigated. Examples of (unbounded) irreducible subnormals, pure subnormals with rich strong symmetric commutants and cyclic subnormals with highly nontrivial strong commutants are discussed.This work was supported by the KBN grant # 2P03A 041 10.     

Almost-invertible Toeplitz operators and K-theory

We consider the question of when a Toeplitz operator with continuous symbol on a connected compact abelian group is almost invertible, and show that this occurs precisely when the symbol is invertible and has zero topological index. The proof uses someK-theory computations.     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

Hankel operators on the Bergman spaces of strongly pseudoconvex domains

We characterize functions fL ²(D) such that the Hankel operators H_f are, respectively, bounded and compact on the Bergman spaces of bounded strongly pseudoconvex domains.Research partially supported by a grant of the National Science Foundation.     

RUC-decompositions in symmetric operator spaces

We show that ifX is a Banach space of type 2 andG is a compact Abelian group, then any system of eigenvectors {x }_G (with respect to a strongly continuous representation ofG onX) is an RUC-system. As an application, we exhibit new examples of RUC-bases in certain symmetric spaces of measurable operators.Research supported by the Australian Research Council     

Spectral stability of general non-negative self-adjoint operators with applications to Neumann-type operators

We prove a stability theorem for the eigenvalues of general non-negative self-adjoint linear operators with compact resolvents and by applying it we prove a sharp stability result for the dependence of the eigenvalues of second order uniformly elliptic linear operators with homogeneous Neumann boundary conditions upon domain perturbation.     

Fredholm weighted composition operators

We characterize the Fredholm weighted composition operators onC(X). In particular, ifX is a set with some regular property like intervals or balls inR ⁿ , our characterization implies that a weighted composition operator is Fredholm if and only if it is invertible. This equivalence is true for weighted composition operators onL ^p (), where is a nonatomic measure (1p<).