A survey on some paranormed sequence spaces

This paper presents a survey of most of the known fundamental results involving the sequence spaces l(p), c0(p), c(p) and l?(p), w0(p), w(p) and w?(p), f0(p) and f (p). These spaces are generalizations of the classical BK spaces lp, c0, c and l?, the spaces wp 0, wp and wp? of sequences that are strongly summable to zero, strongly summable and strongly bounded with index p by the Ces?ro method of order 1, and of almost null and almost convergent sequences, respectively. The results inlude the basic topological properties of the generalized spaces, the complete lists of their known ?-, ?-, ?-, functional and continuous duals, and the characterizations of many classes of matrix transformations between them, in particular, the complete list of characterizations of matrix transformations between the spaces l(p), c0(p), c(p) and l?(p). Furthermore, a great number of interesting special cases are included. The presented results cover a period of four decades. They are intended to inspire the inreasing number of researchers working in related topics, and to provide them with a comprehensive collection of results they may find useful for their work.

Euler-Ces`aro difference spaces of bounded, convergent and null sequences

In this paper, we introduce the spaces $\breve{\ell}_{\infty}$, $\breve{c}$ and $\breve{c}_{0}$ of Euler-Ces`aro bounded, convergent and null difference sequences and prove that the inclusions $\ell_{\infty}\subset\breve{\ell}_{\infty}$, $c\subset\breve{c}$ and $c_{0}\subset\breve{c}_{0}$ strictly hold. We show that the spaces $\breve{c}_{0}$ and $\breve{c}$ turn out to be the separable BK spaces such that $\breve{c}$ does not possess any of the following: AK property and monotonicity. We determine the alpha-, beta- and gamma-duals of the new spaces and characterize the matrix classes $(\breve{c}:\ell_{\infty})$, $(\breve{c}:c)$ and $(\breve{c}:c_0)$.

New approach to some results related to mixed norm sequence spaces

In this paper, the mixed norm sequence spaces ?p,q for 1 ? p,q ? ? are the subject of our research; we establish conditions for an operator T? to be compact, where T? is given by a diagonal matrix. This will be achieved by applying the Hausdorff measure of noncompactness and the theory of BK spaces. This problem was treated and solved in [5, 6], but in a different way, without the application of the theory of infinite matrices and BK spaces. Here, we will present a new approach to the problem. Some of our results are known and others are new.

Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

We study the spacesw0p,wp, andw∞pof sequences that are strongly summable to 0, summable, and bounded with indexp≥1by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaceswpinto the spacesw∞1,w1, andw01. We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators fromw0pandwpintow01andw1.

Measure of noncompactness for compact matrix operators on some BK spaces

In this paper, we characterize the matrix classes (?1, ??p )(1? p < 1). We also obtain estimates for the norms of the bounded linear operators LA defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.