On residualities in the set of Markov operators on

We show that the set of those Markov operators on the Schatten class such that , where is one-dimensional projection, is norm open and dense. If we require that the limit projections must be on strictly positive states, then such operators form a norm dense . Surprisingly, for the strong operator topology operators the situation is quite the opposite.

     

The index of a critical point for densely defined operators of type in Banach spaces

The purpose of this paper is to demonstrate that it is possible to define and compute the index of an isolated critical point for densely defined operators of type acting from a real, reflexive and separable Banach space into This index is defined via a degree theory for such operators which has been recently developed by the authors. The calculation of the index is achieved by the introduction of a special linearization of the nonlinear operator at the critical point. This linearization is a new tool even for continuous everywhere defined operators which are not necessarily Fréchet differentiable. Various cases of operators are considered: unbounded nonlinear operators with unbounded linearization, bounded nonlinear operators with bounded linearization, and operators in Hilbert spaces. Examples and counterexamples are given in 2,$"> illustrating the main results. The associated bifurcation problem for a pair of operators is also considered. The main results of the paper are substantial extensions and improvements of the classical results of Leray and Schauder (for continuous operators of Leray-Schauder type) as well as the results of Skrypnik (for bounded demicontinuous mappings of type Applications to nonlinear Dirichlet problems have appeared elsewhere.

     

Poisson integrals associated to Dunkl operators for dihedral groups

In this paper we study the boundary behavior of Poisson integrals associated to Dunkl differential-difference operators for dihedral groups and the boundary integral representations for functions on the unit disc of annihilated by the Laplace operator corresponding to these differential-difference operators.

     

Data-sparse approximation to a class of operator-valued functions

In earlier papers we developed a method for the data-sparse approximation of the solution operators for elliptic, parabolic, and hyperbolic PDEs based on the Dunford-Cauchy representation to the operator-valued functions of interest combined with the hierarchical matrix approximation of the operator resolvents. In the present paper, we discuss how these techniques can be applied to approximate a hierarchy of the operator-valued functions generated by an elliptic operator .

     

Traces on algebras of parameter dependent pseudodifferential operators and the eta-invariant

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space . For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone , we construct a unique ``symbol valued trace', which extends the -trace on operators of small order. This construction is in the spirit of a trace due to Kontsevich and Vishik in the nonparametric case. Our trace allows us to construct various trace functionals in a systematic way. Furthermore, we study the higher-dimensional eta-invariants on algebras with parameter space . Using Clifford representations we construct for each first order elliptic differential operator a natural family of parametric pseudodifferential operators over . The eta-invariant of this family coincides with the spectral eta-invariant of the operator.

     

Continuity of the maximal operator in Sobolev spaces

We establish the continuity of the Hardy-Littlewood maximal operator on Sobolev spaces , . As an auxiliary tool we prove an explicit formula for the derivative of the maximal function.

     

Power bounded operators and supercyclic vectors II

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators.

For non-reflexive Banach spaces these results remain true; however, the non-supercyclic vector (invariant cone, respectively) relates to the adjoint of the operator.

     

The canonical solution operator to restricted to Bergman spaces

We first show that the canonical solution operator to restricted to -forms with holomorphic coefficients can be expressed by an integral operator using the Bergman kernel. This result is used to prove that in the case of the unit disc in the canonical solution operator to restricted to -forms with holomorphic coefficients is a Hilbert-Schmidt operator. In the sequel we give a direct proof of the last statement using orthonormal bases and show that in the case of the polydisc and the unit ball in 1,$"> the corresponding operator fails to be a Hilbert-Schmidt operator. We also indicate a connection with the theory of Hankel operators.

     

Spectral theory and hypercyclic subspaces

A vector in a Hilbert space is called hypercyclic for a bounded operator if the orbit is dense in . Our main result states that if satisfies the Hypercyclicity Criterion and the essential spectrum intersects the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for . The converse is true even if is a hypercyclic operator which does not satisfy the Hypercyclicity Criterion. As a consequence, other characterizations are obtained for an operator to have an infinite-dimensional closed subspace of hypercyclic vectors. These results apply to most of the hypercyclic operators that have appeared in the literature. In particular, they apply to bilateral and backward weighted shifts, perturbations of the identity by backward weighted shifts, multiplication operators and composition operators. The main result also applies to the differentiation operator and the translation operator defined on certain Hilbert spaces consisting of entire functions. We also obtain a spectral characterization of the norm-closure of the class of hypercyclic operators which have an infinite-dimensional closed subspace of hypercyclic vectors.

