Compact Operators for Almost Conservative and Strongly Conservative Matrices

We obtain the necessary and sufficient conditions for an almost conservative matrix to define a compact operator. We also establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes(f,X), whereX=c,c0,l∞. These results are achieved by applying the Hausdorff measure of noncompactness.

θ-Almost summable sequences

King[3] introduced and examined the concepts of almost A-summable sequence, almost conservative matrix and almost regular matrix By following King, in this paper we introduce and examine the concepts ofθ-almost A-summable sequence,θ-almost conservative matrix andθ-almost regular matrix

$f$-CONSERVATIVE MATRIX SEQUENCES

The main purpose of this paper is to determine the necessary and sufficint conditions on the matrix sequence $\mathcal{A} = (A_p)$ in order that $\mathcal{A}$ contained in one of the classes $(f: f)$, $(f :f_s)$, $(f_s: f)$ and $(f_s: f_s)$, where $f$ and $f_s$ respectively denote the spares of all almost convergent real sequences and series. Our results are more general than those of Duran [3] and Solak [7]. Additionally, theorems of Steinhaus type concerning some subclasses of above matrix classes, are also given.

Submatrices of summability matrices

It is proved that a matrix that mapsℓ1intoℓ1can be obtained from any regular matrix by the deletion of rows. Similarly, a conservative matrix can be obtained by deletion of rows from a matrix that preserves boundedness. These techniques are also used to derive a simple sufficient condition for a matrix to sum an unbounded sequence.

On a summability theorem of Berg, Crawford and Whitley

We recall that a matrix A is said to sum a sequence x if Axε c, the space of all convergent sequences, and that A is conservative if it sums every convergent sequence. If A is conservative, A defines a continuous linear operator on c. Berg (2), Crawford (3)and Whitley (9) have proved the following theorem:Theorem 1. A conservative matrix sums no bounded divergent sequence if and only if,considered as an operator on c, it is range closed and has finite-dimensional null-spac