On Perturbation of Weighted G−Banach Frames in Banach Spaces

In the present paper, we study perturbation of weighted <em>g</em>−Banach frames in Banach spaces and obtain perturbation results for weighted <em>g</em>−Banach frames. Also, sufficient conditions for the perturbation of weighted <em>g</em>−Banach frames by positively confined sequence of scalars and uniformly scaled version of a given weighted <em>g</em>−Banach Bessel sequence have been given. Finally, we give a condition under which the sum of finite number of sequences of operators is a weighted <em>g</em>−Banach frame by comparing each of the sequences with another system of weighted <em>g</em>−Banach frames in Banach spaces.

Some characterizations of fusion Banach frames

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.

ON BI-BANACH FRAMES IN BANACH SPACES

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.

On fusion frames in Banach spaces

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.

FRAMES OF SUBSPACES FOR BANACH SPACES

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 FRAME SYSTEMS IN BANACH SPACES

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.

ON NEAR EXACT BANACH FRAMES IN BANACH SPACES

Near 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.

Some results concerning frames in Banach spaces

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.

GROUP THEORETICAL APPROACH TO QUANTUM ENTANGLEMENT AND TOMOGRAPHY WITH WAVELET TRANSFORM IN BANACH SPACES

The intimate connection between the Banach space wavelet reconstruction method for each unitary representation of a given group and some of well-known quantum tomographies, such as tomography of rotation group, spinor tomography and tomography of unitary group, is established. Also both the atomic decomposition and Banach frame nature of these quantum tomographic examples are revealed in detail. Finally, we consider separability criteria for any state with group theoretical wavelet transform on Banach spaces.

ON PERTURBATION OF BANACH FRAMES

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.