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On sequential properties of Banach spaces, spaces of measures and densities

Piotr Borodulin-Nadzieja, Grzegorz Plebanek (2010)

Czechoslovak Mathematical Journal

We show that a conjunction of Mazur and Gelfand-Phillips properties of a Banach space E can be naturally expressed in terms of weak* continuity of seminorms on the unit ball of E * . We attempt to carry out a construction of a Banach space of the form C ( K ) 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...

On Sobolev spaces of fractional order and ε-families of operators on spaces of homogeneous type

A. Gatto, Stephen Vági (1999)

Studia Mathematica

We introduce Sobolev spaces L α p for 1 < p < ∞ and small positive α on spaces of homogeneous type as the classes of functions f in L p with fractional derivative of order α, D α f , as introduced in [2], in L p . We show that for small α, L α p coincides with the continuous version of the Triebel-Lizorkin space F p α , 2 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...

On the Banach-Stone problem

Jyh-Shyang Jeang, Ngai-Ching Wong (2003)

Studia Mathematica

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 ℓ ₂ ∞ then T can be written as a weighted composition...

On the boundedness of the differentiation operator between weighted spaces of holomorphic functions

Anahit Harutyunyan, Wolfgang Lusky (2008)

Studia Mathematica

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.

On the isomorphism classes of weighted spaces of harmonic and holomorphic functions

Wolfgang Lusky (2006)

Studia Mathematica

Let Ω be either the complex plane or the open unit disc. We completely determine the isomorphism classes of H v = f : Ω → ℂ h o l o m o r p h i c : s u p z ∈ Ω | f ( z ) | v ( z ) < ∞ and investigate some isomorphism classes of h v = f : Ω → ℂ h a r m o n i c : s u p z ∈ Ω | f ( z ) | v ( z ) < ∞ 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 H v ∼ l ∞ or H v ∼ H ∞ , and at least two possibilities for hv, again h v ∼ l ∞ and h v ∼ H ∞ . We also discuss many new examples of weights.

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