Absolute Boundedness and Absolute Convergence in Sequence Spaces

Buntinas, Martin; Tanovic-Miller, Naza

doi:10.2307/2048564

Cited by 3 publications

(10 citation statements)

References 2 publications

(7 reference statements)

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“…PROOF. By Theorem 2, Corollary 2, of[5], an FK-space is solid if and only if it has the property \AB\. This clearly implies [AB] which by Theorem (3.10) yields all but the first equality.…”

mentioning

confidence: 81%

“…This clearly implies [AB] which by Theorem (3.10) yields all but the first equality. But by Theorem 6, Corollary 2, of[5], E\ AK \ = EAD-…”

mentioning

confidence: 95%

“…Thus/ e Ê ffB n £ 1 '°°. Conversely if/ G £cr£ Pi £ lo°, then/ has aB in £ and IM n /|||£| < ||^n/|||^| = \\d n f\\ t i = 0(l)(n->oo), again referring to Theorem 11 in [5]. •…”

mentioning

confidence: 99%

“…• sectional boundedness AB and sectional convergence AK [7], induced by the ordinary convergence / of numerical series; • Cesàro sectional boundedness oB and Cesàro sectional convergence aK [2], induced by Cesàro convergence C\ ; • unrestricted sectional boundedness UAB and unrestricted sectional convergence UAK [11], [12], and absolute boundedness \AB\ and absolute convergence \AK\ [5], The basic definitions are given in Section 2. The results in this paper are given for FKspaces and M'-spaces containing the spaces of finite sequences <j> although generalizations to other topological spaces are possible.…”

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confidence: 99%

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Strong Boundedness and Strong Convergence in Sequence Spaces

Buntinas

Tanović-Miller

1991

Can. j. math.

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Strong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.

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“…PROOF. By Theorem 2, Corollary 2, of[5], an FK-space is solid if and only if it has the property \AB\. This clearly implies [AB] which by Theorem (3.10) yields all but the first equality.…”

mentioning

confidence: 81%

“…This clearly implies [AB] which by Theorem (3.10) yields all but the first equality. But by Theorem 6, Corollary 2, of[5], E\ AK \ = EAD-…”

mentioning

confidence: 95%

“…Thus/ e Ê ffB n £ 1 '°°. Conversely if/ G £cr£ Pi £ lo°, then/ has aB in £ and IM n /|||£| < ||^n/|||^| = \\d n f\\ t i = 0(l)(n->oo), again referring to Theorem 11 in [5]. •…”

mentioning

confidence: 99%

mentioning

confidence: 99%

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Strong Boundedness and Strong Convergence in Sequence Spaces

Buntinas

Tanović-Miller

1991

Can. j. math.

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“…By definition, a BK-space is a vector space of complex sequences f = (f k ) ∞ k=0 endowed with a norm which makes it into a Banach space, such that the coordinate functionals become bounded operators. In the theory of BK-spaces, see [8], the solid hull S BK (X) of a BK-space X is defined as the intersection of all solid BK-spaces containing X.…”

Section: On Solid Hullsmentioning

confidence: 99%

On solid cores and hulls of weighted Bergman spaces $A_μ^1$

Bonet¹,

Lusky²,

Taskinen³

2022

Preprint

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We consider weighted Bergman spaces A 1 µ on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces we characterize the solid core of A 1 µ . Also, as a consequence of a characterization of solid A 1 µ -spaces we show that, in the case of entire functions, there indeed exist solid A 1 µ -spaces. The second part of the paper is restricted to the case of the unit disc and it contains a characterization of the solid hull of A 1 µ , when µ equals the weighted Lebesgue measure with weight v. The results are based on a duality relation of weighted A 1 -and H ∞ -spaces, the validity of which requires the assumption that − log v belongs to the class W 0 , studied in a number of publications; moreover, v has to satisfy condition (b), introduced by the authors. The exponentially decreasing weight v(z) = exp(−1/(1−|z|) provides an example satisfying both assumptions. ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACESWe take a non-atomic positive bounded Borel measure µ on [0, R[ such that µ([r, R[) > 0 for every r > 0 and R 0 r n dµ(r) < ∞ for all n > 0. Put, for 1 ≤ p < ∞,

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ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES $$A_{\mu }^1$$

2022

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We consider weighted Bergman spaces $$A_\mu ^1$$ A μ 1 on the unit disc as well as the corresponding spaces of entire functions, defined using non-atomic Borel measures with radial symmetry. By extending the techniques from the case of reflexive Bergman spaces, we characterize the solid core of $$A_\mu ^1$$ A μ 1 . Also, as a consequence of a characterization of solid $$A_\mu ^1$$ A μ 1 -spaces, we show that, in the case of entire functions, there indeed exist solid $$A_\mu ^1$$ A μ 1 -spaces. The second part of the article is restricted to the case of the unit disc and it contains a characterization of the solid hull of $$A_\mu ^1$$ A μ 1 , when $$\mu$$ μ equals the weighted Lebesgue measure with the weight v. The results are based on the duality relation of the weighted $$A^1$$ A 1 - and $$H^\infty$$ H ∞ -spaces, the validity of which requires the assumption that $$-\log v$$ - log v belongs to the class $$\mathcal {W}_0$$ W 0 , studied in a number of publications; moreover, v has to satisfy the condition (b), introduced by the authors. The exponentially decreasing weight $$v(z) = \exp ( -1 /(1-|z|)$$ v ( z ) = exp ( - 1 / ( 1 - | z | ) provides an example satisfying both assumptions.

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Absolute Boundedness and Absolute Convergence in Sequence Spaces

Cited by 3 publications

References 2 publications

Strong Boundedness and Strong Convergence in Sequence Spaces

Strong Boundedness and Strong Convergence in Sequence Spaces

On solid cores and hulls of weighted Bergman spaces $A_μ^1$

ON SOLID CORES AND HULLS OF WEIGHTED BERGMAN SPACES $$A_{\mu }^1$$

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