Calculations on some sequence spaces.
We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair forms a...
We introduce a class of weights for a which a rich theory of real interpolation can be developed. In particular it led us to extend the commutator theorems associated to this method.
In this paper we use the Calderón-Zygmund operator theory to prove a Calderón type reproducing formula associated with a para-accretive function. Using our Calderón-type reproducing formula we introduce a new class of the Besov and Triebel-Lizorkin spaces and prove a Tb theorem for these new spaces.
We study sufficient conditions on the weight w, in terms of membership in the classes, for the spline wavelet systems to be unconditional bases of the weighted space . The main tool to obtain these results is a very simple theory of regular Calderón-Zygmund operators.
A Banach space has Pełczyński’s property (V) if for every Banach space every unconditionally converging operator is weakly compact. H. Pfitzner proved that -algebras have Pełczyński’s property (V). In the preprint (Krulišová, (2015)) the author explores possible quantifications of the property (V) and shows that spaces for a compact Hausdorff space enjoy a quantitative version of the property (V). In this paper we generalize this result by quantifying Pfitzner’s theorem. Moreover, we...
For a Banach space X such that all quotients only admit direct decompositions with a number of summands smaller than or equal to n, we show that every operator T on X can be identified with an n × n scalar matrix modulo the strictly cosingular operators SC(X). More precisely, we obtain an algebra isomorphism from the Calkin algebra L(X)/SC(X) onto a subalgebra of the algebra of n × n scalar matrices which is triangularizable when X is indecomposable. From this fact we get some information on the...
It is known that is not amenable for p = 1,2,∞, but whether or not is amenable for p ∈ (1,∞) ∖ 2 is an open problem. We show that, if is amenable for p ∈ (1,∞), then so are and . Moreover, if is amenable so is for any index set and for any infinite-dimensional -space E; in particular, if is amenable for p ∈ (1,∞), then so is . We show that is not amenable for p = 1,∞, but also that our methods fail us if p ∈ (1,∞). Finally, for p ∈ (1,2) and a free ultrafilter over ℕ, we exhibit...
It is proved (independently of the result of Holmes [Fund. Math. 140 (1992)]) that the dual space of the uniform closure of the linear span of the maps x ↦ d(x,a) - d(x,b), where d is the metric of the Urysohn space of diameter r, is (isometrically if r = +∞) isomorphic to the space of equivalence classes of all real-valued Lipschitz maps on . The space of all signed (real-valued) Borel measures on is isometrically embedded in the dual space of and it is shown that the image of the embedding...
We introduce by means of reproducing kernel theory and decomposition in orthogonal polynomials canonical correspondences between an interacting Fock space a reproducing kernel Hilbert space and a square integrable functions space w.r.t. a cylindrical measure. Using this correspondences we investigate the structure of the infinite dimensional canonical commutation relations. In particular we construct test functions spaces, distributions spaces and a quantization map which generalized the work of...
The topology and the structure of the set of the canonical extensions of positive, weakly continuous functionals from a von Neumann subalgebra to a von Neumann algebra M are described.