On a Generalization of the Hilbert Frame Generated by the Bilinear Mapping

Abstract

The concept ofb-frame which is a generalization of the frame in Hilbert spaces generated by the bilinear mapping is considered.b-frame operator is defined; analogues of some well-known results of frame theory are obtained in Hilbert spaces. Conditions for the existence ofb-frame in Hilbert spaces are obtained; the relationship between the definite bounded surjective operator andb-frame is also studied. The concept ofb-orthonormalb-basis is introduced.

On Bessel and Riesz-Fisher systems with respect to Banach space of vector-valued sequences

“…k=1 k g k (z)) = ∞ k=1 Se k g k (z) = ∞ k=1 Λ k g k (z).Equality(16) follows immediately from (15). Further we haveS * (f ) = {Λ * k (f )} k∈N , ∀f ∈ Z * .…”

“…The g-frames were also considered in [13,14]. Another generalization of frames in Hilbert spaces was studied in [15,16], where the notion of t-frames was introduced and many properties and noetherian perturbation of such frames were studied. The Banach generalizations of frames were first studied by K. Gröchenig in [17]; in this paper, the notions of Banach frame and atomic decomposition were introduced.…”

“…Thus, {g k } k∈N , U −1 is a BanachX-frame in Z. By Theorem 10 the system {Λ k } k∈N ⊂ L(X, Z), Λ k = U −1 e k formsX * -frame in Z * with bounds 1 B , 1 A and equalities (15) and (16) are fulfilled. It is clear that g k Λ j = δ kj .…”

$K$-$b$-frames for Hilbert spaces and the $b$-adjoint operator

“…Definition 1.2. [9]. {x i } i∈N is said to be a ♭-basis for Z if for every z ∈ Z, there exists a unique sequence…”

“…It is a concept where we are able to generate Hilbert frames by a bilinear mapping b : H 1 × B → H 2 , such that H 1 , H 2 are two Hilbert spaces and B is a Banach space, and via this bilinear mapping we construct a new product that we will call the b−dual product to define a new frame for a Hilbert space, but this time not from the Hilbert space itself; it is from the Banach space B. Moreover M.Ismailov, F.Guliyeva, and Y.Nasibov extended some well known and important results of frame theory existing in the classical case to this more general one (see [9]). Except that the definition was incomplete, and there was no example given.…”