Frames and operators in Hilbert C^✻-modules

Abstract

Abstract. In this paper we introduce the concepts of atomic systems for operators and K -frames in Hilbert C * -modules and we establish some results.Mathematics subject classification (2010): 42C15, 46L05, 46H25.

Controlled $K$-frames in Hilbert $C^*$-modules

“…Definition 2.4. [18] A sequence {ψ j } j∈J of elements in a Hilbert A-module H is said to be a K-frame (K ∈ L(H)) if there exist constants C, D > 0 such that…”

“…If C = I, the C-controlled K-frame {ψ j } j∈J is simply K-frame in H which was discussed in [18]. The sequence {ψ j } j∈J is called a C-controlled Bessel sequence with bound B, if there…”

“…Rahimi [19] defined the concept of controlled K-frames in Hilbert spaces and showed that controlled K-frames are equivalent to K-frames due to which the controlled operator C can be used as preconditions in applications. In [18], Najati et al introduced the concepts of atomic system for operators and K-frames in Hilbert C * -modules. Controlled frames in Hilbert C * -modules were introduced by Rashidi and Rahimi [17], and the authors showed that they share many useful properties with their corresponding notions in a Hilbert space.…”

Continuous $K$-$g$-frames in Hilbert $C^*$-modules

“…Afterwards some generalizations of frames for Hilbert C * -modules have attracted much more attention in recent years ( for example, see [3,12,9] and the bibliography there in). The notion of a g-frame for Hilbert C * -modules was introduced by authors in [5,9] and the notion of a K-frame for Hilbert C * -modules, for an operator K ∈ End * A (H), where End * A (H) denotes the set of all adjointable A-linear maps on a Hilbert space H was introduced by the authors in [11]. In what follows: in section 2, as a generalization by combing these extensions, we consider the fusion of two concepts and introduce continuous K-g-frames (simply, c-K-g-frames) for Hilbert C * -modules.…”

G-frames for operators in Hilbert $C^{\ast}$-modules

“…Recently, Najati et al [13] generalized the notion of frames for operators from Hilbert spaces to Hilbert C * -modules and studied some of their properties. In this paper we give a generalization of g-frames for operators in Hilbert C * -modules and we extend some results in [2] to Hilbert C * -modules.…”