Unconditionally Convergent Series

1997 ◽  
pp. 45-57 ◽  
Author(s):  
Mikhail I. Kadets ◽  
Vladimir M. Kadets
Keyword(s):  
Convergent Series ◽  
Unconditionally Convergent Series

Quasi-reflexivity, weakly unconditionally convergent series and weak compactness

1969 ◽  
Vol 182 (1) ◽  
pp. 45-54 ◽  
Author(s):  
J. R. Retherford ◽  
R. J. Fleming
Keyword(s):  
Convergent Series ◽  
Weak Compactness ◽  
Unconditionally Convergent Series

Quasi-reflexivity and weakly unconditionally convergent series

1967 ◽  
Vol 172 (2) ◽  
pp. 135-138 ◽  
Author(s):  
J. R. Retherford
Keyword(s):  
Convergent Series ◽  
Unconditionally Convergent Series

scholarly journals UNCONDITIONALLY CONVERGENT SERIES OF OPERATORS AND NARROW OPERATORS ON $L_1$

2005 ◽  
Vol 37 (02) ◽  
pp. 265-274 ◽  
Author(s):  
VLADIMIR KADETS ◽  
NIGEL KALTON ◽  
DIRK WERNER
Keyword(s):  
Convergent Series ◽  
Unconditionally Convergent Series

scholarly journals Completeness of a normed space via strong p-Cesàro summability

Filomat ◽  
2019 ◽  
Vol 33 (10) ◽  
pp. 3013-3022 ◽  
Author(s):  
F. Léon-Saavedra ◽  
S. Moreno-Pulido ◽  
A. Sala-Pérez
Keyword(s):  
Normed Space ◽  
Convergent Series ◽  
Cesàro Summability ◽  
Cesaro Summability ◽  
Weakly Unconditionally Cauchy Series ◽  
Unconditionally Convergent Series ◽  
Cauchy Series

In this paper we will characterize the completeness and barrelledness of a normed space through the strong p-Ces?ro summability of series. A new characterization of weakly unconditionally Cauchy series and unconditionally convergent series through the strong p-Ces?ro summability is obtained.

scholarly journals Banach Actions Preserving Unconditional Convergence

Axioms ◽  
2021 ◽  
Vol 11 (1) ◽  
pp. 13
Author(s):  
Taras Banakh ◽  
Vladimir Kadets
Keyword(s):  
Hilbert Space ◽  
Linear Functional ◽  
Orthonormal Basis ◽  
Convergent Series ◽  
Absolute Summability ◽  
Bounded Sequence ◽  
Unconditional Convergence ◽  
Diagonal Operators ◽  
The Absolute ◽  
Unconditionally Convergent Series

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

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