On fusion frames in Banach spaces

2010 ◽  
Vol 18 (1) ◽  
pp. 121-130
Author(s):  
Shiv K. Kaushik ◽  
Varinder Kumar
Keyword(s):  
Banach Spaces ◽  
Complete Sequence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Fusion Frames ◽  
Banach Frame ◽  
Necessary And Sufficient

Abstract A necessary and sufficient condition for a complete sequence of subspaces to be a fusion Banach frame for E is given. Also, we introduce fusion Banach frame sequences and give a characterization for a complete sequence of subspaces of E to be a fusion Banach frame for E in terms of fusion Banach frame sequences. Finally, along with other results, we characterize fusion Banach frames in terms of Banach frames.

ON BI-BANACH FRAMES IN BANACH SPACES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450015 ◽  
Author(s):  
SHALU SHARMA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Banach Frame ◽  
Necessary And Sufficient

Bi-Banach frames in Banach spaces have been defined and studied. A necessary and sufficient condition under which a Banach space has a Bi-Banach frame has been given. Finally, Pseudo exact retro Banach frames have been defined and studied.

scholarly journals Some results concerning frames in Banach spaces

2007 ◽  
Vol 38 (3) ◽  
pp. 267-276 ◽  
Author(s):  
S. K. Kaushik
Keyword(s):  
Finite Number ◽  
Banach Spaces ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Linearly Independent ◽  
Banach Frame ◽  
Minimal Sequence ◽  
Definition Of ◽  
Necessary And Sufficient

A necessary and sufficient condition for the associated sequence of functionals to a complete minimal sequence to be a Banach frame has been given. We give the definition of a weak-exact Banach frame, and observe that an exact Banach frame is weak-exact. An example of a weak-exact Banach frame which is not exact has been given. A necessary and sufficient condition for a Banach frame to be a weak-exact Banach frame has been obtained. Finally, a necessary condition for the perturbation of a retro Banach frame by a finite number of linearly independent vectors to be a retro Banach frame has been given.

ON FRAME SYSTEMS IN BANACH SPACES

2009 ◽  
Vol 07 (01) ◽  
pp. 1-7 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
NISHA GUPTA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Bessel Sequence ◽  
Banach Frame ◽  
Frame System ◽  
Frame Systems ◽  
Necessary And Sufficient

Banach frame systems in Banach spaces have been defined and studied. A sufficient condition under which a Banach space, having a Banach frame, has a Banach frame system has been given. Also, it has been proved that a Banach space E is separable if and only if E* has a Banach frame ({φn},T) with each φn weak*-continuous. Finally, a necessary and sufficient condition for a Banach Bessel sequence to be a Banach frame has been given.

FRAMES OF SUBSPACES FOR BANACH SPACES

2010 ◽  
Vol 08 (02) ◽  
pp. 243-252 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
VARINDER KUMAR
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Compactly Generated

Frames of subspaces for Banach spaces have been introduced and studied. Examples and counter-examples to distinguish various types of frames of subspaces have been given. It has been proved that if a Banach space has a Banach frame, then it also has a frame of subspaces. Also, a necessary and sufficient condition for a sequence of projections, associated with a frame of subspaces, to be unique has been given. Finally, we consider complete frame of subspaces and prove that every weakly compactly generated Banach space has a complete frame of subspaces.

ON PERTURBATION OF BANACH FRAMES

2006 ◽  
Vol 04 (03) ◽  
pp. 559-565 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
L. K. VASHISHT
Keyword(s):  
Linear Combination ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Banach Frames ◽  
Finite Linear Combination ◽  
Linearly Independent ◽  
Banach Frame ◽  
Necessary And Sufficient ◽  
Finite Linear

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

scholarly journals A further necessary and sufficient condition for strong convergence of nonlinear contraction semigroups and of iterative methods for accretive operators in Banach spaces

1995 ◽  
Vol 38 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Zong-Ben Xu ◽  
Yao-Lin Jiang ◽  
G. F. Roach
Keyword(s):  
Banach Space ◽  
Strong Convergence ◽  
Banach Spaces ◽  
Iterative Methods ◽  
Accretive Operator ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Accretive Operators ◽  
Uniformly Smooth ◽  
Necessary And Sufficient

Let A be a quasi-accretive operator defined in a uniformly smooth Banach space. We present a necessary and sufficient condition for the strong convergence of the semigroups generated by – A and of the steepest descent methods to a zero of A.

ON SOME CLASSES OF SCHAUDER FRAMES IN BANACH SPACES

2014 ◽  
Vol 12 (02) ◽  
pp. 1450022
Author(s):  
KHOLE TIMOTHY POUMAI
Keyword(s):  
Banach Spaces ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Schauder Frame

Various types of Schauder frames have been defined and studied. A necessary and sufficient condition for each type of Schauder frame is given. Finally, we give some theoretical applications of these types of Schauder frames.

scholarly journals ON NEAR EXACT BANACH FRAMES IN BANACH SPACES

2008 ◽  
Vol 78 (2) ◽  
pp. 335-342 ◽  
Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
NISHA GUPTA
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Sufficient Condition ◽  
Banach Frames ◽  
Banach Frame

AbstractNear exact Banach frames are introduced and studied, and examples demonstrating the existence of near exact Banach frames are given. Also, a sufficient condition for a Banach frame to be near exact is obtained. Further, we consider block perturbation of retro Banach frames, and prove that a block perturbation of a retro Banach frame is also a retro Banach frame. Finally, it is proved that if E and F are both Banach spaces having Banach frames, then the product space E×F has an exact Banach frame.

Block sequences and g-frames

2015 ◽  
Vol 13 (02) ◽  
pp. 1550009
Author(s):  
Renu Chugh ◽  
S. K. Sharma ◽  
Shashank Goel
Keyword(s):  
Hilbert Spaces ◽  
Operator Theory ◽  
American Mathematical Society ◽  
Mathematical Society ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Block Sequence ◽  
Fusion Frames ◽  
Frame System ◽  
Necessary And Sufficient

Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.

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