Weak Orlicz space and its convergence theorems

In this article, a class of weak Orlicz function spaces is defined and their basic properties are discused. In particular, for the sequences in weak Orlicz space, we establish several basic convergence theorems including bounded convergence theorem, control convergence theorem and Vitali-type convergence theorem and so on. Moreover, the conditional compactness of its subsets is also discussed.     

Relative weak compactness of orbits in Banach spaces associated with locally compact groups

We study analogues of weak almost periodicity in Banach spaces on locally compact groups.

i) If is a continous measure on the locally compact abelian group and , then is not relatively weakly compact.

ii) If is a discrete abelian group and , then is not relatively weakly compact if has non-empty interior. That result will follow from an existence theorem for -sets, as follows.

iii) Every infinite subset of a discrete abelian group contains an infinite -set such that for every neighbourhood of the identity of the interpolation (except at a finite subset depending on ) can be done using at most 4 point masses.

iv) A new proof that for abelian groups is given that identifies the weak limits of translates of Fourier-Stieltjes transforms.

v) Analogous results for , , and are given.

vi) Semigroup compactifications of groups are studied, both abelian and non-abelian: the weak* closure of in , for abelian ; and when is a continuous homomorphism of the locally compact group into the unitary elements of a von Neumann algebra , the weak* closure of is studied.

     

Compact operators on a Banach space into the space of almost periodic functions

LetS be a topological semigroup andAP(S) the space of continous complex almost periodic functions onS. We obtain characterizations of compact and weakly compact operators from a Banach spaceX into AP(S). For this we use the almost periodic compactification ofS obtained through uniform spaces. For a bounded linear operatorT fromX into AP(S), letT ₅, be the translate ofT bys inS defined byT ₅(x)=(Tx) ₅ . We define topologies on the space of bounded linear operators fromX into AP(S) and obtain the necessary and sufficient conditions for an operatorT to be compact or weakly compact in terms of the uniform continuity of the maps→T ₅. IfS is a Hausdorff topological semigroup, we also obtain characterizations of compact and weakly compact multipliers on AP(S) in terms of the uniform continuity of the map S→μ_s, where μ_s denotes the unique vector measure corresponding to the operatorT ₅.     

The algebra of almost periodic functions has infinite topological stable rank

We show that if is the uniform algebra of almost periodic functions, then the set cannot be dense in for any positive integer .

     

On weak convergence of random fields

We provide a characterization of compactness in the spaceD of functions of two variables defined on a unit square. The functions fromD have the property that their discontinuity points lie on smooth curves. Conditions for the tightness of probability measures inD and conditions for weak convergence of random fields with trajectories inD are derived. Vilnius Gediminas Technical University, Saulétekio 11; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 2, pp 169–184, April–June, 1999. Translated by R. Banys     

Bochner’s theorem and Stepanov almost periodic functions

Motivated by a renewed interest in generalizations of classical almost periodicity (originally due to Harald Bohr), we develop a theorem of Bochner within the framework of almost periodic functions in the sense of Stepanov. As a result we establish some conditions that guarantee the existence of Stepanov almost periodic solutions to differential equations with Stepanov almost periodic coefficients. Finally, we extend a now classic theorem of Favard originally stated for classical almost periodic functions to the Stepanov almost periodic case.      

Fredholmness and invertibility of Toeplitz operators with matrix almost periodic symbols

We consider Toeplitz operators with symbols that are almost periodic matrix functions of several variables. It is shown that under certain conditions on the group generated by the Fourier support of the symbol, a Toeplitz operator is Fredholm if and only if it is invertible.

     

Weak compactness in

We characterize weak compactness and weak conditional compactness of subsets of in terms of regular methods of summability. We also study when these results still hold using only convergence in the sense of Cesàro.

