We study the spaces where Ω is a disc with radius R and μ is a given probability measure on [0,R[. We show that, depending on μ, is either isomorphic to l₁ or to . Here Aₙ is the space of all polynomials of degree ≤ n endowed with the L₁-norm on the unit sphere.
We introduce and examine the notion of dense weak openness. In particular we show that multiplication in C(X) is densely weakly open whenever X is an interval in ℝ.
We investigate isomorphic embeddings T: C(K) → C(L) between Banach spaces of continuous functions. We show that if such an embedding T is a positive operator then K is the image of L under an upper semicontinuous set-function having finite values. Moreover we show that K has a π-base of sets whose closures are continuous images of compact subspaces of L. Our results imply in particular that if C(K) can be positively embedded into C(L) then some topological properties of L, such as countable...
We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space can be naturally expressed in terms of weak* continuity of seminorms on the unit ball of . We attempt to carry out a construction of a Banach space of the form which has the Mazur property but does not have the Gelfand-Phillips property. For this purpose we analyze the compact spaces on which all regular measures lie in the weak* sequential closure of atomic measures, and the set-theoretic properties of generalized...
We introduce Sobolev spaces for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in with fractional derivative of order α, , as introduced in [2], in . We show that for small α, coincides with the continuous version of the Triebel-Lizorkin space as defined by Y. S. Han and E. T. Sawyer in [4]. To prove this result we give a more general definition of ε-families of operators on spaces of homogeneous type, in which the identity operator is...
We describe the Banach envelopes of Hardy-Orlicz spaces of analytic functions on an annulus in the complex plane generated by Orlicz functions well-estimated by power-type functions.
Let X and Y be locally compact Hausdorff spaces, let E and F be Banach spaces, and let T be a linear isometry from C₀(X,E) into C₀(Y,F). We provide three new answers to the Banach-Stone problem: (1) T can always be written as a generalized weighted composition operator if and only if F is strictly convex; (2) if T is onto then T can be written as a weighted composition operator in a weak sense; and (3) if T is onto and F does not contain a copy of then T can be written as a weighted composition...
We give necessary and sufficient conditions on the weights v and w such that the differentiation operator D: Hv(Ω) → Hw(Ω) between two weighted spaces of holomorphic functions is bounded and onto. Here Ω = ℂ or Ω = 𝔻. In particular we characterize all weights v such that D: Hv(Ω) → Hw(Ω) is bounded and onto where w(r) = v(r)(1-r) if Ω = 𝔻 and w = v if Ω = ℂ. This leads to a new description of normal weights.