A consequence of Littlewood’s Subordination Principle [10] is the (not at all obvious) fact that every composition operator restricts to a bounded operator on the Hardy space H2 (see also [5, Theorem 1.7, page 10] or [16, pp. 13–15]), and this in turn has inspired a lively enterprise connecting complex function theory with operator theory, the goal being to understand how properties Cφ are rela...

We provide a direct proof that the Haagerup estimate on the completely bounded norm of elementary operators is best possible in the case of B(H) via a generalisation of a theorem of Stampfli. We show that for an elementary operator T of length `, the completely bounded norm is equal to the k-norm for k = `. A C*-algebra A has the property that the completely bounded norm of every elementary ope...

It is proved that the maximal operator σ # of the triangular-Fejér-means of a two-dimensional Walsh–Fourier series is bounded from the dyadic Hardy space Hp to Lp for all 1/2 < p ≤ ∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular-Fejér-means σ 2n of a function f ∈ L1 converge a.e. to f . The maximal operator σ # is bounded from the Hardy space H1/2 to t...

In this paper we give a simple characterization of weighted Sobolev spaces (with piecewise monotonous weights) such that the multiplication operator is bounded: it is bounded if and only if the support of μ0 is large enough. We also prove some basic properties of the appropriate weighted Sobolev spaces. To have bounded multiplication operator has important consequences in Approximation Theory: ...

In this paper, we prove that the topological dual of the Banach space of bounded measurable functions with values in the space of nuclear operators, furnished with the natural topology, is isometrically isomorphic to the space of finitely additive linear operator-valued measures having bounded variation in a Banach space containing the space of bounded linear operators. This is then applied to ...

In this note we describe the dual and the completion of the space of finite linear combinations of (p,∞)-atoms, 0 < p ≤ 1. As an application, we show an extension result for operators uniformly bounded on (p,∞)-atoms, 0 < p < 1, whose analogue for p = 1 is known to be false. Let 0 < p < 1 and let T be a linear operator defined on the space of finite linear combinations of (p,∞)-atoms, 0 < p < 1...

For f ∈ S(R), we consider the Bochner-Riesz operator R of index δ > 0 defined by R̂δf(ξ) = (1− |ξ|)+ f̂(ξ). Then we prove the Bochner-Riesz conjecture which states that if δ > max{d|1/p − 1/2| − 1/2, 0} and p > 1 then R is a bounded operator from L(R) into L(R); moreover, if δ(p) = d(1/p − 1/2) − 1/2 and 1 < p < 2d/(d + 1), then Rδ(p) is a bounded operator from L(R) into L(R).

For a polynomial P (ξ) in ξ in R with constant complex coefficients, the operator defined by R(z)f = F((P (·)−z)f̂), where ∧ denotes the Fourier transform and F denotes its inverse, is not bounded from L to L when z is in the spectrum of P (D). What are suitable spaces B and C so that R(z) is bounded from B to C? When P (ξ) is simply characteristic, we prove that the operator R(z) is bounded fro...

Johnson and Zippin recently showed that if X is a weak∗-closed subspace of 1 and T : X → C(K ) is any bounded operator then T can be extended to a bounded operator T̃ : 1 → C(K ). We give a converse result: if X is a subspace of 1 such that 1/X has an unconditional finitedimensional decomposition (UFDD) and every operator T : X → C(K ) can be extended to 1 then there is an automorphism τ of 1 su...