Norm closure and extension of the symbolic calculus for the cone algebra

We investigate the symbolic structure of an algebra of pseudodifferential operators on manifolds with conical singularities which has been introduced by B.-W. Schulze. Our main objective is the extension of the symbolic calculus of this algebra to its norm closure in an adapted scale of Sobolev spaces. This procedure yields Banach algebras and Fréchet algebras of singular integral operators with continuous principal symbols.     

On Fredholm Operators Between Non-archimedean Fréchet Spaces

It is proved that the index of a Fredholm operator between non-Archimedean Fréchet spaces is preserved under compact perturbations. A similar result is shown for Fredholm operators between non-Archimedean polar regular LF-spaces.     

A short proof of existence of disjoint hypercyclic operators

We give a short proof of existence of disjoint hypercyclic tuples of operators of any given length on any separable infinite dimensional Fréchet space. Similar argument provides disjoint dual hypercyclic tuples of operators of any length on any infinite dimensional Banach space with separable dual.     

On some hyperbolic type equations and weighted anisotropic Hardy operators

We introduce a version of weighted anisotropic Morrey spaces and anisotropic Hardy operators. We find conditions for boundedness of these operators in such spaces. We also reveal the role of these operators in solving some classes of degenerate hyperbolic partial differential equations. Copyright © 2016 John Wiley & Sons, Ltd.     

Continuity Properties for Modulation Spaces,with Applications to Pseudo-Differential Calculus,II

We discuss continuity for weighted modulation spaces, andprove that many such spaces can be obtained in a canonicalway from the corresponding standard modulation spaces. We also discussthe trace operator aa(0, ·) acting on modulationspaces. The results are used to get inclusions betweenmodulation spaces and Besov spaces, and proving continuityfor pseudo-differential operators and Toeplitz operators.     

Hypercyclic operators on non-locally convex spaces

We transfer a number of fundamental results about hypercyclic operators on locally convex spaces (due to Ansari, Bès, Bourdon, Costakis, Feldman, and Peris) to the non-locally convex situation. This answers a problem posed by A. Peris [Multi-hypercyclic operators are hypercyclic, Math. Z. 236 (2001), 779-786].

     

Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces

The behavior of certain weighted Hardy-type operators on rearrangement-invariant function spaces is thoroughly studied. Emphasis is put on the optimality of the obtained results. First, the optimal rearrangement-invariant function spaces guaranteeing the boundedness of the operators from/to a given rearrangement-invariant function space are described. Second, the optimal rearrangement-invariant function norms being sometimes complicated, the question of whether and how they can be simplified to more manageable expressions is addressed. Next, the relation between optimal rearrangement-invariant function spaces and interpolation spaces is investigated. Last, iterated weighted Hardy-type operators are also studied.     

Composition operators on weighted Dirichlet spaces

We characterize bounded and compact composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function for the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. We also present several examples and counter-examples that point out the borderlines of the result and its connections to other themes.

     

Integral mappings between Banach spaces

We consider the classes of “Grothendieck-integral” (G-integral) and “Pietsch-integral” (P-integral) linear and multilinear operators (see definitions below), and we prove that a multilinear operator between Banach spaces is G-integral (resp. P-integral) if and only if its linearization is G-integral (resp. P-integral) on the injective tensor product of the spaces, together with some related results concerning certain canonically associated linear operators. As an application we give a new proof of a result on the Radon-Nikodym property of the dual of the injective tensor product of Banach spaces. Moreover, we give a simple proof of a characterization of the G-integral operators on C(K,X) spaces and we also give a partial characterization of P-integral operators on C(K,X) spaces.     

A Sum Operator with Applications to Self-Improving Properties of Poincaré Inequalities in Metric Spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.     

Vector lattices of weakly compact operators on Banach lattices

A result of Aliprantis and Burkinshaw shows that weakly compact operators from an AL-space into a KB-space have a weakly compact modulus. Groenewegen characterised the largest class of range spaces for which this remains true whenever the domain is an AL-space and Schmidt proved a dual result. Both of these authors used vector-valued integration in their proofs. We give elementary proofs of both results and also characterise the largest class of domains for which the conclusion remains true whenever the range space is a KB-space. We conclude by studying the order structure of spaces of weakly compact operators between Banach lattices to prove results analogous to earlier results of one of the authors for spaces of compact operators.

     

Weighted Norm Inequalities for Paraproducts and Bilinear Pseudodifferential Operators with Mild Regularity

We establish boundedness properties on products of weighted Lebesgue, Hardy, and amalgam spaces of certain paraproducts and bilinear pseudodifferential operators with mild regularity. We do so by showing that these operators can be realized as generalized bilinear Calderón–Zygmund operators.      

Diagonal multilinear operators on Köthe sequence spaces

We analyse the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also define and study some properties of the ideal of (E, p)-summing multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.     

Hypercyclic convolution operators on Fréchet spaces of analytic functions

A result of Godefroy and Shapiro states that the convolution operators on the space of entire functions on Cn, which are not multiples of identity, are hypercyclic. Analogues of this result have appeared for some spaces of holomorphic functions on a Banach space. In this work, we define the space holomorphic functions associated to a sequence of spaces of polynomials and determine conditions on this sequence that assure hypercyclicity of convolution operators. Some known results come out as particular cases of this setting. We also consider holomorphic functions associated to minimal ideals of polynomials and to polynomials of the Schatten-von Neumann class.     

Elementary operators on algebras of unbounded operators

Summary Structural results about elementary operators of length one, local elementary operators and injectivity preserving maps are proved. These are generalizations of results concerning algebras of bounded operators on Banach spaces to algebras of unbounded operators on Hilbert spaces.     

On Almost-Periodic Operators in the Spaces of Sequences

We study some properties of almost-periodic operators in the spaces l ^p and the spaces of Bohr and Besicovich almost-periodic sequences.     

Yosida-Hewitt type decompositions for weakly compact operators and operator-valued measures

We derive Yosida-Hewitt type decompositions for weakly compact operators from Köthe-Bochner function spaces to Banach spaces. As an application, we obtain a Yosida-Hewitt type decomposition for strongly bounded operator-valued measures.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.