The bounded approximation property for spaces of holomorphic mappings on infinite dimensional spaces

We study the bounded approximation property for spaces of holomorphic functions. We show that if U is a balanced open subset of a Fréchet–Schwartz space or (DFM )‐space E , then the space ??(U ) of holomorphic mappings on U , with the compact‐open topology, has the bounded approximation property if and only if E has the bounded approximation property. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On well posedness of best simultaneous approximation problems in Banach spaces

The well posedness of best simultaneous approximation problems is considered. We establish the generic results on the well posedness of the best simultaneous approximation problems for any closed weakly compact nonempty subset in a strictly convex Kadec Banach space. Further, we prove that the set of all points inE(G) such that the best simultaneous approximation problems are not well posed is a u- porous set inE(G) whenX is a uniformly convex Banach space. In addition, we also investigate the generic property of the ambiguous loci of the best simultaneous approximation.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous.     

Duality and Reflexivity of Spaces of Approximable Polynomials on Locally Convex Spaces

 We develop a duality theory for spaces of approximable n-homogeneous polynomials on locally convex spaces, generalising results previously obtained for Banach spaces. For E a Fréchet space with its dual having the approximation property and with E′ _b locally Asplund we show that the space of n-homogeneous polynomials on (E′ _b )′ _b is the inductive dual of the space of boundedly weakly continuous n-homogeneous polynomials on E. We show that when E is a reflexive Fréchet space, the space of n-homogeneous polynomials on E is reflexive if and only if every n-homogeneous polynomial on E is boundedly weakly continuous. (Received 24 March 1999; in final form 14 February 2000)     

New criterion of the approximation property

This paper is concerned with the approximation property which is an important property in Banach space theory. We show that a Banach space X has the approximation property if (and only if), for every Banach space Y, the set of finite rank operators from X to Y is dense in the corresponding space of compact operators, in the usual topology of uniform convergence on compact sets.     

Weakly Compact and Absolutely Summing Polynomials

This paper shows that, contrary to the case of linear operators, absolutely summing homogeneous polynomials are not always weakly compact. It is also shown that, regardless of the infinite dimensional Banach space E and the positive integer n, there exists an n-homogeneous polynomial P from E to E that plays the role of the identity operator in the sense that P is neither compact nor absolutely r-summing for any r, and P is weakly compact if and only if E is reflexive.     

Three-space problems for the bounded compact approximation property

In this paper, the notion of the bounded compact approximation property (BCAP) of a pair [Banach space and its subspace] is used to prove that if X is a closed subspace of L∞ with the BCAP, then L∞/X has the BCAP. We also show that X* has the λ-BCAP with conjugate operators if and only if the pair (X, Y) has the λ-BCAP for each finite codimensional subspace Y∈X. Let M be a closed subspace of X such that M⊥ is complemented in X*. If X has the (bounded) approximation property of order p, then M has the (bounded) approximation property of order p.     

Behavior of zeros of polynomials of near best approximation

The purpose of this paper is to study the asymptotic behavior of the zeros of polynomials of near best approximation to continuous functions f on a compact set E in the case when f is analytic on the interior of E but not everywhere on the boundary. For example, suppose E is a finite union of compact intervals of the real line and f is a continuous function on E, but is not analytic on E; then we show (cf. Corollary 2.2) that every point of E is a limit point of zeros of the polynomials of best uniform approximation to f on E. This fact answers a question posed by P. Borwein who showed that, for the case when E is a single interval and f is real-valued, then the above hypotheses on f imply that at least one point of E is the limit point of zeros of such polynomials.     

