### Convolution of Operators and Applications.

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Let X be a complex Banach space and let Bloch(X) denote the space of X-valued analytic functions on the unit disc such that $su{p}_{\left|z\right|<1}(1-|z\left|\xb2\right)\left|\right|{f}^{\text{'}}\left(z\right)\left|\right|<\infty $. A sequence (Tₙ)ₙ of bounded operators between two Banach spaces X and Y is said to be an operator-valued multiplier between Bloch(X) and ℓ₁(Y) if the map ${\sum}_{n=0}^{\infty}x\u2099z\u207f\to \left(T\u2099\left(x\u2099\right)\right)\u2099$ defines a bounded linear operator from Bloch(X) into ℓ₁(Y). It is shown that if X is a Hilbert space then (Tₙ)ₙ is a multiplier from Bloch(X) into ℓ₁(Y) if and only if $su{p}_{k}{\sum}_{n={2}^{k}}^{{2}^{k+1}}\left|\right|T\u2099\left|\right|\xb2<\infty $. Several results about Taylor coefficients of vector-valued...

The duality between H^{1} and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained.
In this paper we shall study such space in little more detail and we shall consider the H^{1}-BMO duality for vector-valued functions in the more general setting of spaces of homogeneous type (see [CW]).

We develop the notion of the $({X}_{1},{X}_{2})$-summing power-norm based on a Banach space $E$, where ${X}_{1}$ and ${X}_{2}$ are symmetric sequence spaces. We study the particular case when ${X}_{1}$ and ${X}_{2}$ are Orlicz spaces ${\ell}_{\Phi}$ and ${\ell}_{\Psi}$ respectively and analyze under which conditions the $(\Phi ,\Psi )$-summing power-norm becomes a multinorm. In the case when $E$ is also a symmetric sequence space $L$, we compute the precise value of $\parallel ({\delta}_{1},\cdots ,{\delta}_{n}){\parallel}_{n}^{({X}_{1},{X}_{2})}$ where $\left({\delta}_{k}\right)$ stands for the canonical basis of $L$, extending known results for the $(p,q)$-summing power-norm based on the space ${\ell}_{r}$ which...

We find necessary and sufficient conditions on radial weights w on the unit disc so that the Bergman type projections of Forelli-Rudin are bounded on L¹(w) and in the Herz spaces ${K}_{p}^{q}\left(w\right)$.

The purpose of this note is to announce some results related to Hardy spaces of vector valued functions and to show that some properties on B have to be required if we want that the classical theorems to remain valid in the B-valued setting.

We consider (p,q)-multi-norms and standard t-multi-norms based on Banach spaces of the form ${L}^{r}\left(\Omega \right)$, and resolve some question about the mutual equivalence of two such multi-norms. We introduce a new multi-norm, called the [p,q]-concave multi-norm, and relate it to the standard t-multi-norm.

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