Weak-polynomial convergence on a Banach space

Abstract

Abstract.We show that any super-reflexive Banach space is a A-space (i.e., the weak-polynomial convergence for sequences implies the norm convergence). We introduce the notion of /c-space (i.e., a Banach space where the weakpolynomial convergence for sequences is different from the weak convergence) and we prove that if a dual Banach space Z is a k-space with the approximation property, then the uniform algebra A(B) on the unit ball of Z generated by the weak-star continuous polynomials is not tight.We shall be concerned in this note with some questions, posed by Carne, Cole, and Gamelin in [3], involving the weak-polynomial convergence and its relation to the tightness of certain algebras of analytic functions on a Banach space.Let A' be a (real or complex) Banach space. For m = 1,2, ... let 3°(mX) denote the Banach space of all continuous m-homogeneous polynomials on X, endowed with its usual norm; that is, each P e 3B(mX) is a map of the form P(x) = A(x, ... ,m) x), where A is a continuous, m-linear (scalar-valued) form on Xx---m]xX.Also, let S6(X) denote the space of all continuous polynomials on X; that is, each P e 3°(X) is a finite sum P -Pq + Px-\-\-Pn , where P0 is constant and each Pm e 3B(mX) for m= 1,2, ... , n.In [3] a sequence (Xj) c X is said to be weak-polynomial convergent to x e X if P(xj) -> P(x) for all P e 36(X); and the space X is defined to be a A-space, if whenever (xj) is a sequence in X that is weak-polynomial convergent to 0, then ||jc,-|| -»■ 0.It is shown in [3] that lp is a A-space for 1 < p < co ; it is also shown that Lp(p) is a A-space for 2 < p < oo and Lx[0, 1] is not a A-space, and the question is posed as to whether Lp(p) is a A-space for 1 < p < 2. Our next result will provide an affirmative answer to this question.First we recall that super-reflexive Banach spaces can be defined as those spaces that admit an equivalent uniformly convex norm. In particular, spaces Lp(p) are super-reflexive for 1 < p < oo and any measure p (see, e.g., [8, Chapter 3]).We shall use the following fact: If X is super-reflexive, then there exists

Bourgain algebras and tightness in some algebras of analytic functions

“…Cole and T. Gamelin proved that the polydisk algebras are not tight for n 2 in [3] (see [9] too). Further results on tightness of algebras of analytic functions were shown by J. Jaramillo and A. Prieto [8]. Now, we may obtain that the polydisk algebras are not tight as a corollary of Proposition 2 since we have that weakly compact operators on a space with the DPP are completely continuous and, therefore, the Bourgain algebras would be the whole space C(K).…”

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

“…A Banach space E is called a Λ-space, if all null sequences in (E, σ(E, P (E))) are norm convergent, and hence, convergent sequences in (E, σ(E, P (E))) are also norm convergent. All superreflexive spaces and 1 are Λ-spaces [15].…”

Upper lp-estimates in vector sequence spaces, with some applications

“…In [2], Carne, Cole, and Gamelin defined a A-space as a Banach space such that every sequence weakpolynomial convergent to zero is norm null, and conjectured that, for 1 <p < oo, L p -spaces are A-spaces, proving themselves the conjecture for p ^ 2. In [8], Jaramillo and Prieto proved, using results of [3], that super-reflexive spaces are Aspaces. Usually, it is not easy to prove that a Banach space is a A-space.…”