Controlled generalized fusion frame in the tensor product of Hilbert spaces

We present controlled by operators generalized fusion frame in the tensor product of Hilbert spaces and discuss some of its properties. We also describe the frame operator for a pair of controlled $g$-fusion Bessel sequences in the tensor product of Hilbert spaces.

Binary Operations in Metric Spaces Satisfying Side Inequalities

The theory of metric spaces is a convenient and very powerful way of examining the behavior of numerous mathematical models. In a previous paper, a new operation between functions on a compact real interval called fractal convolution has been introduced. The construction was done in the framework of iterated function systems and fractal theory. In this article we extract the main features of this association, and consider binary operations in metric spaces satisfying properties as idempotency and inequalities related to the distance between operated elements with the same right or left factor (side inequalities). Important examples are the logical disjunction and conjunction in the set of integers modulo 2 and the union of compact sets, besides the aforementioned fractal convolution. The operations described are called in the present paper convolutions of two elements of a metric space E. We deduce several properties of these associations, coming from the considered initial conditions. Thereafter, we define self-operators (maps) on E by using the convolution with a fixed component. When E is a Banach or Hilbert space, we add some hypotheses inspired in the fractal convolution of maps, and construct in this way convolved Schauder and Riesz bases, Bessel sequences and frames for the space.

Some Properties of Canonical Dual K-Bessel Sequences for Parseval K-Frames

The concept of canonical dual K-Bessel sequences was recently introduced, a deep study of which is helpful in further developing and enriching the duality theory of K-frames. In this paper we pay attention to investigating the structure of the canonical dual K-Bessel sequence of a Parseval K-frame and some derived properties. We present the exact form of the canonical dual K-Bessel sequence of a Parseval K-frame, and a necessary and sufficient condition for a dual K-Bessel sequence of a given Parseval K-frame to be the canonical dual K-Bessel sequence is investigated. We also give a necessary and sufficient condition for a Parseval K-frame to have a unique dual K-Bessel sequence and equivalently characterize the condition under which the canonical dual K-Bessel sequence of a Parseval K-frame admits a unique dual K⁎-Bessel sequence. Finally, we obtain a minimal norm property on expansion coefficients of elements in the range of K resulting from the canonical dual K-Bessel sequence of a Parseval K-frame.

K-g-frames and their dual

This paper is devoted to the study of the dual [Formula: see text]-g-Bessel sequences of [Formula: see text]-g-frames. We firstly make use of the g-preframe operators of a g-Bessel sequence to investigate the constructions of [Formula: see text]-g-frames. And then, taking the g-preframe operators into account, we present several necessary and sufficient conditions under which a g-Bessel sequence is the dual [Formula: see text]-g-Bessel sequence of a given [Formula: see text]-g-frame. We also give a simple example to show that the duality of [Formula: see text]-g-frames is not transitive.

Frames, biorthogonal dual and other properties associated with wavelet system on ℝ

It is well known that the system of translates in [Formula: see text] has been characterized as Bessel sequences, frames and Riesz bases in terms of the Fourier transform. The aim of this paper is to obtain similar types of characterization for the wavelet system emerging out of integer translations and dyadic dilations. The existence of the biorthogonal dual system and nonredundancy properties of the wavelet system are also investigated here.

More on Inequalities for Weaving Frames in Hilbert Spaces

In this paper, we present several new inequalities for weaving frames in Hilbert spaces from the point of view of operator theory, which are related to a linear bounded operator induced by three Bessel sequences and a scalar in the set of real numbers. It is indicated that our results are more general and cover the corresponding results recently obtained by Li and Leng. We also give a triangle inequality for weaving frames in Hilbert spaces, which is structurally different from previous ones.

Some Inequalities for g-Frames in Hilbert C*-Modules

In this paper, we obtain new inequalities for g-frames in Hilbert C * -modules by using operator theory methods, which are related to a scalar λ ∈ R and an adjointable operator with respect to two g-Bessel sequences. It is demonstrated that our results can lead to several known results on this topic when suitable scalars and g-Bessel sequences are chosen.