Asplund Spaces, Stegall Variational Principle and the RNP

Given a pair of Banach spaces X and Y such that one is the dual of the other, we study the relationships between generic Fréchet differentiability of convex continuous functions on Y (Asplund property), generic existence of linear perturbations for lower semicontinuous functions on X to have a strong minimum (Stegall variational principle), and dentability of bounded subsets of X (Radon-Nikodym Property).      

Optimality of function spaces for kernel integral operators

We explore boundedness properties of kernel integral operators acting on rearrangement-invariant (r.i.) spaces. In particular, for a given r.i. space X we characterize its optimal range partner, that is, the smallest r.i. space Y such that the operator is bounded from X to Y. We apply the general results to Lorentz spaces to illustrate their strength.     

Range universal spaces

We characterize those topological spaces Y for which the Isbell and finest splitting topologies on the set C(X,Y) of all continuous functions from X into Y coincide for all topological spaces X. We also consider the same question for the coincidence of the restriction of the finest splitting topology on the upper semicontinuous set-valued functions to C(X,Y) and the finest splitting topology on C(X,Y). In the first case, the spaces in question are, after identifying points that are in each others closures, subsets of the two point Sierpiński space, which gives a converse and generalization of a result of S. Dolecki, G.H. Greco, and A. Lechicki. In the second case, the spaces in question are, after identifying points that are in each others closures, order bases for bounded complete continuous DCPOs with the Scott topology.     

Priestley duality for order-preserving maps into distributive lattices

The category of bounded distributive lattices with order-preserving maps is shown to be dually equivalent to the category of Priestley spaces with Priestley multirelations. The Priestley dual space of the ideal lattice L of a bounded distributive lattice L is described in terms of the dual space of L. A variant of the Nachbin-Stone-ech compactification is developed for bitopological and ordered spaces. Let X be a poset and Y an ordered space; X ^Y denotes the poset of continuous order-preserving maps from Y to X with the discrete topology. The Priestley dual of L ^P is determined, where P is a poset and L a bounded distributive lattice.     

On the continuity of the support of bimodules over maximal abelian self-adjoint algebras

We relate the convergence of a net of maximal Abelian selfadjoint algebras (masas) to that of the net of their corresponding supports. This is achieved by using a family of capacities on the collection of subsets of X × Y (where the masas are realized as collections of operators of multiplication by essentially bounded functions on the measure spaces X and Y), which extends a capacity studied previously by Haydon and Shulman.     

Measure of Noncompactness of Linear Operators between Spaces of Sequences That Are (N,q) Summable or Bounded

In this paper we investigate linear operators between arbitrary BK spaces X and spaces Y of sequences that are summable or bounded. We give necessary and sufficient conditions for infinite matrices A to map X into Y. Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for A to be a compact operator.     

Mikhlin's Theorem for Operator–Valued Fourier Multipliers in n Variables

An operator–valued Mikhlin theorem is proved for multipliers of the form M : ℝⁿ → ℒ︁(X, Y) where X and Y are UMD spaces. The usual norm bounds of the classical Mikhlin condition are replaced by R–bounds. Furthermore, the concept of R–bounded variation is introduced to generalize the Marcinkiewicz Fourier multiplier Theorem to the operator–valued setting.     

Absolute continuity of information channels

Let [X, v, Y] be an abstract information channel with the input X = (X, ) and the output Y = (Y, ) which are measurable spaces, and denote by L(Y) = L(Y, ) the Banach space of all bounded signed measures with finite total variation as norm. The channel distribution ν(·,·) is considered as a function defined on (X, ) and valued in L(Y). It will be proved that, if the measurable space (Y, ) is countably generated, then the is a strongly measurable function from X into L(Y) if and only if there exists a probability measure μ on (Y, ) which dominates every measure ν(x, ·) (x X). Furthermore, under this condition, the Radon-Nikodym derivative ν(x, dy)/μ(dy) is jointly measurable with respect to the product measure space (X, , m) (Y, , μ) where m is any but fixed probability measure of (X, ). As an application, it will be shown that the channel given as above is uniformly approximated by channels of Hibert-Schmidt type.     

Some approximation properties of Banach spaces and Banach lattices

The notion of the bounded approximation property = BAP (resp. the uniform approximation property = UAP) of a pair [Banach space, its subspace] is used to prove that if X is a ℒ _∞-space, Y a subspace with the BAP (resp. UAP), then the quotient X/Y has the BAP (resp. UAP). If Q: X → Z is a surjection, X is a ℒ ₁-space and Z is a ℒ _p -space (1 ≤ p ≤ ∞), then ker Q has the UAP. A complemented subspace of a weakly sequentially complete Banach lattice has the separable complementation property = SCP. A criterion for a space with GL-l.u.st. to have the SCP is given. Spaces which are quotients of weakly sequentially complete lattices and are uncomplemented in their second duals are studied. Examples are given of spaces with the SCP which have subspaces that fail the SCP. The results are applied to spaces of measures on a compact Abelian group orthogonal to a fixed Sidon set and to Sobolev spaces of functions of bounded variation on ℝ ⁿ .     

On set-valued cone absolutely summing maps

Spaces of cone absolutely summing maps are generalizations of Bochner spaces L ^p (μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L ¹(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] $ \mathcal{L}^1 \left[ {\sum ,cbf(X)} \right] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.     

