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Compactness by the Hausdorff measure of noncompactness. (English) Zbl 1211.47061
A linear subspace $X$ of the space of all complex sequences, denoted by $w$, is called a $BK$-space if it is a Banach space with continuous coordinates $p_{n}: X \to \mathbb{C}$ $(n\in \mathbb{N})$, where $\mathbb{C}$ is the complex field and $p_{n}(x)=x_{n}$ for all $x=(x_{k})\in X$. Let $A$ be an infinite matrix with complex entries $a_{nk}$ $(n,k\in \mathbb{N})$ and let $A_{n}=(a_{nk}) _{k=0}^{\infty }$ be the sequence in the $n$th row of $A$ for every $n\in \mathbb{N}$. If $x=(x_{k}) \in w$, then the $A$-transform of $x$ is the sequence $Ax=( A_{n}( x))_{n=0}^{\infty }$, where $A_{n}(x)= \sum_{k=0}^\infty a_{nk} x_{k}$ $(n\in \mathbb{N})$, provided that the series on the right converges for each $n\in {\mathbb{N}}$. Let $X$ and $Y$ be subsets of $w$. Then $A$ defines a matrix mapping from $X$ into $Y$ if $A(x)$ exists and is in $Y$ for all $x\in X$. Let $\phi$ be the set of all finite complex sequences that terminate in zeros. If $X\supset \phi$ and $Y$ are $BK$-spaces, then every infinite matrix $A$ that maps $X$ into $Y$ defines a continuous linear operator $L_{A}:X$ $\rightarrow$ $Y$ by $L_{A}(x)= A(x)$ for all $x\in X$. Let $(X,\Vert \cdot\Vert _{X})$ be a $BK$-space, then the matrix domain $X_{T}=\{ x\in w:Ax\in X\}$ is also a $BK$-space with the norm $\Vert x\Vert _{X_{T}}=\Vert Tx\Vert _{X}$ for all $x\in X_{T}$. In the paper under review, the authors prove some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain operators $L_{A}$ that map an arbitrary $BK$-space $X\supset \phi$ into the $BK$-spaces $c_{0},c,l_{\infty }$ and $l_{1},$ and into the matrix domains $c_{0_{T}},c_{T},l_{\infty _{T}}l_{1_{T}}$ of infinite triangles matrices $T$, i.e., such that the complex entries of $T$ satisfy $t_{nn}\neq 0$ and $t_{nk}=0$ for all $k>n$ $( n\in \mathbb{N})$. Further, the authors give necessary and sufficient (or only sufficient) conditions for such operators to be compact.

##### MSC:
 47B37 Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47H08 Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 46B15 Summability and bases in normed spaces 46B45 Banach sequence spaces
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