Starting with a general formula, precise but difficult to use, for the adjoint of a composition operator on a functional Hilbert space, we compute an explicit formula on the classical Hardy Hilbert space for the adjoint of a composition operator with rational symbol. To provide a foundation for this formula, we study an extension to the definitions of composition, weighted composition, and Toep...

Let E be a nonempty finite set, and let V be a real or complex vector space. In these notes we write F(E, V ) for the vector space of V -valued functions on E. Suppose that E1, E2 are nonempty finite sets, and that V is the vector space of real or complex-valued functions on E2. In this case we can identify F(E1, V ) with the vector space of real or complex-valued functions on the Cartesian pro...

A dx. Given a Banach space (X, ‖ · ‖) and 1 ≤ p < ∞ we shall denote by L(R, X) the space of Bochner p-integrable functions endowed with the norm ‖f‖Lp(Rn,X) = ( ∫ Rn ‖f(x)‖ dx), by Lc (R , X) the closure of the compactly supported functions in L(R, X) and by Lweak,α(R , X) the space of measurable functions such that |{x ∈ R : ‖f(x)‖ > λ}| ≤ α(λ) where α : R → R is a non increasing function. We ...

The solid hulls of the Hardy–Lorentz spaces Hp,q , 0 < p < 1, 0 < q 6∞ and Hp,∞ 0 , 0 < p < 1, as well as of the mixed norm space H p,∞,α 0 , 0 < p 6 1, 0 < α <∞, are determined. Introduction In [JP1] the solid hull of the Hardy space H, 0 < p < 1, is determined. In this article we determine the solid hulls of the Hardy–Lorentz spaces H, 0 < p < 1, 0 < q 6 ∞ and H 0 , 0 < p < 1, as well as of t...

We have already studied several kinds of functions of several variables: vector valued functions of a scalar variable – i.e., parameterized curves in the plane and in space; scalar-valued functions of two or more variables (the case of two variables z = f(x, y) giving the graph of a surface in space; and functions from the plane to space, parameterized surfaces. Here we consider functions V fro...

We establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. As an application, we obtain the $(L^p, L^q)$-norm inequality for vector-valued multilinear operators.

we establish a sharp maximal function estimate for some vector-valued multilinear singular integral operators. as an application, we obtain the $(l^p, l^q)$-norm inequality for vector-valued multilinear operators.

We give a necessary and a sufficient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L 2 a(C ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [4]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin t...