On nonseperable simplex spaces.
Let D be the open unit disc and μ a positive bounded measure on [0,1]. Extending results of Mateljević/Pavlović and Shields/Williams we give Banach-space descriptions of the classes of all harmonic (holomorphic) functions f: D → ℂ satisfying .
We show that every separable complex L₁-predual space X is contractively complemented in the CAR-algebra. As an application we deduce that the open unit ball of X is a bounded homogeneous symmetric domain.
Let be a commuting approximating sequence of the Banach space X leaving the closed subspace A ⊂ X invariant. Then we prove three-space results of the following kind: If the operators Rₙ induce basis projections on X/A, and X or A is an -space, then both X and A have bases. We apply these results to show that the spaces and have bases whenever Λ ⊂ ℤ and ℤ∖Λ is a Sidon set.
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.
It is proved that for any , where is the Poulsen simplex, so that and , are smooth points, there is a rotation of carrying in .
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.
We study Toeplitz operators with radial symbols in weighted Bergman spaces , 1 < p < ∞, on the disc. Using a decomposition of into finite-dimensional subspaces the operator can be considered as a coefficient multiplier. This leads to new results on boundedness of and also shows a connection with Hardy space multipliers. Using another method we also prove a necessary and sufficient condition for the boundedness of for a satisfying an assumption on the positivity of certain indefinite...
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.
Let v be a standard weight on the upper half-plane , i.e. v: → ]0,∞[ is continuous and satisfies v(w) = v(i Im w), w ∈ , v(it) ≥ v(is) if t ≥ s > 0 and . Put v₁(w) = Im wv(w), w ∈ . We characterize boundedness and surjectivity of the differentiation operator D: Hv() → Hv₁(). For example we show that D is bounded if and only if v is at most of moderate growth. We also study composition operators on Hv().
This work is an introduction to anisotropic spaces of holomorphic functions, which have -weight and are generalizations of Bloch spaces on a unit ball. We describe the holomorphic Bloch space in terms of the corresponding space. We establish a description of via the Bloch classes for all .
The -weighted Besov spaces of holomorphic functions on the unit ball in are introduced as follows. Given a function of regular variation and , a function holomorphic in is said to belong to the Besov space if where is the volume measure on and stands for the fractional derivative of . The holomorphic Besov space is described in the terms of the corresponding space. Some projection theorems and theorems on existence of the inversions of these projections are proved. Also,...
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