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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.
Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of
and investigate some isomorphism classes of
where v is a given radial weight function. Our main results show that, without any further condition on v, there are only two possibilities for Hv, namely either or , and at least two possibilities for hv, again and . We also discuss many new examples of weights.
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