A note about Volterra operators on weighted Banach spaces of entire functions

We characterize boundedness, compactness and weak compactness of Volterra operators acting between different weighted Banach spaces of entire functions with sup‐norms in terms of the symbol g; thus we complement recent work by Bassallote, Contreras, Hernández‐Mancera, Martín and Paul 3 for spaces of holomorphic functions on the disc and by Constantin and Peláez 16 for reflexive weighted Fock spaces.     

Convolution operators and zeros of entire functions

Let be a real entire function of order less than with only real zeros. Then we classify certain distribution functions such that the convolution has only real zeros.

     

Existence of hypercyclic polynomials on complex Fréchet spaces

We show that every complex separable infinite dimensional Fréchet space admits hypercyclic polynomials of any degree. This result complements the analogous one for the linear case, due to Ansari, Bernal, Bonet and Peris.     

Pontryagin spaces of entire functions II

We continue the study of a generalization of L. de Branges's theory of Hilbert spaces of entire functions to the Pontryagin space setting. In this-second-part we investigate isometric embeddings of spaces of entire functions into spacesL ² () understood in a distributional sense and consider Weyl coefficients of matrix chains. The main task is to give a proof of an indefinite version of the inverse spectral theorem for Nevanlinna functions. Our methods use the theory developed by L. de Branges and the theory of extensions of symmetric operators of M.G.Krein.     

Hermitian indefinite functions and Pontryagin spaces of entire functions

We establish and investigate a connection between hermitian indefinite continuous functions with finitely many negative squares defined on a finite interval and so-called de Branges spaces of entire functions. This enables us to relate to any hermitian indefinite continuous function on the real axis a certain chain of 2×2-matrix valued entire functions, which are in the positive definite case tightly connected with canonical systems of differential equations.     

Pontryagin spaces of entire functions I

We give a generalization of L.de Branges theory of Hilbert spaces of entire functions to the Pontryagin space setting. The aim of this-first-part is to provide some basic results and to investigate subspaces of Pontryagin spaces of entire functions. Our method makes strong use of L.de Branges's results and of the extension theory of symmetric operators as developed by M.G.Krein.     

Complete systems of partial derivatives of entire functions and frequently hypercyclic operators

We find some sufficient conditions for a system of partial derivatives of an entire function to be complete in the space H(C^d)$H({\mathbb{C}}^d)$ of all entire functions of d   variables. As an application of this result we describe new classes of frequently hypercyclic operators on H(C^d)$H({\mathbb{C}}^d)$.     

Polynomial automorphisms and hypercyclic operators on spaces of analytic functions

We consider hypercyclic composition operators on which can be obtained from the translation operator using polynomial automorphisms of . In particular we show that if C _S is a hypercyclic operator for an affine automorphism S on , then for some polynomial automorphism Θ and vectors a and b, where I is the identity operator. Finally, we prove the hypercyclicity of “symmetric translations” on a space of symmetric analytic functions on ℓ₁. Received: 8 June 2006 Revised: 26 September 2006     

Fredholm composition operators on spaces of holomorphic functions

Composition operators on vector spaces of holomorphic functions are considered. Necessary conditions that range of the operator is of a finite codimension are given. As a corollary of the result it is shown that a composition operatorC on a certain Banach space of holomorphic functions on a strictly pseudoconvex domain withC ² boundary or a polydisc or a compact bordered Riemann surface or a bounded domainD such that intD = D is invertible if and only if it is a Fredholm operator if and only if is a holomorphic automorphism.     

Some properties of maximal monotone operators on nonreflexive Banach spaces

Important properties of maximal monotone operators on reflexive Banach spaces remain open questions in the nonreflexive case. The aim of this paper is to investigate some of these questions for the proper subclass of locally maximal monotone operators. (This coincides with the class of maximal monotone operators in reflexive spaces.) Some relationships are established with the maximal monotone operators of dense type, which were introduced by J.-P. Gossez for the same purpose.     

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsW _, on certain weighted locally convex spacesCV _o(X, E) of vector-valued continuous functions induced by self maps ofX and the operator-valued mappings XB(E).The work of this author was supported in part by CSIR Grant 9/100/92-EMR-IThe work of this author was supported in part by UGC Grant F.8-7/91 (RBB-II)     

Defect operators, defect functions and defect indices for analytic submodules

This paper mainly concerns defect operators and defect functions of Hardy submodules, Bergman submodules over the unit ball, and Hardy submodules over the polydisk. The defect operator (function) carries key information about operator theory (function theory) and structure of analytic submodules. The problem when a submodule has finite defect is attacked for both Hardy submodules and Bergman submodules. Our interest will be in submodules generated by polynomials. The reason for choosing such submodules is to understand the interaction of operator theory, function theory and algebraic geometry.     

Composition operators on Orlicz spaces

In this paper we characterize the composition operators on Orlicz spaces and study some of their properties.Research supported by NBHM DDF No. 40/11/95 (R&D-II)/1429     

Dynamics of the differentiation operator on weighted spaces of entire functions

The continuity of the differentiation operator on weighted Banach spaces of entire functions with sup-norm has been characterized recently by Harutyunyan and Lusky. We give necessary and sufficient conditions to ensure that the differentiation operator on these weighted Banach spaces of entire functions is hypercyclic or chaotic, when it is continuous. This research was partially supported by MEC and FEDER Project MTM2007-62643.     

Reducing subspaces of compressed analytic Toeplitz operators on the Hardy space

In this paper we discuss necessary conditions and sufficient conditions for the compression of an analytic Toeplitz operator onto a shift coinvariant subspace to have nontrivial reducing subspaces. We give necessary and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial and obtain examples of reducing subspaces from these kernels. Motivated by this result we give necessary conditions and sufficient conditions for the kernel of a Toeplitz operator whose symbol is the quotient of two inner functions to be nontrivial in terms of the supports of the two inner functions. By studying the commutant of a compression, we are able to give a necessary condition for the existence of reducing subspaces on certain shift coinvariant subspaces.     

Behavior of entire functions on balls in a Banach space

In this paper we prove that given any two disjoint balls in an infinite dimensional complex Banach space, there exists an entire function which is bounded on one and unbounded on the other.     

Grothendieck space ideals and weak continuity of polynomials on locally convex spaces

We introduce the classes of locally convex spaces with the local Dunford-Pettis property and locally dual Schur spaces. We examine their properties and their relationship to other classes of locally convex spaces. In the class of locally convex spaces with the local Dunford-Pettis property all polynomials are weakly sequentially continuous whereas in the class of locally dual Schur spaces all polynomials are weakly continuous on bounded sets. Research supported by Science Foundation Ireland, Basic Research Grant 2004.     

Canonical matrices of isometric operators on indefinite inner product spaces

We give canonical matrices of a pair (A,B) consisting of a nondegenerate form B and a linear operator A satisfying B(Ax,Ay)=B(x,y) on a vector space over F in the following cases:

•

F is an algebraically closed field of characteristic different from 2 or a real closed field, and B is symmetric or skew-symmetric;

•

F is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field, and B is Hermitian or skew-Hermitian with respect to any nonidentity involution on F.

These classification problems are wild if B may be degenerate. We use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver. This method was described in [V.V. Sergeichuk, Classification problems for systems of forms and linear mappings, Math. USSR-Izv. 31 (3) (1988) 481-501].