Hankel operators on the Dirichlet space

We study (small) Hankel operators on the Dirichlet space D with symbols in a class of function space, and show that such (small) Hankel operators are closely related to the corresponding Hankel operators on the Bergman space and the Hardy space H².     

Sub-Bergman Hilbert spaces in the unit disk, II

For a finite Blaschke product B let T_B denote the analytic multiplication operator (also called a Toeplitz operator) on the Bergman space of the unit disk. We show that the defect operators and both map the Bergman space to the Hardy space and the Hardy space to the Dirichlet space.     

Hopf-Rinow theorem in the Sato Grassmannian

Let U₂(H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit

     

Hankel operators and the Stieltjes moment problem

Let s be a non-vanishing Stieltjes moment sequence and let μ be a representing measure of it. We denote by μn the image measure in Cn of μ⊗σn under the map , where σn is the rotation invariant probability measure on the unit sphere. We show that the closure of holomorphic polynomials in L²(μn) is a reproducing kernel Hilbert space of analytic functions and describe various spectral properties of the corresponding Hankel operators with anti-holomorphic symbols. In particular, if n=1, we prove that there are nontrivial Hilbert-Schmidt Hankel operators with anti-holomorphic symbols if and only if s is exponentially bounded. In this case, the space of symbols of such operators is shown to be the classical Dirichlet space. We mention that the classical weighted Bergman spaces, the Hardy space and Fock type spaces fall in this setting.     

Ideals of operators and the metric approximation property

We prove that a Banach space X has the metric approximation property if and only if , the space of all finite rank operators, is an ideal in , the space of all bounded operators, for every Banach space Y. Moreover, X has the shrinking metric approximation property if and only if is an ideal in for every Banach space Y.Similar results are obtained for u-ideals and the corresponding unconditional metric approximation properties.     

Trace formula on the p-adic upper half-plane

This article aims at showing a p-adic analogue of Selberg's trace formula, which describes a duality between the spectrum of a Hilbert-Schmidt operator and the length of prime geodesics appearing in the p-adic upper half-plane associated with a hyperbolic discontinuous subgroup of . Then we construct Markov processes on the fundamental domain relative to such subgroups, to whose transition operators the trace formula applied and a p-adic analogue of prime geodesic theorem is proved.     

Eigenvalue asymptotics for Sturm-Liouville operators with singular potentials

We derive eigenvalue asymptotics for Sturm-Liouville operators with singular complex-valued potentials from the space , α∈[0,1], and Dirichlet or Neumann-Dirichlet boundary conditions. We also give application of the obtained results to the inverse spectral problem of recovering the potential from these two spectra.     

Hardy spaces that support no compact composition operators

We consider, for G a simply connected domain and 0<p<∞, the Hardy space formed by fixing a Riemann map τ of the unit disc onto G, and demanding of functions F holomorphic on G that the integrals of |F|^p over the curves τ({|z|=r}) be bounded for 0<r<1. The resulting space is usually not the one obtained from the classical Hardy space of the unit disc by conformal mapping. This is reflected in our Main Theorem: supports compact composition operators if and only if∂Ghas finite one-dimensional Hausdorff measure. Our work is inspired by an earlier result of Matache (Proc. Amer. Math. Soc. 127 (1999) 1483), who showed that the spaces of half-planes support no compact composition operators. Our methods provide a lower bound for the essential spectral radius which shows that the same result holds with “compact” replaced by “Riesz.” We prove similar results for Bergman spaces, with the Hardy-space condition “∂G has finite Hausdorff 1-measure” replaced by “G has finite area.” Finally, we characterize those domains G for which every composition operator on either the Hardy or the Bergman spaces is bounded.     

On the Dirichlet semigroup for Ornstein-Uhlenbeck operators in subsets of Hilbert spaces

We consider a family of self-adjoint Ornstein-Uhlenbeck operators Lα in an infinite dimensional Hilbert space H having the same gaussian invariant measure μ for all α∈[0,1]. We study the Dirichlet problem for the equation λφ−Lαφ=f in a closed set K, with f∈L²(K,μ). We first prove that the variational solution, trivially provided by the Lax-Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution φ (which is by definition in a Sobolev space ) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to . In the second case we exploit the Malliavin's theory of surface integrals which is recalled in Appendix A of the paper, then we are able to give a meaning to the trace of φ at ∂K and to show that it vanishes, as it is natural.     

