Some summability invariants

Beekmann, Wolfgang; Chang, Shih Ching

doi:10.1007/bf02320700

Cited by 5 publications

(5 citation statements)

References 11 publications

(3 reference statements)

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“…On the other hand, ifA is #-unique, then #± N #÷ = O, and our examples (see [2], Example 1 and below) show that (E) and (E') are indeed independent•…”

Section: Corollary 1 I/ a Is Non-#-unique Then (E) And(e') Are Equimentioning

confidence: 80%

“…Here, as usual, In [2], we showed that (E') is an invariant property, i.e. if (E') is true for A, then it holds for every D with c D = c A.…”

Section: Introductionmentioning

confidence: 95%

“…But there are matrices with property (E') which do not have property (E). The example in [2] is a replaceable #,unique matrix. No such matrix can have property (E), see proof of Corollary 2 below.…”

Section: Introductionmentioning

confidence: 99%

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An example in summability

Beekmann

Chang

1983

Period Math Hung

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“…On the other hand, ifA is #-unique, then #± N #÷ = O, and our examples (see [2], Example 1 and below) show that (E) and (E') are indeed independent•…”

Section: Corollary 1 I/ a Is Non-#-unique Then (E) And(e') Are Equimentioning

confidence: 80%

“…Here, as usual, In [2], we showed that (E') is an invariant property, i.e. if (E') is true for A, then it holds for every D with c D = c A.…”

Section: Introductionmentioning

confidence: 95%

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An example in summability

Beekmann

Chang

1983

Period Math Hung

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“…It will turn out that the three conditions axe all equivalent. The corresponding theory for CA is given in [5], [1], [2]. We shall refer later to the sets This proves the theorem.…”

Section: {E"mentioning

confidence: 86%

Some Properties of Absolute Summability Domains

Macphail¹,

Orhan²

1989

Analysis

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Certain sets IA, and properties E, E' have been defined for the summability domain CA of a matrix A. We maJce corresponding definitions for the absolute summability domain ^A-We show that for CA, E and E' are equivalent, and that if either IA or A^ is invariant they both are, and then IA = A;;^ = CA-AMS 1980 Classification 40H05. DEFINITIONS. We use ip, c, m respectively for the set of finitely non-zero sequences, convergent sequences and bounded sequences, and 7, £ respectively for the set of sequences (xfc) with Exfc convergent, S|sfc| convergent. liA = (a"fc) is an infinite matrix and x = (xk) a sequence, we write {Ax)" = Ejta"fcXit if this exists, Ax = ((Ax)") and £A = '• Ax G i}.We do not need to assume £A D I, but we assume throughout that £A D f, that is, the columns of A belong to £. We denote the colunrn sums S"a"fc by ak and similarly for matrices denoted by other letters. We write e* = (0,0,..., 0,1,0,...) (1 in the Ä;-th place).It is known that £A can be made into an FK space and that any f G. £'A, the continuous Brought to you by | University of Pennsylvania Authenticated Download Date | 7/21/15 8:45 PM

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“…In the meantime several authors have studied this Situation with some added conditions (such as (v) and (vi)), see [2], [3] and [4], …”

Section: Invariant and F^w Then Is Invariant;mentioning

confidence: 98%

An Invariance Problem in Summability

Beekmann¹,

Chang²

1981

Analysis

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Abstiact. In 1971, Grahame Bennett proved that for an infinite complex matrix^ with convergent cofumns, if the distinguished subset is invariant in the convergence domain, then A^ is invariant also. Seme partial answers to the question of whether the converse of the above theorem holds, have been given by Bennett, Macphail and Wilansky and the present authors. In this note, we prove that, in general, the invariance of A-*-is indeed equivalent to the invariance of/. We also prove that if the distinguished subset B is closed in c^, then B = P. This result is essential for the proof of the above equivalence theorem.

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Some summability invariants

Cited by 5 publications

References 11 publications

An example in summability

An example in summability

Some Properties of Absolute Summability Domains

An Invariance Problem in Summability

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