M. H. Faroughi

In this paper we introduce the generalized continuous version of fusion frame, namely $gc$-fusion frame. Also we get some new results about Bessel mappings and perturbation in this case.

Wenchang Sun in his paper [Wenchang Sun, $G$-frames and $g$-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437--452] has introduced $g$-frames which are generalized frames and include ordinary frames and many recent generalizations of frames, e.g., bounded quasi-projectors and frames of subspaces. In this paper we develop the $g$-frame theory for separable Hilbert spaces and give characterizations of $g$-frames and we show that $g$-frames share many useful properties with frames. We present a version of the Paley-Wiener Theorem for $g$-frames which is in spirit close to results for frames, due to Ole Christensen.

In the present paper we are going to introduce an operator-valued integral of a square modulus weakly integrable mappings the ranges of which are Hilbert spaces, as bounded operators. Then, we shall show that each operator-valued integrable mapping of the index set of an orthonormal basis of a Hilbert space $H$ into $H$ can be written as a multiple of a sum of three orthonormal bases.

For extending the concepts of $p$-frame, frame for Banach spaces and atomic decomposition, we will define the concept of $pg$-frame and $g$-frame for Banach spaces, by which each $f\in X$ ($X$ is a Banach space) can be represented by an unconditionally convergent series $f=\sum g_{i}\Lambda_{i},$ where $\{\Lambda_{i}\}_{i\in J}$ is a $pg$-frame, $\{g_{i}\}\in(\sum\oplus Y_{i}^{*})_{l_q}$ and $\frac{1}{p}+\frac{1}{q}=1$. In fact, a $pg$-frame $\{\Lambda_{i}\}$ is a kind of an overcomplete basis for $X^{*}.$ We also show that every separable Banach space $X$ has a $g$-Banach frame with bounds equal to $1.$

A variety of Banach algebras is a non-empty class of Banach algebras, for which there exists a family of laws such that its elements satisfy all of the laws. Each variety has a unique core (see [3]) which is generated by it. Each Banach algebra is not a core but, in this paper, we show that for each Banach algebra there exists a cardinal number (quantum of that Banach algebra) which shows the elevation of that Banach algebra for bearing a core. The class of all cores has interesting properties. Also, in this paper, we shall show that each core of a variety is generated by essential elements and each algebraic law of essential elements permeates to all of the elements of all of the Banach algebras belonging to that variety, which shows the existence of considerable structures in the cores.