Some distinguished subspaces of domains of operator valued matrices

Boos, Johann; Leiger, Toivo

doi:10.1007/bf03322472

Cited by 7 publications

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“…From [2] and [7] it is known that C(Y)A is an FK(X)-space and, if Y is sequentially <T(y,y')-complete, that C{Y)a^ , ^"(y)^ and are FK(X)-spaces, too. Now, we assume C C(Y)A which is equivalent to…”

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 97%

“…Furthermore, in [7, Section 3] the distinguished subspaces LA , IA and A^ of C(Y)A are considered. In this connection, we'll make use of Theorems 3.3-3.5 of [7]. If A' = y then A is called regulär if c(X) C C(X)A and lim^x = limx for each x 6 c(X).…”

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 98%

“…We remark that the limit functions of X are essential. In the sequel we'll make use of the remarks behind of the definition 2.5 in [7]. k are considered in [7].…”

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 99%

“…In the sequel we'll make use of the remarks behind of the definition 2.5 in [7]. k are considered in [7]. Furthermore, in this paper we deal with the weak application domain In the special cases of the domains C(Y)A , C(Y)A, , C'"{Y)A and we use for the limit functions belonging to the considered domain the Rotations lim^ : = lim oA, lim^^ : = lim oAz , w-limx : = w-lim oA, and w-lim^^ : = w-lim oAz , respectively.…”

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 99%

“…Proving the above theorem of Mazur-Orlicz-type we made use of the special structure of WB aad the representation of the continuous linear functionals on the FK(X)-space C{Y)B (because we applied Theorem 3.4 of[7]). In case of any weak domain we don't know whether it is an FK(X)-space and whether a theorem like Theorem 3.4 of[7] is true in general because we don't know a simple representation of continuous linear functionals on weak domains. However, for the derivation of consistency theorems it is sufficient to prove theorems of Mazur-Orlicz-type of the kindM n w^E c c'"(y)B.…”

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Consistency theory for operator valued matrices

Boos¹,

Leiger²

1991

Analysis

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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

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Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 97%

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 98%

“…We remark that the limit functions of X are essential. In the sequel we'll make use of the remarks behind of the definition 2.5 in [7]. k are considered in [7].…”

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 99%

Section: Now Let Us Recall That U>{x) With the Topology R^ Of Coordimentioning

confidence: 99%

mentioning

confidence: 99%

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Consistency theory for operator valued matrices

Boos¹,

Leiger²

1991

Analysis

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The gliding hump property in vector sequence spaces

Swartz

1993

Monatshefte f�r Mathematik

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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. 9kn-l+l'' ''' kn "

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Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series

Swartz

2004

Proyecciones

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Let X, Y be locally convex spaces and L(X, Y) the space of continuous linear operators from X into Y. We consider 2 types of mul-tiplier convergent theorems for a series T k in L(X, Y). First, if λ is a scalar sequence space, we say that the series T k is λ multiplier convergent for a locally convex topology τ on L(X, Y) if the series t k T k is τ convergent for every t = {t k } ∈ λ. We establish conditions on λ which guarantee that a λ multiplier convergent series in the weak or strong operator topology is λ multiplier convergent in the topology of uniform convergence on the bounded subsets of X. Second , we consider vector valued multipliers. If E is a sequence space of X valued sequences, the series T k is E multiplier convergent in a locally convex topology η on Y if the series T k x k is η convergent for every x = {x k } ∈ E. We consider a gliding hump property on E which guarantees that a series T k which is E multiplier convergent for the weak topology of Y is E multiplier convergent for the strong topology of Y. 136 Charles Swartz

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Some distinguished subspaces of domains of operator valued matrices

Cited by 7 publications

References 11 publications

Consistency theory for operator valued matrices

Consistency theory for operator valued matrices

The gliding hump property in vector sequence spaces

Orlicz - Pettis Theorems for Multiplier Convergent Operator Valued Series

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