A new notion of frames, called the relay fusion frames, for Hilbert spaces has been introduced by the authors. It provides a mathematical framework for applications that require the transmission of signals over long distances or need to expand the coverage of wireless networks. The technique described in this paper is not only a natural technique suitable for the applications of relay communication systems, but also can be regarded as a natural generalization of fusion frames or even g-frames. We transfer some common properties in general frames and fusion frames to relay fusion frames with the definition of the relay fusion frames and their operators. In particular, besides canonical duality, we obtain two new dualities of the relay fusion frames. Moreover, we prove that relay fusion frames are stable under small perturbations.

The relay fusion frames have been recently introduced in Hilbert spaces to model sensor relay networks and distributed sensor relay systems, which are deeply connected with compressed sensing. In this article, we introduce the notions of relay fusion frames and Banach relay fusion frames in Banach spaces and study certain attractive properties of relay fusion frames in this more general setting. In a particular sense, Schauder frames can be shown to be a special case of relay fusion frames. Moreover, the stability issue of relay fusion frames and Banach relay fusion frames will be addressed.

For two given bounded adjointable operators T and S between Hilbert C*-modules, it is well known that an operator Moore-Penrose inverse exists iff the operator has closed range. In this paper, we give some formulas for the Moore-Penrose inverses of products T S.

No abstract

In this paper, we mainly give some quadratic refinements of Young type inequalities. Namely:$$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-v{{\sum\limits_{j=1}^N}}2^{j}\Big(b- \sqrt[2^{j}]{ab^{2^{j}-1} }\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+v^{2}(a-b)^{2} \end{array}$$for v ∉ [0, $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$], N ∈ ℕ, a, b > 0; and$$\begin{array}{} \displaystyle (va+(1-v)b)^{2}-(1-v){{\sum\limits_{j=1}^N}}2^{j}\Big(a- \sqrt[2^{j}]{a^{2^{j}-1}b}\, \Big)^{2}\leq(a^{v}b^{1-v})^{2}+(1-v)^{2}(a-b)^{2} \end{array}$$for v ∉ [1 − $\begin{array}{} \displaystyle \frac{1}{2^{N+1}} \end{array}$, 1], N ∈ ℕ, a, b > 0. As an application of these scalars results, we obtain some matrix inequalities for operators and Hilbert-Schmidt norms.