Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic Banach space. The construction of this operator relies on the existence of transfinite -spreading models in the dual of the space.

     

Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

We prove that the norm of a weighted composition operator on the Hardy space of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard function spaces on the unit ball.

     

Composition operators with maximal norm on weighted Bergman spaces

We prove that any composition operator with maximal norm on one of the weighted Bergman spaces (in particular, on the space ) is induced by a disk automorphism or a map that fixes the origin. This result demonstrates a major difference between the weighted Bergman spaces and the Hardy space , where every inner function induces a composition operator with maximal norm.

     

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.

     

Weighted norm inequalities for integral operators

We consider a large class of positive integral operators acting on functions which are defined on a space of homogeneous type with a group structure. We show that any such operator has a discrete (dyadic) version which is always essentially equivalent in norm to the original operator. As an application, we study conditions of ``testing type,' like those initially introduced by E. Sawyer in relation to the Hardy-Littlewood maximal function, which determine when a positive integral operator satisfies two-weight weak-type or strong-type estimates. We show that in such a space it is possible to characterize these estimates by testing them only over ``cubes'. We also study some pointwise conditions which are sufficient for strong-type estimates and have applications to solvability of certain nonlinear equations.

     

Weyl's theorem for perturbations of paranormal operators

A bounded linear operator on a Banach space is said to satisfy ``Weyl's theorem' if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if is a paranormal operator on a Hilbert space, then satisfies Weyl's theorem for every algebraic operator which commutes with .

     

On wandering vector multipliers for unitary groups

A wandering vector multiplier is a unitary operator which maps the set of wandering vectors for a unitary system acting on a separable Hilbert space into itself. It is proved that the wandering vector multipliers for a unitary group form a group, which gives a positive answer for a problem of Han and Larson. Furthermore, non-abelian unitary groups of order 6 are considered. We prove that the wandering vector multipliers of such a unitary group can not generate . This negatively answers another of their problems.

     

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.

     

Equilibrium points of logarithmic potentials induced by positive charge distributions. I. Generalized de Bruijn-Springer relations

A notion of weighted multivariate majorization is defined as a preorder on sequences of vectors in Euclidean space induced by the Choquet ordering for atomic probability measures. We characterize this preorder both in terms of stochastic matrices and convex functions and use it to describe the distribution of equilibrium points of logarithmic potentials generated by discrete planar charge configurations. In the case of positive charges we prove that the equilibrium points satisfy weighted majorization relations and are uniquely determined by such relations. It is further shown that the Hausdorff geometry of the equilibrium points and the charged particles is controlled by the weighted standard deviation of the latter. By using finite-rank perturbations of compact normal Hilbert space operators we establish similar relations for infinite charge distributions. We also discuss a hierarchy of weighted de Bruijn-Springer relations and inertia laws, the existence of zeros of Borel series with positive -coefficients, and an operator version of the Clunie-Eremenko-Rossi conjecture.

     

The complete isomorphism class of an operator space

Suppose is an infinite-dimensional operator space and is a positive integer. We prove that for every there exists an operator space such that the formal identity map is a complete isomorphism, is an isometry, and . This provides a non-commutative counterpart to a recent result of W. Johnson and E. Odell.

     

A theorem on reflexive large rank operator spaces

If every nonzero operator in an -dimensional operator space has rank , then is reflexive.

     

The behavior of the heat operator on weighted Sobolev spaces

Denoting by the heat operator in , we investigate its properties as a bounded operator from one weighted Sobolev space to another. Our main result gives conditions on the weights under which is an injection, a surjection, or an isomorphism. We also describe the range and kernel of in all the cases. Our results are analogous to those obtained by R. C. McOwen for the Laplace operator in .

     

On self-adjointness of a Schrödinger operator on differential forms

Let be a complete Riemannian manifold and let denote the space of differential forms on . Let be the exterior differential operator and let be the Laplacian. We establish a sufficient condition for the Schrödinger operator (where the potential is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by I. Oleinik about self-adjointness of a Schrödinger operator which acts on the space of scalar valued functions.

     

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.

     

The backward shift on the space of Cauchy transforms

This note examines the subspaces of the space of Cauchy transforms of measures on the unit circle that are invariant under the backward shift operator . We examine this question when the space of Cauchy transforms is endowed with both the norm and weak topologies.

     

On the class of norm limits of nilpotents

It is known that every operator on a Hilbert space whose invariant subspace lattice is possibly is a norm-limit of a sequence of nilpotent operators. In this note we study properties of such approximating sequences.

     

Quasitriangular + small compact = strongly irreducible

Let be a bounded linear operator acting on a separable infinite dimensional Hilbert space. Let be a positive number. In this article, we prove that the perturbation of by a compact operator with can be strongly irreducible if is a quasitriangular operator with the spectrum connected. The Main Theorem of this article nearly answers the question below posed by D. A. Herrero.

Suppose that is a bounded linear operator acting on a separable infinite dimensional Hilbert space with connected. Let be given. Is there a compact operator with such that is strongly irreducible?

     

Characterization of chaotic order and its application to furuta inequality

In this note, we give a simple characterization of the chaotic order among positive invertible operators on a Hilbert space. As an application, we discuss Furuta's type operator inequality.

     

The backward shift on Dirichlet-type spaces

We study the backward shift operator on Hilbert spaces (for ) which are norm equivalent to the Dirichlet-type spaces . Although these operators are unitarily equivalent to the adjoints of the forward shift operator on certain weighted Bergman spaces, our approach is direct and completely independent of the standard Cauchy duality. We employ only the classical Hardy space theory and an elementary formula expressing the inner product on in terms of a weighted superposition of backward shifts.

     

Linear maps preserving the isomorphism class of lattices of invariant subspaces

Let be an -dimensional complex linear space and the algebra of all linear transformations on . We prove that every linear map on , which maps every operator into an operator with isomorphic lattice of invariant subspaces, is an inner automorphism or an inner antiautomorphism multiplied by a nonzero constant and additively perturbed by a scalar type operator. The same result holds if we replace the lattice of invariant subspaces by the lattice of hyperinvariant subspaces or the set of reducing subspaces. Some of these results are extended to linear transformations of finite-dimensional linear spaces over fields other than the complex numbers. We also characterize linear bijective maps on the algebra of linear bounded operators on an infinite-dimensional complex Hilbert space which have similar properties with respect to the lattice of all invariant subpaces (not necessarily closed).