THE COMPLETENESS PROBLEM IN SPACES OF PETTIS INTEGRABLE FUNCTIONS

Abstract

Two subspaces of the space of Banach space valued Pettis integrable functions are considered: the space P(μ, X, var) of Pettis integrable functions with integrals of finite variation in a Banach space X and LLN(μ,X,var), the space of functions satisfying the law of large numbers. It is proved that LLN(μ,X*,var) is always complete and P(μ, X*,var) is complete if Martin's axiom and the perfectness of μ are assumed. Moreover, a non-trivial example of a non-conjugate Banach space X with non-complete P(μ, X, var) is presented.     

COMPLEMENTED COPIES OF c 0 IN THE SPACE OF PETTIS INTEGRABLE FUNCTIONS

Abstract

Let X be a Banach space containing a copy of c0, then the space of Pettis integrable functions defined from any perfect atomless measure space to X, contains a complemented copy of c0.     

NOWHERE WEAK DIFFERENTIABILITY OF THE PETTIS INTEGRAL

Abstract

For an arbitrary infinite-dimensional Banach space 𝔛, we construct examples of strongly-measurable 𝔛-valued Pettis integrable functions whose indefinite Pettis integrals are nowhere weakly differentiable; thus, for these functions the Lebesgue Differentiation Theorem fails rather spectacularly. We also relate the degree of nondifferentia-bility of the indefinite Pettis integral to the cotype of 𝔛, from which it follows that our examples are reasonably sharp.     

THE LIMIT SET OF RIEMANN SUMS OF A VECTOR STIELTJES INTEGRAL

Abstract

Conditions for convexity of the limit set of Riemann sums of a vector integral are investigated.     

THE WEAK RADON-NIKODYM PROPERTY IN SPACES OF NUCLEAR OPERATORS

Abstract

We prove that if X and Y are Banach spaces such that X* has the weak Radon-Nikodym property (WRNP), Y has the Radon-Nikodym property (RNP) and Y is complemented in its bidual, then the space N(X,Y) of all nuclear operators from X to Y has the WRNP. If moreover X* has the RNP, then N(X,Y) has the RNP.     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).     

On the Radon-Nikodym property for operator valued measures

A theorem about the Radon-Nikodym property and the convergence of bounded martingales is proved for a bilinear integral in locally convex spaces.     

Henstock-Kurzweil可积函数空间的紧性特征

本文研究Henstock-Kurzweil可积(HK可积)函数空间中的一个经典问题.文章通过研究分布Henstock-Kurzweil积分(DHK积分)的性质,给出了该问题的否定答案.进一步,利用收敛性获得了函数HK可积的一个充分必要条件.最后,在上述结论的基础上刻画了HK可积函数空间的紧性.所得结果丰富和推广了HK可...     

Some types of vector measures

Letx be a metrizable locally convex space with a Schauder basis and letB(T) be a -ring generated by the compact subsets of a locally compact Hausdorff spaceT. We prove that any vector measure :B(T)X which has an antiregular relative is antimonogenic (Theorem 16) and that can be uniquely decomposable, = ₁ + ₂, where ₁ is monogenic and ₂ has an antiregular relative (Theorem 19). These results are due to R. A. Johnshon [6] for the case whereX is the real line.     

Optimal Domains and Integral Representations of Convolution Operators in <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>(<Emphasis Type="Italic">G</Emphasis>)

Given , a compact abelian group G and a function , we identify the maximal (i.e. optimal) domain of the convolution operator (as an operator from L^p(G) to itself). This is the largest Banach function space (with order continuous norm) into which L^p(G) is embedded and to which has a continuous extension, still with values in L^p(G). Of course, the optimal domain depends on p and g. Whereas is compact, this is not always so for the extension of to its optimal domain. Several characterizations of precisely when this is the case are presented.     

Banach lattices with the fatou property and optimal domains of kernel operators

New features of the Banach function space L¹_w(v), that is, the space of all v-scalarly integrable functions (with v any vector measure), are exposed. The Fatou property plays an essential role and leads to a new representation theorem for a large class of abstract Banach lattices. Applications are also given to the optimal domain of kernel operators taking their values in a Banach function space.     

ON THE VITALI-HAHN-SAKS-NIKODYM THEOREM

Abstract

The classical Vitali-Hahn-Saks-Nikodym Theorem [5, Thm. I.4.8] gives a limit criterion for when a sequence of strongly additive vector measures on a σ-field of sets having their range in a Banach space can be expected to be uniformly strongly additive. In [16, Cor. 8], Saeki proved that the limit condition on the sequence of vector measures could be substantially weakened as long as the Banach space in play is “good enough”. Saeki's result was based upon his work on a class of set functions too large to have Rosenthal's Lemma at his disposal. In Section 2, we prove Saeki's result with Rosenthal's Lemma at the basis of our work and then augment our characterization of Banach spaces enjoying Saeki's result in [1] with another natural equivalent condition. In Section 3 we extend Saeki's result to Boolean algebras having the Subsequential Interpolation property.     

The Canonical Spectral Measure in Köthe Echelon Spaces

Operator and measure theoretic properties of the canonical spectral measure acting in K?the echelon sequence spaces X are characterized via topological and geometric properties of X (such as being nuclear, Montel, satisfying the density condition, etc.).     

ON A THEOREM OF A.A. LYAPUNOV

Abstract

In this note we obtain some extensions and an approximation of the Lyapunov convexity theorem by means of the bilinear integration of a set-valued function. The integration is performed successively with respect to a non-atomic, a direct sum and a Darboux vector measure. The necessary counterexamples are provided.     

Summability and estimates for polynomials and multilinear mappings

In this paper we extend and generalize several known estimates for homogeneous polynomials and multilinear mappings on Banach spaces. Applying the theory of absolutely summing nonlinear mappings, we prove that estimates which are known for mappings on ?_p spaces in fact hold true for mappings on arbitrary Banach spaces.     

THE UNIQUENESS OF OPERATIONAL QUANTITIES IN VON NEUMANN ALGEBRAS

Abstract

Since 1970 a number of operational quantities, characteristic of either the semi-Fredholm operators or of some “ideal” of compact-like operators, have been introduced in the theory of bounded operators between Banach spaces and applied successfully to for example perturbation theory. More recently such quantities have been introduced even in the abstract setting of Fredholm theory in a von Neumann algebra relative to some closed two-sided ideal. We show that in this fairly general setting there is only one “reasonable” set of such quantities—a result which in its present form is to the best of our knowledge new even in the case of B(H), the algebra of all bounded operators on a Hilbert space H. We accomplish this by first of all introducing the concept of a (reduced) minimum modulus in the setting of C*-algebras and developing the relevant techniques. In the process we generalise a result of Nikaido [N].