     

On injective or dense-range operators leaving a given chain of subspaces invariant

In this paper we prove the existence of dense-range or one-to-one compact operators on a separable Banach space leaving a given finite chain of subspaces invariant. We use this result to prove that a semigroup of bounded operators is reducible if and only if there exists an appropriate one-to-one compact operator such that the collection of compact operators is reducible.

     

Interpolation operators associated with sub-frame sets

Interpolation operators associated with wavelets sets introduced by Dai and Larson play an important role in their operator algebraic approach to wavelet theory. These operators are also related to the von Neumann subalgebras in the ``local commutant' space, which provides the parametrizations of wavelets. It is a particularly interesting question of how to construct operators which are in the local commutant but not in the commutant. Motivated by some questions about interpolation family and C*-algebras in the local commutant, we investigate the interpolation partial isometry operators induced by sub-frame sets. In particular we introduce the -congruence domain of the associated mapping between two sub-frame sets and use it to characterize these partial isometries in the local commutant. As an application, we obtain that if two wavelet sets have the same -congruence domain, then one is a multiresolution analysis (MRA) wavelet set which implies that the other is also an MRA wavelet set.

     

Anti-Wick quantization with symbols in spaces

We give a classification of pseudo-differential operators with anti-Wick symbols belonging to spaces: if the corresponding operator belongs to trace classes; if we get Hilbert-Schmidt operators; finally, if , the operator is compact. This classification cannot be improved, as shown by some examples.

     

The Berezin transform and radial operators

We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial symbol.

     

Continuity of the support of a state

Let be a finite von Neumann algebra and a projection. It is well known that the map which assigns its support projection to a positive normal functional of is not continuous. In this note it is shown that if one restricts to the set of positive normal functionals with support equivalent to a fixed , endowed with the norm topology, and the set of projections of is considered with the strong operator topology, then the support map is continuous. Moreover, it is shown that the support map defines a homotopy equivalence between these spaces. This fact together with previous work implies that, for example, the set of projections of the hyperfinite II factor, in the strong operator topology, has trivial homotopy groups of all orders .

     

Zeta forms and the local family index theorem

For a family of elliptic pseudodifferential operators we show there is a natural zeta-form and zeta-determinant form in the ring of smooth differential forms on the parameterizing manifold, generalizing the classical single operator zeta-function and zeta-determinant. We show that the zeta forms extend the Atiyah-Bott-Seeley formula for the index of an elliptic operator to a family of elliptic operators, while the zeta-determinant form leads to a graded Chern class form for the index bundle. Globally, the zeta-form and zeta-determinant form exist only at the level of -theory as maps to cohomology.

     

Spectral radii and eigenvalues of subdivision operators

This paper discusses the spectra of matrix subdivision operators. We establish some formulas for spectral radii of subdivision operators on various invariant subspaces in . A formula for the spectral radius of a subdivision operator, in terms of the moduli of eigenvalues, is derived under a mild condition. The results are even new in the scalar case. In this case, we show that the subdivision operator has no eigenvector in if the corresponding subdivision scheme converges for some .

     

On the commutant and orbits of conjugation

In this note, we analyse the relationship between the commutant of a bounded linear operator and the algebra of similarity that was introduced in the late 70s as a characterization of nest algebras. Necessary and sufficient conditions are also obtained for an operator to commute with real scalar generalized operators in the sense of Colojoara-Foias in Banach spaces. In the second part, we analyse the relationship between the generalized inverse, the generalized commutant and the orbits of conjugation.

     

Regularity of operators on essential extensions of the compacts

A semiregular operator on a Hilbert -module, or equivalently, on the -algebra of `compact' operators on it, is a closable densely defined operator whose adjoint is also densely defined. It is shown that for operators on extensions of compacts by unital or abelian -algebras, semiregularity leads to regularity. Two examples coming from quantum groups are discussed.

     

On area integral estimates for solutions to parabolic systems in time-varying and non-smooth cylinders

In this paper we prove results relating the (parabolic) non-tangential maximum operator and appropriate square functions in for solutions to general second order, symmetric and strongly elliptic parabolic systems with real valued and constant coefficients in the setting of a class of time-varying, non-smooth infinite cylinders . In particular we prove a global as well as a local and scale invariant equivalence between the parabolic non-tangential maximal operator and appropriate square functions for solutions of our system. The novelty of our approach is that it is not based on singular integrals, the prevailing tool in the analysis of systems in non-smooth domains. Instead the methods explored have recently proved useful in the analysis of elliptic measure associated to non-symmetric operators through the work of Kenig-Koch-Pipher-Toro and in the analysis of caloric measure without the use of layer potentials.