     

Conditional expectations and weak and strong compactness in spaces of Bochner integrable functions

In [6, theorem IV.8.18], relatively norm compact sets K in L^p(μ) are characterized by means of strong convergence of conditional expectations, E_πf → f in L^p(μ), uniformly for f ∈ K, where (E_π) is the family of conditional expectations corresponding to the net of all finite measurable partitions.In this paper we extend the above result in several ways: we consider nets of not necessarily finite partitions; we consider spaces $\text{L}{}_{\mathrm{E}}{}^{\mathrm{p}}\text{(μ)}$ of vector valued pth power Bochner integrable functions (and spaces M(Σ, E) of vector valued measures with finite variation); we characterize relatively strong compact sets K in $\text{L}{}_{\mathrm{E}}{}^{\mathrm{p}}\text{(μ)}$ by means of uniform strong convergence E_πf → f, as well as relatively weak compact sets K by means of uniform weak convergence E_πf → f. Previously, in [4], uniform strong convergence (together with some other conditions) was proved to be sufficient (but not necessary) for relative weak compactness.     

Algebras of almost periodic functions with Bohr-Fourier spectrum in a semigroup: Hermite property and its applications

It is proved that the unital Banach algebra of almost periodic functions of several variables with Bohr-Fourier spectrum in a given additive semigroup is an Hermite ring. The same property holds for the Wiener algebra of functions that in addition have absolutely convergent Bohr-Fourier series. As applications of the Hermite property of these algebras, we study factorizations of Wiener-Hopf type of rectangular matrix functions and the Toeplitz corona problem in the context of almost periodic functions of several variables.     

Weak and point-wise convergence in for filter convergence

We study those filters on for which weak -convergence of bounded sequences in C(K) is equivalent to point-wise -convergence. We show that it is sufficient to require this property only for C[0,1] and that the filter-analogue of the Rainwater extremal test theorem arises from it. There are ultrafilters which do not have this property and under the continuum hypothesis there are ultrafilters which have it. This implies that the validity of the Lebesgue dominated convergence theorem for -convergence is more restrictive than the property which we study.     

Weak convergence to the fractional Brownian sheet in Besov spaces

In this paper we study the problem of the approximation in law of the fractional Brownian sheet in the topology of the anisotropic Besov spaces. We prove the convergence in law of two families of processes to the fractional Brownian sheet: the first family is constructed from a Poisson procces in the plane and the second family is defined by the partial sums of two sequences of real independent fractional brownian motions.     

Subspaces of Hilbert-generated Banach spaces and the quantification of super weak compactness

We introduce a measure of super weak noncompactness Γ defined for bounded subsets and bounded linear operators in Banach spaces that allows to state and prove a characterization of the Banach spaces which are subspaces of a Hilbert-generated space. The use of super weak compactness and Γ casts light on the structure of these Banach spaces and complements the work of Argyros, Fabian, Farmaki, Godefroy, Hájek, Montesinos, Troyanski and Zizler on this subject. A particular kind of relatively super weakly compact sets, namely uniformly weakly null sets, plays an important role and exhibits connections with Banach-Saks type properties.     

Carathéodory-Toeplitz and Nehari problems for matrix valued almost periodic functions

In this paper the positive and strictly contractive extension problems for almost periodic matrix functions are treated. We present necessary and sufficient conditions for the existence of extensions in terms of Toeplitz and Hankel operators on Besicovitch spaces and Lebesgue spaces. Furthermore, when a solution exists a special extension (the band extension) is constructed which enjoys a maximum entropy property. A linear fractional parameterization of the set of all extensions is also provided. The techniques used in the proofs include factorizations of matrix valued almost periodic functions and a general algebraic scheme called the band method.

     

Transformation semigroup compactifications and norm continuity of weakly almost periodic functions

We prove if there exists a separately continuous action of a topologically right simple semitopological semigroupS on a topological spaceX and ifS acts topologically surjective onX then each weakly almost periodic function onX, with respect toS, is left norm continuous.     

Approximation of semigroups and cosine functions in spaces of periodic functions

In this article we furnish a representation of the solutions of some classes of first-order and second-order evolution problems as limit of iterates of classical sequences of approximating operators. The method is based on Trotter's theorem on the approximation of semigroups which is applied here also for the approximation of groups and cosine functions. We apply this method in spaces of continuous periodic functions and using some classical sequences of trigonometric polynomials.