Inverse problems of the theory of separately continuous mappings

The present paper investigates the problem of constructing a separately continuous function defined on the product of two topological spaces that possesses a specified set of points of discontinuity and the related special problem of constructing a pointwise convergent sequence of continuous functions that possesses a specified set of points of nonuniform convergence and set of points of discontinuity of a limit function. In the metrizable case the former problem is solved for separable F-sets whose projections onto every cofactor is of the first category. The second problem is solved for a pair of embedded F.Translated from Ukrayins'kyy Matematychnyy Zhurnal, Vol. 44, No. 9, pp. 1209–1220, September, 1992.     

On isomorphisms of some Köthe function <Emphasis Type="Italic">F</Emphasis>-spaces

We prove that if Köthe F-spaces X and Y on finite atomless measure spaces (Ω _X ; Σ _X , µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ), respectively, with absolute continuous norms are isomorphic and have the property

$\mathop {\lim }\limits_{\mu (A) \to 0} \left\| {\mu (A)^{ - 1} 1_A } \right\| = 0$

(for µ = µ _X and µ = µ _Y , respectively) then the measure spaces (Ω _X ; Σ _X ; µ _X ) and (Ω _Y ; Σ _Y ; µ _Y ) are isomorphic, up to some positive multiples. This theorem extends a result of A. Plichko and M. Popov concerning isomorphic classification of L _p (µ)-spaces for 0 < p < 1. We also provide a new class of F-spaces having no nonzero separable quotient space.

     

Diagonals of separately pointwise Lipschitz mappings

It is proved that, for a metric space X and a normed space Z, the diagonals of pointwise Lipschitz mappings f : X ²? →?Z are exactly stable pointwise limits of pointwise Lipschitz mappings. The joint Lipschitz property of separately pointwise Lipschitz mappings f : X?×?Y?→?Z, where X, Y, and Z are metric spaces, is investigated.     

On the sum of a narrow and a compact operators

Our main technical tool is a principally new property of compact narrow operators which works for a domain space without an absolutely continuous norm. It is proved that for every Köthe F-space X and for every locally convex F-space Y   the sum _T1+_T2$T_1+T_2$ of a narrow operator _T1:X→Y$T_1:X\to Y$ and a compact narrow operator _T2:X→Y$T_2:X\to Y$ is a narrow operator. This gives a positive answers to questions asked by M. Popov and B. Randrianantoanina [6, Problems 5.6 and 11.63].     

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.     

Metrizable compacta in the space of continuous functions with the topology of pointwise convergence

We prove that every point-finite family of nonempty functionally open sets in a topological space X has the cardinality at most an infinite cardinal κ if and only if w(X) ≦ κ for every Valdvia compact space Y C _p (X). Correspondingly a Valdivia compact space Y has the weight at most an infinite cardinal κ if and only if every point-finite family of nonempty open sets in C _p (Y) has the cardinality at most κ, that is p(C _p (Y)) ≦ κ. Besides, it was proved that w(Y) = p(C _p (Y)) for every linearly ordered compact Y. In particular, a Valdivia compact space or linearly ordered compact space Y is metrizable if and only if p(C _p (Y)) = ℵ₀. This gives answer to a question of O. Okunev and V. Tkachuk.      

On order structure and operators in L ∞(μ)

It is known that there is a continuous linear functional on L _∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L _∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L _∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L _∞(μ) to c ₀(Γ) is narrow while not every such an operator is AM-compact.     

Narrow and ℓ2-strictly singular operators from L p

In the first part of the paper we prove that for 2 < p, r < ∞ every operator T: L _p → ? _r is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L _p → X is narrow. Next, using similar methods we prove that every ?₂-strictly singular operator from L _p , 1 < p < ∞, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator T from L ₁[0, 1] to itself satisfies the assumption that for each measurable set A ? [0, 1] the restriction \(T{|_{{L_1}(A)}}\) is not an isomorphic embedding, then T is narrow. (Here L ₁(A) = {x ∈ L ₁ : supp x ? A}.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being ?₂-strictly singular, for operators from L _p [0, 1] to itself, 1 < p < 2, to be narrow. We define a notion of a “gentle” growth of a function and we prove that for 1 < p < 2 every operator T from L _p to itself which, for every A ? [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.     

Structure changes in Al80Ni15Y5 amorphous alloy

The structure of Al₈₀Ni₁₅Y₅ amorphous alloy at various temperatures have been studied with X-ray diffraction methods. The obtained scattered intensities, structure factors, pair correlation functions and main structure parameters have been analyzed. Temperature dependences of the parameters suggest formation of Al, Al₃Ni and Al₂₃Ni₆Y₄ phases upon crystallization of an amorphous alloy.     

Some Geometric Aspects of Operators Acting from L 1

We study some properties of the space (L₁,X) of all continuous linear operators acting from L₁ to a Banach space X. It is proved that every operator T ∈ (L₁, X) ``almost' attains its norm at the entire positive cone of functions supported at some suitable measurable subset , μ(A) > 0. Using this fact and a new elementary technique we prove that every operator T∈ (L₁) = (L₁, L₁) is uniquely represented in the form T= R+S, R, S∈ (L₁) , where R is representable and S possess a special property (*). Moreover, this representation generates a decomposition of the space (L₁) into complemented subspaces by means of contractive projections (the fact that the subspace of all representable operators is complemented in (L₁) was proved before by Z. Liu).