Consistency theory for operator valued matrices

Abstract

The authors proved in [6] that the implication (*) MOWECF => MnWECWr holds for every separable FK-space F, for every FK-space E containing the set (p of all finite (real er complex valued) sequences, and for each sequence space M having suitable factor sequences; thereby We denotes the set of all elements of E being weakly sectionally convergent. This result was proved by the first author [4] under the additional assumption that M is an FK-AB-space and by both authors [5] under the same assumption and in the special case that is a summability domain. For the "history" of such theorems of Mazur-Orlicz type we refer for example to [6]. On the base of the examinations of distinguished subspaces of FK-spaces over an F-space X (FK(X)-spaces) in [7] (see also [2] and [10]) we prove in section 2 of the present paper that the implication (*) remains true if, in addition, E is an FK(X)-space eind F is a summability domain of a suitable operator valued matrix. Proving an inclusion theorem we show in section 3 that a light variant of implication (») remains true in case of weak domains of operator valued matrices. In section 4 we get -quite similar to the classical case -as immediate corollaries of the Theorems of Mazur-Orlicz-type some consistency theorems containing, for example, operator bounded consistency theorems due to A. Alexiewicz and W. Orlicz (see [1] and [9, Theorem 6.27]). We complete the paper with an example of operator valued matrices related to almost convergence which proves that it is of mathematical interest to consider weak domains of operator valued matrices. AMS 1980 Classification: 40J05, 40H05, 40C99, 40D99, 46A45 Notations and preliminariesFor basic definitions, notations and preliminaries, especially for the notion of FK(X)-spaces, we refer to [7]. Throughout the whole paper we assume that (X,TX) and {Y,TY) are (locally convex) Frechet spaces (F-spaces) and, as usually, B{X, Y) denotes the set of all continuous linear maps from X ' to y. Furthermore we assume that Z is a linear subspace of V such that (V, Z) is a total dual pair. Apart from the sequence spaces , m(X) , c(A'), co(X), ip(X) and 1{X') defined in [7] we consider in the current paper the following sequence spaces (over X):Brought to you by | University of Pennsylvania Authenticated Download Date | 6/14/15 7:46 PM

Some New Classes in Topological Sequence Spaces Related to $L_r$-Spaces and an Inclusion Theorem for K(X)-Spaces

“…Now, it is obvious to ask whether (1) remains true in the more general setting of sequence spaces over an F-space and, in particular, for separable FK(X)-spaces. With the aim of providing a positive answer to this question the authors proved in [6], on the basis of the proof of the corresponding clascicai result, that (1) reniaius even true if F is the domain of an operator valued matrix. To reformulate this theorem in detail we need the definition of a special class of '(scalar) factor sequences' and the 'gliding humps property' of sequence spaces (over 1K), see Definitions 2.1 and 2.2 in [6].…”

“…Now, we recall the main result of [6] which generalizes Theorem 1 of [4]. Using Theorem 2.4 we may prove the validity of (1) in case of separable FK(X)-spaces but the obtained theorem would not contain Theorem 2.4 since domains of operator valued matrices are not necessarily separable FK(X)-spaces as the following simple example shows: the domain c(m)J of the identity matrix I (together with its FK( m )-topology) is not separable as the BK-space m is not separable.…”

“…where ço := hmB oek (e X') and ' := (k) E G. By (6) is dense in (F',u(F',F)). Since (G,c(G'3,G)) is sequentially complete, DA , is a(F',F)sequentially closed (see [9,Lemma 2.2]).…”

Gliding hump properties and some applications

“…See for example, [8] for extensions of Schur's theorems, ([4], [5], [20]) fo Mazur--Orlicz type theorems and ( [5], [14]) for weak sequential cmnpleteness results. The gliding hump technique has also proven to be a key ingredient in the solution to problems related to the Wilansky Property ([1], [21], [15]).…”

“…We give examples in section 5 to show that most of these implications are strict and they are, in some sense, affording a structure to the set of sequence spaces between the solid spaces and those with weakly sequentially complete -duals. E has the pointwise gliding hump property (p_ghp) if for each z E, any block sequence (y(")) satisfying sup I1-11 < o0 nd any monotonicly increasing sequence (n) of integers there exists nell subsequence (mk) of (n) with E zy(",) E (pointwise sum).…”