Let k ∈ ℕ and let E be a Banach space such that every k-homogeneous polynomial defined on a subspace of E has an extension to E. We prove that every norm one k-homogeneous polynomial, defined on a subspace, has an extension with a uniformly bounded norm. The analogous result for holomorphic functions of bounded type is obtained. We also prove that given an arbitrary subspace F ⊂ E, there exists a continuous morphism φk, F: (kF) → (kE) satisfying φk, F(P)|F = P, if and only E is isomorphic to a Hilbert space.
Questions concerning extensions of polynomials or analytic functions from a Banach space E to its bidual E", and others about reflexivity and related properties on spaces of homogeneous polynomials Pk(E), have recently spurred interest regarding the structure of the bidual of a space of polynomials [4, 14, 22].In this paper we study the relationship between the bidual of Pk(E) and the space of polynomials over E". We define a map through which elements of the bidual of Pk(E) may be viewed as polynomials over E"formula hereand study this map to obtain information about Pk(E)". We have tried as far as possible to avoid imposing restricting conditions on E.
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
We study the existence of hypercyclic algebras for convolution operators Φ(D) on the space of entire functions whose symbol Φ has unimodular constant term. In particular, we provide new eigenvalue criteria for the existence of densely strongly algebrable sets of hypercyclic vectors. Particular attention has been given to this question for the case where X = H(C) is the algebra of entire functions on the complex plane, endowed with the compact-open topology, and where T is a convolution operator, that is, an operator that commutes with all translations. Godefroy and Shapiro [13] showed that such operators are precisely the ones that commute with the operator D of complex differentiation, that they coincide with the operators of the form T = Φ(D) with Φ ∈ H(C) of exponential type, and that a convolution operator supports hypercyclic vectors precisely when it is not a scalar multiple of the identity. Aron et al [2] first noted that any translation τ a , which is the operator Φ(D) with Φ(z) = e az , fails to support a hypercyclic algebra in a dramatic way: given any f ∈ H(C), the multiplicity of any zero of an element in the orbit of f p must be divisible by p. They also stopped short from giving a positive answer for D, showing that for the generic element f of H(C), each power f n (n ∈ N) is a hypercyclic vector for D. Shkarin [17], and independently Bayart and Matheron [4] finally showed that D supports a hypercyclic algebra, prompting the question: Question 1. Which convolution operators support a hypercyclic algebra? In other words, for which Φ ∈ H(C) of exponential type does Φ(D) support a hypercyclic algebra?Several new positive examples were obtained since then by various authors [8,9,6], including for instance when Φ is of subexponential growth having zero constant
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
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