Narrow Results

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.

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 ({φ_n},T) with each φ_n 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 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.

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.

Near exact fusion Banach frames have been discussed with the help of examples. Further, it has been proved that if a Banach space has a fusion Banach frame then it has a normalized tight and exact fusion Banach frame. In the sequel, we consider block perturbation of fusion Banach frames and proved that a block perturbation of a fusion Banach frame is also a fusion Banach frame. Some stability results for fusion Banach frames have also been obtained. Finally, we give an application of near exact fusion Banach frame.

In this paper we consider perturbation of several frame-concepts in separable Banach spaces. We determine stability conditions with sharp bounds and discuss the necessity of some of them. Further, we investigate equivalence between several perturbation conditions. In particular, a result for tight frames for Hilbert spaces is obtained.