The role of the angle in supercyclic behavior

A bounded operator T acting on a Hilbert space is said to be supercyclic if there is a vector such that the projective orbit and is dense in . We use a new method based on a very simple geometric idea that allows us to decide whether an operator is supercyclic or not. The method is applied to obtain the following result: A composition operator acting on the Hardy space whose inducing symbol is a parabolic linear-fractional map of the disk onto a proper subdisk is not supercyclic. This result finishes the characterization of the supercyclic behavior of composition operators induced by linear fractional maps and, thus, completes previous work of Bourdon and Shapiro.     

Selfadjoint operators in S-spaces

We study S-spaces and operators therein. An S-space is a Hilbert space with an additional inner product given by , where U is a unitary operator in . We investigate spectral properties of selfadjoint operators in S-spaces. We show that their spectrum is symmetric with respect to the real axis. As a main result we prove that for each selfadjoint operator A in an S-space we find an inner product which turns S into a Krein space and A into a selfadjoint operator therein. As a consequence we get a new simple condition for the existence of invariant subspaces of selfadjoint operators in Krein spaces, which provides a different insight into this well-know and in general unsolved problem.     

A Besov class functional calculus for bounded holomorphic semigroups

It is well-known that -sectorial operators generally do not admit a bounded H^∞ calculus over the right half-plane. In contrast to this, we prove that the H^∞ calculus is bounded over any class of functions whose Fourier spectrum is contained in some interval [ε,σ] with 0<ε<σ<∞. The constant bounding this calculus grows as as and this growth is sharp over all Banach space operators of the class under consideration. It follows from these estimates that -sectorial operators admit a bounded calculus over the Besov algebra of the right half-plane. We also discuss the link between -sectorial operators and bounded Tadmor-Ritt operators.     

Compact differences of weighted composition operators from weighted Bergman spaces to weighted-type spaces

We characterize the compactness of differences of weighted composition operators from the weighted Bergman space , 0 < p < ∞, α > −1, to the weighted-type space of analytic functions on the open unit disk D in terms of inducing symbols and . For the case 1 < p < ∞ we find an asymptotically equivalent expression to the essential norm of these operators.     

A note on the boundedness of Calderón-Zygmund operators on Hardy spaces

It was well known that Calderón-Zygmund operators T are bounded on Hp for provided T^∗(1)=0. A new Hardy space , where b is a para-accretive function, was introduced in [Y. Han, M. Lee, C. Lin, Hardy spaces and the Tb-theorem, J. Geom. Anal. 14 (2004) 291-318] and the authors proved that Calderón-Zygmund operators T are bounded from the classical Hardy space Hp to the new Hardy space if T^∗(b)=0. In this note, we give a simple and direct proof of the boundedness of Calderón-Zygmund operators via the vector-valued singular integral operator theory.     

COMPOSITION OPERATORS BETWEEN SOME WEIGHTED HARDY SPACES

Let D be the unit disc and H(D) be the set of all analytic functions on D. In [2], C. Cowen defined a space H = f ∈ H(D) : f(z) =sum from k=o to ∞ ak(z + 1)k, z∈ D, ‖f‖2 = sum from k=o to ∞ |ak|24k < ∞In this article, the authors consider the similar Hardy spaces with arbitrary weights and discuss some properties of them. Boundedness and compactness of composition operators between such spaces are also studied.     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

On closability of classical Dirichlet forms

We construct a measure μ on such that the classical Dirichlet form is closable, but the partial Dirichlet form is not. This proves the well-known conjecture of M. Röckner.     

Tensor products of holomorphic representations and bilinear differential operators

Let be the weighted Bergman space on a bounded symmetric domain D=G/K. It has analytic continuation in the weight ν and for ν in the so-called Wallach set still forms unitary irreducible (projective) representations of G. We give the irreducible decomposition of the tensor product of the representations for any two unitary weights ν and we find the highest weight vectors of the irreducible components. We find also certain bilinear differential intertwining operators realizing the decomposition, and they generalize the classical transvectants in invariant theory of . As applications, we find a generalization of the Bol's lemma and we characterize the multiplication operators by the coordinate functions on the quotient space of the tensor product modulo the subspace of functions vanishing of certain degree on the diagonal.     

Trace-preserving homomorphisms of semigroups

We consider those homomorphisms φ of semigroups of trace-class operators on a Hilbert space that preserve trace. If φ is a spatially induced isomorphism on a semigroup , that is φ(S)T=TS for an invertible operator T and for all S in , then φ clearly has this property. More generally, if T in the relation above is a densely defined, closed, injective operator with dense image, φ still preserves trace. We prove the converse of this statement under certain conditions. Using these results we prove simultaneous similarity theorems for semigroups of operators (on finite or infinite-dimensional spaces) whose members are individually similar to unitary or J-unitary operators.