Equiconnected spaces and Baire classification of separately continuous functions and their analogs

We investigate the Baire classification of mappings f: X × Y → Z, where X belongs to a wide class of spaces which includes all metrizable spaces, Y is a topological space, Z is an equiconnected space, which are continuous in the first variable. We show that for a dense set in X these mappings are functions of a Baire class α in the second variable.     

Characterizations of new Cohen summing bilinear operators

We prove two characterizations of new Cohen summing bilinear operators. The first one is: Let X, Y and Z be Banach spaces, 1 < p < ∞, V : X × Y → Z a bounded linear operator and n ≥ 2 a natural number. Then V is new Cohen p-summing if and only if for all Banach spaces X₁,?…?, X_n and all p-summing operators U : X₁ × · · · × X_n → X, the operator V ? (U, I_Y) : X₁ × · · · × X_n × Y → Z is -summing. The second result is: Let H be a Hilbert space,, Y, Z Banach spaces and V : H × Y → Z a bounded bilinear operator and 1 < p < ∞. Then V is new Cohen p-summing if and only if for all Banach spaces E and all p-summing operators U : E → H, the operator V ? (U, I_Y) is (p, p*)-dominated.     

Remarks on Linear Homeomorphisms of Function Spaces

It is proved that whenever X and Y are completely regular -spaces of pointwise countable type and the spaces C _p(X) and C _p(Y) of real-valued continuous functions on X and Y, respectively, endowed with the topology of pointwise convergence, are linearly homeomorphic, the X is locally compact iff Y is locally compact. This extends the McCoy and Ntantu result.     

Special uniform approximations of continuous vector-valued functions. Part II: special approximations in CX(T)CY(S)

In this paper, which is a continuation of Timofte (J. Approx. Theory 119 (2002) 291–299, we give special uniform approximations of functions from C_XY(T×S) and C_∞(T×S,XY) by elements of the tensor products C_X(T)C_Y(S), respectively C₀(T,X)C₀(S,Y), for topological spaces T,S and Γ-locally convex spaces X,Y (all four being Hausdorff).     

A New Uniform Convergence for Partial Functions

Suppose X, Y are topological spaces. In this paper maps are not necessarily continuous. A map f from a non-empty subset of X to Y is called a partial map. Partial maps occur as inverse functions in elementary analysis, as solution of ordinary differential equations, as utility functions in mathematical economics, etc. In many applications, X and Y are metric spaces and there is a need to have a uniform convergence on a family of partial functions. Since partial maps do not have a common domain, the usual uniform convergence (u.c.) is not available. Noting that in many situations, all maps of a family under consideration, have a common range, we define a new uniform convergence (n.u.c.) that is complementary to the usual one. This n.u.c. does not preserve continuity but preserves (uniform) openness. Its usefulness stems from the fact that it can be used when u.c. cannot be defined. Moreover, in some situations where both u.c. and n.u.c. are available, the latter satisfies our intuition but not the former. We give applications to ODE's and throw some light on earlier literature.     

SOME ASPECTS OF THE LIFTING PROPERTY OF THE SPACE £∞ (μ, X)

Abstract

Using a lifting of £_∞ (μ, X) ([5],[6]), we construct a lifting ρ x of the seminormed vector space £_∞ (μ, X) of measurable, essentially bounded X-valued functions. We show that in a certain sense such a lifting always exists. If μ is Lebesgue measure on (0, 1) we show that ρ x exists as map from £_∞ ((O, 1), X) → £_∞,((0, l), X) if and only if X is reflexive. In general the lifted function takes its values in X **. Therefore we investigate the question, when f ε £_∞ (μ, X) is strictly liftable in the sense that the lifted function is a map with values even in X.

As an application we introduce the space £_∞ ^strong (μ, L (X, Y**)), a subspace of the space of strongly measurable, essentially bounded L (X, Y, **)-valued functions, and the associated quotient space £_∞ ^strong (μ, L (X,Y**)). We show that this space is a Banach space because there is a kind of a Dunford-Pettis Theorem for a subspace of L (X, £_∞(μ Y**)). Finally we investigate the measurability property of functions in £_∞(μ Y**)) und see that there exists a connection to the Radon-Nikodym property of the space L (X, Y).     

Operators on <Emphasis Type="Italic">C</Emphasis>[0,1] preserving copies of asymptotic ℓ<Subscript>1</Subscript> spaces

Given separable Banach spaces X, Y, Z and a bounded linear operator T:X→Y, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such that the restriction of T to E is an into isomorphism. It is proved that every operator on C([0,1]) which preserves a copy of an asymptotic ℓ₁ space also preserves a copy of C([0,1]).     

The Sum and Chain Rules for Maximal Monotone Operators

This paper is primarily concerned with the problem of maximality for the sum A + B and composition L* ML in non-reflexive Banach space settings under qualifications constraints involving the domains of A, B, M. Here X, Y are Banach spaces with duals X*, Y*, A, B: X ⇉ X*, M: Y ⇉ Y* are multi-valued maximal monotone operators, and L: X → Y is linear bounded. Based on the Fitzpatrick function, new characterizations for the maximality of an operator as well as simpler proofs, improvements of previously known results, and several new results on the topic are presented.      

Approximately spectrum-preserving maps

Let X and Y be superreflexive complex Banach spaces and let B(X) and B(Y) be the Banach algebras of all bounded linear operators on X and Y, respectively. If a bijective linear map Φ:B(X)→B(Y) almost preserves the spectra, then it is almost multiplicative or anti-multiplicative. Furthermore, in the case where X=Y is a separable complex Hilbert space, such a map is a small perturbation of an automorphism or an anti-automorphism.