Density of multivariate homogeneous polynomials on star like domains

The famous Weierstrass theorem asserts that every continuous function on a compact set in ${\mathbb{R}}^d$ can be uniformly approximated by algebraic polynomials. A related interesting problem consists in studying the same question for the important subclass of homogeneous polynomials containing only monomials of the same degree. The corresponding conjecture claims that every continuous function on the boundary of convex 0-symmetric bodies can be uniformly approximated by pairs of homogeneous polynomials. The main objective of the present paper is to review the recent progress on this conjecture and provide a new unified treatment of the same problem on non convex star like domains. It will be shown that the boundary of every 0-symmetric non convex star like domain contains an exceptional zero set so that a continuous function can be uniformly approximated on the boundary of the domain by a sum of two homogeneous polynomials if and only if the function vanishes on this zero set. Thus the Weierstrass type approximation problem for homogeneous polynomials on non convex star like domains amounts to the study of these exceptional zero sets. We will also present an extension of a theorem of Varjú which describes the exceptional zero sets for intersections of star like domains. These results combined with certain transformations of the underlying region will lead to the discovery of some new classes of convex and non convex domains for which the Weierstrass type approximation result holds for homogeneous polynomials.     

A Gelfand-Phillips Property with Respect to the Weak Topology

We consider a Gelfand-Phillips type property for the weak topology. The main results that we obtain are (1) for certain Banach spaces, E^?_? F inherits this property from E and F, and (2) the spaces L^p(μ, E) have this property when E does. A subset A of a Banach space E is a limited set if every (bounded linear) operator T:E → c₀ maps A onto a relatively compact subset of c₀. The Banach space E has the Gelfand-Phillips property if every limited set is relatively compact. In this note, we study the analogous notions set in the weak topology. Thus we say that A ? E is a Grothendieck set if every T: E → c₀ maps A onto a relatively weakly compact set; and E is said to have the weak type GP property if every Grothendieck set in E is relatively weakly compact. In the papers [3, 4 and 6], it is shown among other results that the ?-tensor product E and the spaces L^p(μ, E) inherit the Gelfand-Phillips property from E and F. In this paper, we study the same questions for the weak type GP property. It is easily verified that continuous linear images of Grothendieck sets are Grothendieck and that the weak type GP property is inherited by subspaces. Among the spaces with the weak type GP property one easily finds the separable spaces, and more generally, spaces with a weak* sequentially compact dual ball. Also, C(K) spaces where K is (DCSC) are weak type GP (see [3] and the discussion before Corollary 4 below). A Grothendieck space (a Banach space whose unit ball is a Grothendieck set) has the weak type GP if and only if it is reflexive.     

The Alternative Dunford–Pettis Property for Subspaces of the Compact Operators

A Banach space X has the alternative Dunford–Pettis property if for every weakly convergent sequences (x_n) → x in X and (x_n*) → 0 in X* with ||x_n|| = ||x||= 1 we have (x_n*(x_n)) → 0. We get a characterization of certain operator spaces having the alternative Dunford–Pettis property. As a consequence of this result, if H is a Hilbert space we show that a closed subspace M of the compact operators on H has the alternative Dunford–Pettis property if, and only if, for any h ∈ H, the evaluation operators from M to H given by S ↦ Sh, S ↦ S^th are DP1 operators, that is, they apply weakly convergent sequences in the unit sphere whose limits are also in the unit sphere into norm convergent sequences. We also prove a characterization of certain closed subalgebras of K(H) having the alternative Dunford-Pettis property by assuming that the multiplication operators are DP1.     

Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ H_b(U, F), where U ? E is a balanced open subset, and E, F are complex Banach spaces, let A : H_b(F) ∈ H_b(U) be the homomorphism defined by A(f) = fog for all f ∈ H_b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if g is weakly compact; (b) A is completely continuous if and only if g(W) is a Dunford-Pettis set for every U-bounded subset W ? U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize the reflexivity of the space H_b(U,F) and prove that E' and F have the Schur property if and only if H_b(U, F) has the Schur property. As an application, we obtain some results on linearization of holomorphic mappings.     

