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\title[Compact Operators]
      {Compact Operators on Spaces of Strongly Bounded and Summable Sequences}

\author{Eberhard Malkowsky}
\address{Department of Mathematics, Faculty of Sciences, Fatih University,
         \newline
         34500 B\"uy\"uk\c{c}ekmece, Istanbul, Turkey}
\email{ema@Bankerinter.net}

\subjclass[2000]{Primary: 46A45 ; Secondary: 40H05}

\keywords{Sequence spaces, matrix transformations, Hausdorff measure of 
          noncompactness, compact and Fredholm operators}

\thanks{Research supported by the pesearch project $\#1440033$ of the 
        Serbian Ministry of Science, Technology and Development.}

\begin{abstract}
The sets $w_{0}^{p}$, $w^{p}$, and $w_{\infty}^{p}$ of sequences that are
strongly summable to zero, strongly summable, and strongly bounded, with
index $p$ for $1\le p<\infty$, by the Ces\`{a}ro method of order $1$ were
first introduced and studied by Maddox (\cite{emaabstr.MadKuttner}) who
characterised the class $(w^{p},c)$ of all matrix transformations from
$w^{p}$ into the space $c$ of all convergent sequences. We present the
complete list of characterisations of the classes of matrix transformations
from each one of Maddox's spaces into the classical sequence spaces, that is, 
into the sets $c_{0}$ of null sequences, $c$, $\ell_{\infty}$ of bounded 
sequences, and $\ell_{1}$, the set of all absolutely convergent series. 
(\cite{emmabastr.wpdelta,emaabstr.wpt}). We also give the representations
of the general linear operators from $w^{p}$ into any of the classical 
sequence spaces. Finally we compute or estimate the Hausdorff measure of 
noncompactness of linear operators between those spaces, and apply the results
to characterise the respective classes of compact operators.  
\end{abstract}

\maketitle

\begin{thebibliography}{99}

\bibitem{emmabastr.wpdelta}
B.Altay, F.Ba\c{s}ar, E. Malkowsky, 
Matrix transformations on some sequence spaces related to 
strong Ces\`{a}ro summability and boundedness,
{\it Applied Mathematics and Computation }{\bf 211(2) }(2009), 255--264

\bibitem{emaabstr.BM}
F. Ba\c{c}sar, E. Malkowsky, The characterisation of compact operators
on spaces of strongly summable and bounded sequences, 
{\it under communication}

\bibitem{emaabstr.wpt}
F.Ba\c{s}ar, E. Malkowsky, B.Altay, 
Matrix transformations on the matrix domains of triangles in the spaces of 
strongly $C_{1}-$ summable and bounded sequences, {\it Publ. Math. Debrecen}
{\bf 73/1-2 }(2008), 193--213

\bibitem{emaabstr.MadKuttner}
I.J.Maddox, On Kuttner's theorem, {\it J.London Math. Soc. }{\bf 43 }(1968), 
285--290
\end{thebibliography}

\end{document}
