The gliding hump property in vector sequence spaces

It is shown that vector sequence spaces with a gliding hump property have many of the properties of complete spaces. For example, it is shown that the -dual of certain vector sequence spaces with a gliding hump property are sequentially complete with respect to the topology of pointwise convergence and also versions of the Banach-Steinhaus Theorem are established for such spaces.     

Sequence Spaces with Oscillating Properties

In this note we consider various types of oscillating properties for a sequence spaceEbeing motivated by an oscillating property introduced by Snyder and by recent papers dealing with theorems of Mazur–Orlicz type and gliding hump properties. Our main tools, two summability theorems, allow us to identify two such oscillating properties for a sequence spaceEone of which provides a sufficient condition forEFto implyEW_Fwhile the other affords a sufficient condition forEFto implyES_F. HereFis anyL-space, a class of spaces which includes the class of separable FK-spaces,S_Fdenotes the elements ofFhaving sectional convergence, andW_Fdenotes the elements ofFhaving weak sectional convergence. This, in turn, is applied to yield improvements on some other theorems of Mazur–Orlicz type and to obtain a general consistency theorem. Furthermore, combining the above observations with the work of Bennett and Kalton we obtain the first oscillating property on a sequence spaceEas a sufficient condition forE^β, the β-dual ofE, to be σ(E^β, E) sequentially complete whereas the second assures both the weak sequential completeness ofE^βand the AK-property forEwith the Mackey topology of the dual pair (E, E^β).     

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.     

一类无限矩阵拓扑代数

Let λ and μ be sequence spaces and have both the signed-weak gliding hump property, (λ,μ) the algebra of the infinite matrix operators which transform λ into μ. In this paper, it is proved that if λ and μ are β-spaces and λ^β and ,μ^β have also the signed-weak gliding hump property, then for any polar topology τ, ((λ,μ),τ) is always sequentially complete locally convex topological algebra.     

Function algebras on which homomorphisms are point evaluations on sequences

In the study of the spectrum of a subalgebraA ofC(X), whereX is a completely regular Hausdorff space, a key question is, whether each homomorphism ?:A→R has the point evaluation property for sequences inA, that is whether, for each sequence (f _n ) inA, there exists a pointa inX such that ?(f _n )=f _n (a) for alln. In this paper it is proved that all algebras, which are closed under composition with functions inC ^∞ (R) and have a certain local property, have the point evaluation property for sequences. Such algebras are, for instance, the spaceC ^m (E) (m=0,1,...,∞) ofC ^m -functions on any real locally convex spaceE. This result yields in a trivial manner that each homomorphism ? onA is a point evaluation, ifX is Lindelöf or ifA contains a sequence which separates points inX. Further, also a well known result as well as some new ones are obtained as a consequence of the main theorem.     

On Gliding Hump Properties of Matrix Domains

In this note, we establish several results concerning the gliding hump properties of matrix domains. In order to discuss F-WGHP, we introduce the UAK-property and find that this sort of property has close relationship with F-WGHP. In the course of discussing F-WGHP and WGHP of （C0）cn, we discuss the F-WGHP and WGHP of the almost-null sequence space f0.     

Point Evaluations and Polynomial Approximation in the Mean

We construct an example of a Borel subset E of the unit disk such that area measure restricted to E, which we denote A _E , has the property that the set of bounded point evaluations for the polynomials with respect to the L ^t (A _E ) norm varies with t. We further show that E can be chosen to be a simply connected region. In the context of smooth measures, like area measure, examples of this type were unexpected.      

Infinite Matrix Rings on Spaces with Signed-Weakly Gliding Hump Property

Ｌｅｔωｂｅｔｈｅｓｐａｃｅｏｆａｌｌｓｃａｌａｒｖａｌｕｅｄｓｅｑｕｅｎｃｅｓ,ａｎｄ φｉｔｓｓｕｂｓｐａｃｅｗｉｔｈｏｎｌｙｆｉｎｉｔｅｌｙｍａｎｙｎｏｎ ｚｅｒｏｃｏｏｒｄｉｎａｔｅｓ.ＡｌｉｎｅａｒｓｕｂｓｐａｃｅＥｏｆωｉｓｃａｌｌｅｄａｓｅｑｕｅｎｃｅｓｐａｃｅ.　Ｗｅｓａｙｔｈａｔａｎｏｎ ｚｅｒｏｖｅｃｔｏｒｓｅｑｕｅｎｃｅ {ｚ(ｎ) }ｉｎωｉｓａｂｌｏｃｋｓｅｑｕｅｎｃｅｉｆｔｈｅｒｅｅｘｉｓｔｓａｓｔｒｉｃｔｌｙｉｎｃｒｅａｓｉｎｇｓｅｑｕｅｎｃｅｏｆｐｏｓｉｔｉｖｅｉｎｔｅｇｅｒ…     

A banach space containing non-trivial limited sets but no non-trivial bounding sets

An example of a Banach spaceE is given with the following properties: Every bounding setA⊂E (i.e.f(A) is bounded for each holomorphic functionf:E →C) is relatively compact but there are relatively non-compact limited setsA (i.e.T(A) is relatively compact for each bounded linear mapT:E →c ₀).     

FLATNESS AND THE RING OF QUASI-ENDOMORPHISMS

Abstract

This paper investigates torsion-free abelian groups A which are Q E-flat, i.e. for which Q A is flat as an Q E(A)-module. It is shown that a torsion-free A has this property iff Tor¹ (M, A) is torsion for all right E(A)-modules M. Furthermore, a torsion-free group of rank 4 is constructed which is Q E-flat but not quasi-isomorphic to an E-flat group. This gives a negative response to a question of R. Pierce. The paper concludes with a discussion of the structure of torsion-free groups of finite rank which are Q E-flat.