Compact and Weakly Compact Homomorphisms on Fréchet Algebras of Holomorphic Functions

We study homomorphisms between Fréchet algebras of holomorphic functions of bounded type. In this setting we prove that any pointwise bounded homomorphism into the space of entire functions of bounded type is rank one. We characterize up to the approximation property of the underlying Banach space, the weakly compact composition operators on H_b(V), V absolutely convex open set.     

Weak and quasi approximation properties in Banach spaces

In this paper, we introduce weak versions (the weak approximation property, the bounded weak approximation property, and the quasi approximation property) of the approximation property and derive various characterizations of these properties. And we show that if the dual of a Banach space X has the weak approximation property (respectively the bounded weak approximation property), then X itself has the weak approximation property (respectively the bounded weak approximation property). Also we observe that the bounded weak approximation property is closely related to the quasi approximation property.     

The weak metric approximation property and ideals of operators

We study the weak metric approximation property introduced by Lima and Oja. We show that a Banach space X has the weak metric approximation property if and only if F(Y,X), the space of finite rank operators, is an ideal in W(Y,X∗∗), the space of weakly compact operators for all Banach spaces Y.     

Sequences with equi-distributed extreme points in uniform polynomial approximation

Let E be a compact set in with connected complement and positive logarithmic capacity. For any f continuous on E and analytic in the interior of E, we consider the distribution of extreme points of the error of best uniform polynomial approximation on E. Let Λ=(n_j) be a subsequence of such that n_j+1/n_j→1. If, for nΛ, A_n( f)∂E denotes the set of extreme points of the error function, we prove that there is a subsequence Λ′ of Λ such that the distribution of any (n+2)th Fekete point set of A_n( f) tends weakly to the equilibrium distribution on E as n→∞ in Λ′. Furthermore, we prove a discrepancy result for the distribution of the point sets if the boundary of E is smooth enough.     

Strong Minkowski Separation and Co-Drop Property

In the framework of topological vector spaces, we give a characterization of strong Minkowski separation, introduced by Cheng, et al., in terms of convex body separation. From this, several results on strong Minkowski separation are deduced. Using the results, we prove a drop theorem involving weakly countably compact sets in locally convex spaces. Moreover, we introduce the notion of the co-drop property and show that every weakly countably compact set has the co-drop property. If the underlying locally convex space is quasi-complete, then a bounded weakly closed set has the co-drop property if and only if it is weakly countably compact.     

A polynomial Hutton theorem with applications

Extending a classical linear result due to Hutton to a nonlinear setting, we prove that a continuous homogeneous polynomial between Banach spaces can be approximated by finite rank polynomials if and only if its adjoint can be approximated by finite rank linear operators. Among other consequences, we apply this result to generalize a classical result due to Aron and Schottenloher about the approximation property on spaces of polynomials and a recent result due to Çaliskan and Rueda about the quasi-approximation property on projective symmetric tensor products.     

Compact adjoint operators and approximation properties

This paper is concerned with the space of all compact adjoint operators from dual spaces of Banach spaces into dual spaces of Banach spaces and approximation properties. For some topology on the space of all bounded linear operators from separable dual spaces of Banach spaces into dual spaces of Banach spaces, it is shown that if a bounded linear operator is approximated by a net of compact adjoint operators, then the operator can be approximated by a sequence of compact adjoint operators whose operator norms are less than or equal to the operator norm of the operator. Also we obtain applications of the theory and, in particular, apply the theory to approximation properties.     

Dual problems for weak and quasi approximation properties

It is shown that for the separable dual X^∗ of a Banach space X if X^∗ has the weak approximation property, then X has the metric quasi approximation property. Using this it is shown that for the separable dual X^∗ of a Banach space X the quasi approximation property and metric quasi approximation property are inherited from X^∗ to X and for a separable and reflexive Banach space X, X having the weak approximation property, bounded weak approximation property, quasi approximation property, metric weak approximation property, and metric quasi approximation property are equivalent. Also it is shown that the weak approximation property, bounded weak approximation property, and quasi approximation property are not inherited from a Banach space X to X^∗.