FRAMES OF SUBSPACES FOR BANACH SPACES

Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
VARINDER KUMAR

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.

Author(s):  
SHALU SHARMA

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.


Author(s):  
P. K. JAIN ◽  
S. K. KAUSHIK ◽  
NISHA GUPTA

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.


2007 ◽  
Vol 38 (3) ◽  
pp. 267-276 ◽  
Author(s):  
S. K. Kaushik

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.


2010 ◽  
Vol 18 (1) ◽  
pp. 121-130
Author(s):  
Shiv K. Kaushik ◽  
Varinder Kumar

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.


1995 ◽  
Vol 38 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Zong-Ben Xu ◽  
Yao-Lin Jiang ◽  
G. F. Roach

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.


1986 ◽  
Vol 34 (1) ◽  
pp. 87-92
Author(s):  
M. A. Ariño

Necessary and sufficient condition are given for an infinite dimensional subspace of a p-Banach space X with basis to contain a basic sequence which can be extended to a basis of X.


Author(s):  
Khole Timothy Poumai ◽  
Shah Jahan

Gavruta [L. Gavruta, Frames for operators, Appl. Comput. Harmon. Anal. 32 (2012) 139–144] introduced the notion of [Formula: see text]-frame and atomic system for an operator [Formula: see text] in Hilbert spaces. We extend these notions to Banach spaces and obtain various new results. A necessary and sufficient condition for the existence of an atomic system for an operator [Formula: see text] in a Banach space is given. Also, a characterization for the family of local atoms of subspaces of Banach spaces has been given. Further, we give methods to construct an atomic system for an operator [Formula: see text] from a given Bessel sequence and an [Formula: see text]-Bessel sequence. Finally, a result related to stability of atomic system for an operator [Formula: see text] in a Banach space has been given.


2004 ◽  
Vol 69 (1) ◽  
pp. 1-18 ◽  
Author(s):  
Tomonari Suzuki

In this paper, we discuss a necessary and sufficient condition for common fixed points of two nonexpansive mappings. We then prove a convergence theorem to a common fixed point. Finally, we discuss the existence of a nonexpansive retraction onto the set of common fixed points of nonexpansive mappings. In these theorems, we do not assume the strict (uniform) convexity of the norm of the Banach space.


2005 ◽  
Vol 178 ◽  
pp. 55-61 ◽  
Author(s):  
Guantie Deng

Let α be a nonnegative continuous function on ℝ. In this paper, the author obtains a necessary and sufficient condition for polynomials with gaps to be dense in Cα, where Cα is the weighted Banach space of complex continuous functions ƒ on ℝ with ƒ(t) exp(−α(t)) vanishing at infinity.


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