Lattice-universal Orlicz spaces on probability spaces

Lattice-universal Orlicz function spacesL ^F _α,β[0, 1] with prefixed Boyd indices are constructed. Namely, given 0<α<β<∞ arbitrary there exists Orlicz function spacesL ^F _α,β[0, 1] with indices α and β such that every Orlicz function spaceL ^G [0, 1] with indices between α and β is lattice-isomorphic to a sublattice ofL ^F _α,β[0, 1]. The existence of classes of universal Orlicz spacesl ^Fα,β(I) with uncountable symmetric basis and prefixed indices α and β is also proved in the uncountable discrete case. Partially supported by BFM2001-1284.     

On weak convergence of stocastic processes with Lusin path spaces

Under the assumption that a sequence of stochastic processes has paths in a Lusin function space we can prove the following. If convergence in the path space implies stochastic convergence, then tightness and convergence of the finite dimensional distributions of the stochastic processes are sufficient for weak convergence. The result in many cases implies a unification of the weak convergence proof. Demonstrably, such cases are C, D, Lip, L_p and , the space of distribution functions of finite measures.     

Factorizing operators on Banach function spaces through spaces of multiplication operators

In order to extend the theory of optimal domains for continuous operators on a Banach function space X(μ) over a finite measure μ, we consider operators T satisfying other type of inequalities than the one given by the continuity which occur in several well-known factorization theorems (for instance, Pisier Factorization Theorem through Lorentz spaces, pth-power factorable operators …). We prove that such a T factorizes through a space of multiplication operators which can be understood in a certain sense as the optimal domain for T. Our extended optimal domain technique does not need necessarily the equivalence between μ and the measure defined by the operator T and, by using δ-rings, μ is allowed to be infinite. Classical and new examples and applications of our results are also given, including some new results on the Hardy operator and a factorization theorem through Hilbert spaces.     

Random fixed points for random multivalued operators defined on unbounded sets in banach spaces

Random fixed points for nonexpansive and pseudocontractive random multivalued operators defined on unbounded subsets of a Banach space are obtained     

A remark on probability spaces

The object of this paper is to exhibit some aspects of Kolmogorov's axiomatic theory of probability related to its phenomenological interpretation.J. P. Gabriel has informed the editor that P. Milasevic was killed in a mountain accident on October 27, 1989.     

Sublinear operators on block-type spaces

This study establishes the boundedness of sublinear operators on block spaces built on Banach function spaces. These results are used to study the boundedness of the Marcinkiewicz integrals, singular integral operators and fractional integral operators with homogeneous kernels on block-type spaces.     

Hypercyclic operators on quasi-Mazur spaces

Modifying the method of Ansari, we give some criteria for hypercyclicity of quasi-Mazur spaces. They can be applied to judging hypercyclicity of non-complete and non-metrizable locally convex spaces. For some special locally convex spaces, for example, Köthe (LF)-sequence spaces and countable inductive limits of quasi-Mazur spaces, we investigate their hypercyclicity. As we see, bounded biorthogonal systems play an important role in the construction of Ansari. Moreover, we obtain characteristic conditions respectively for locally convex spaces having bounded sequences with dense linear spans and for locally convex spaces having bounded absorbing sets, which are useful in judging the existence of bounded biorthogonal systems.     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.     

Narrow operators on lattice-normed spaces

The aim of this article is to extend results of Maslyuchenko, Mykhaylyuk and Popov about narrow operators on vector lattices. We give a new definition of a narrow operator, where a vector lattice as the domain space of a narrow operator is replaced with a lattice-normed space. We prove that every GAM-compact (bo)-norm continuous linear operator from a Banach-Kantorovich space V to a Banach lattice Y is narrow. Then we show that, under some mild conditions, a continuous dominated operator is narrow if and only if its exact dominant is so.     

Multiplication operators on Lorentz-Karamata-Bochner spaces

In this small note, we will characterize the boundedness, compactness and closedness of the range of the multiplication operators on Lorentz-Karamata-Bochner spaces L _p,q;b (Ω,X) for p, q ∈ (0,∞].     

Positive operators on Krein spaces

A Krein operator is a positive operator, acting on a partially ordered Banach space, that carries positive elements to strong units. The purpose of this paper is to present a survey of the remarkable spectral properties (most of which were established by M.G. Krein) of these operators. The proofs presented here seem to be simpler than the ones existing in the literature. Some new results are also obtained. For instance, it is shown that every positive operator on a Krein space which is not a multiple of the identity operator has a nontrivial hyperinvariant subspace. Dedicated to the memory of M.G. Krein (1907–1989)     

Composition operators on Dirichlet-type spaces

The Dirichlet-type space ) is the Banach space of functions analytic in the unit disc with derivatives belonging to the Bergman space . Let be an analytic self-map of the disc and define for . The operator is bounded (respectively, compact) if and only if a related measure is Carleson (respectively, compact Carleson). If is bounded (or compact) on , then the same behavior holds on ) and on the weighted Dirichlet space . Compactness on implies that is compact on the Hardy spaces and the angular derivative exists nowhere on the unit circle. Conditions are given which, together with the angular derivative condition, imply compactness on the space . Inner functions which induce bounded composition operators on are discussed briefly.

     

Superstable operators on Banach spaces

It has been shown by W. Arendt—C.J.K. Batty and Yu.I. Lyubich— V.Q. Phong that the powers of a linear contraction on a reflexive Banach space converge strongly to zero if the boundary spectrum is countable and contains no eigenvalues. In this paper we characterize the countability of the boundary spectrum through a stronger convergence property in terms of ultrapower extensions. This paper is part of a research project supported by the Deutsche Forschungsgemeinschaft DFG     

Composition operators on Lizorkin-Triebel spaces

This paper is devoted to the study of the composition operator Tf(g):=f○g on Lizorkin-Triebel spaces . In case s>1+(1/p), 1<p<∞, and 1?q?∞ we will prove the following: the operator Tf takes to itself if and only if f(0)=0 and f belongs locally to .     

Classical operators on BLOCH spaces

Let \mathbbD \mathbb{D} ⁿ denote the unit polydisk and let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥1. We study weighted composition operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } ( \mathbbD \mathbb{D} ⁿ ), α > 1. We also study Cesàro type operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } (B _n ), α > 0. Bibliography: 15 titles.     

Pseudodifferential operators on ultra-modulation spaces

The boundedness of pseudodifferential operators on modulation spaces defined by the means of almost exponential weights is studied. The results are applied to symbol class with almost exponential bounds including polynomial and ultra-polynomial symbols. The Weyl correspondence is used and it is noted that the results can be transferred to the operators with appropriate anti-Wick symbols. It is proved that a class of elliptic pseudodifferential operators can be almost diagonalized by the elements of Wilson bases, and estimates for their eigenvalues are given. Furthermore, it is shown that the same can be done by using Gabor frames.