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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.

47B37Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
47H08Measures of noncompactness and condensing mappings, $K$-set contractions, etc.
46B15Summability and bases in normed spaces
46B45Banach sequence spaces
Full Text: DOI
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