Galerkin method for Wiener-Hopf operators with piecewise continuous symbol

This paper is concerned with the applicability of maximum defect polynomial (Galerkin) spline approximation methods with graded meshes to Wiener-Hopf operators with matrix-valued piecewise continuous generating function defined on R. For this, an algebra of sequences is introduced, which contains the approximating sequences we are interested in. There is a direct relationship between the stability of the approximation method for a given operator and invertibility of the corresponding sequence in this algebra. Exploring this relationship, the methods of essentialization, localization and identification of the local algebras are used in order to derive stability criteria for the approximation sequences.Supported by grant Praxis XXI/BD/4501/94 from FCT.Partly supported by FCT/BMFT grant 423.     

Products of Bergman space Toeplitz operators on the polydisk

Motivated by recent works of Ahern and uković on the disk, we study the generalized zero product problem for Toeplitz operators acting on the Bergman space of the polydisk. First, we extend the results to the polydisk. Next, we study the generalized compact product problem. Our results are new even on the disk. As a consequence on higher dimensional polydisks, we show that the generalized zero and compact product properties are the same for Toeplitz operators in a certain case.The first three authors were partially supported by KOSEF(R01-2003-000-10243-0) and the last author was partially supported by the National Science Foundation.     

The invertibility of the product of unbounded Toeplitz operators

In this note I give necessary and sufficient conditions on outer functionsf andg for the operator to be bounded and invertible on H². I also discuss the relationship of this question to two open questions in operator theory and weighted norm inequalities.     

On the Determinant of a Certain Wiener-Hopf + Hankel Operator

We establish an asymptotic formula for determinants of truncated Wiener-Hopf+Hankel operators with symbol equal to the exponential of a constant times the characteristic function of an interval. This is done by reducing it to the corresponding (known) asymptotics for truncated Toeplitz+Hankel operators. The determinants in question arise in random matrix theory in determining the limiting distribution for the number of eigenvalues in an interval for a scaled Laguerre ensemble of positive Hermitian matrices.     

Weighted norm inequalities for convolution operators and links with the Weiss Conjecture

We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section 3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section 4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture (for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness results for certain convolution operators.     

Wiener-Hopf operators with oscillating symbols and convolution operators on a union of intervals

The present work was sponsored by JNICT (Portugal) under grant n. 87422/MATM and Programa Ciência.     

On Toeplitz operators with quasihomogeneous symbols

In this paper, we give some basic results concerning Toeplitz operators whose symbol is of the form e^{i p θ}ϕ, where ϕ is a radial function, then use these results to characterize all Toeplitz operators which commute with them.Received: 12 June 2004; revised: 27 January 2005     

Finiteness of the discrete spectrum of some block Toeplitz operators

Sufficient conditions are given for the finiteness of the discrete spectrum of the block Toeplitz operatorT _A generated in the spaceH ₂ ⁿ by self-adjoint matrix functionA(t)(|t|=1). These results are obtained by means of theorems concerning the spectrum of a perturbed self-adjoint operators.     

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     

An index theorem for Wiener-Hopf operators

We study the multivariate generalisation of the classical Wiener-Hopf algebra, which is the C^∗-algebra generated by the Wiener-Hopf operators, given by convolutions restricted to convex cones. By the work of Muhly and Renault, this C^∗-algebra is known to be isomorphic to the reduced C^∗-algebra of a certain restricted action groupoid. It admits a composition series, and therefore, a ‘symbol’ calculus. Using groupoid methods, we obtain, in the framework of Kasparov's bivariant KK-theory, a topological expression of the index maps associated to these symbol maps in terms of geometric-topological data of the underlying convex cone. This generalises an index theorem by Upmeier concerning Wiener-Hopf operators on symmetric cones. Our result covers a wide class of cones containing polyhedral and homogeneous cones.     

Pathological hypercyclic operators

We exhibit a hypercyclic operator whose square is not hypercyclic. Our operator is necessarily unbounded since a result of S. Ansari asserts that powers of a hypercyclic bounded operator are also hypercyclic. We also exhibit an unbounded Hilbert space operator whose non-zero vectors are hypercyclic. Received: 19 March 2005; revised: 18 July 2005     

Polynomial sequences of bounded operators

The basic results of spectral theory are obtained using the sequence of powers of a bounded linear operator T,T²,…,Tⁿ,…. In this paper, we replace the powers Tⁿ by certain polynomials p_n(T), and make use of special properties of the polynomial sequence to derive some new results concerning operators. For example, using an arbitrary polynomial sequence , we obtain “binomial” spectral radii and semidistances, which reduce, in the case of the sequence of powers, to the usual spectral radius and semidistance.     

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.     

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.     

Toeplitz and singular integral operators on Carleson curves with logarithmic whirl points

We consider Toeplitz operators with piecewise continuous symbols and singular integral operators with piecewise continuous coefficients onL ^p (,w) where 1<p<,w is a Muckenhoupt weight and belongs to a large class of Carleson curves. This class includes curves with corners and cusps as well as curves that look locally like two logarithmic spirals scrolling up at the same point. Our main result says that the essential spectrum of a Toeplitz operator is obtained from the essential range of its symbol by joining the endpoints of each jump by a certain spiralic horn, which may degenerate to a usual horn, a logarithmic spiral, a circular arc or a line segment if the curve and the weightw behave sufficiently well at the point where the symbol has a jump. This result implies a symbol calculus for the closed algebra of singular integral operators with piecewise continuous coefficients onL ^p (,w).Research supported by the Alfried Krupp Förderpreis für junge Hochschullehrer of the Krupp Foundation.     

Semi-commutators of Toeplitz operators on the Bergman space

In this paper several necessary and sufficient conditions are obtained for the semi-commutator of Toeplitz operators andT _g with bounded pluriharmonic symbols on the unit ball to be compact on the Bergman space. Using -harmonic function theory on the unit ball we show that with bounded pluriharmonic symbolsf andg is zero on the Bergman space of the unit ball or the Hardy space of the unit sphere if and only if eitherf org is holomorphic.The author was supported in part by the National Science Foundation.