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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 ${p}_{n}:X\to ℂ$ $\left(n\in ℕ\right)$, where $ℂ$ is the complex field and ${p}_{n}\left(x\right)={x}_{n}$ for all $x=\left({x}_{k}\right)\in X$. Let $A$ be an infinite matrix with complex entries ${a}_{nk}$ $\left(n,k\in ℕ\right)$ and let ${A}_{n}={\left({a}_{nk}\right)}_{k=0}^{\infty }$ be the sequence in the $n$th row of $A$ for every $n\in ℕ$. If $x=\left({x}_{k}\right)\in w$, then the $A$-transform of $x$ is the sequence $Ax={\left({A}_{n}\left(x\right)\right)}_{n=0}^{\infty }$, where ${A}_{n}\left(x\right)={\sum }_{k=0}^{\infty }{a}_{nk}{x}_{k}$ $\left(n\in ℕ\right)$, provided that the series on the right converges for each $n\in ℕ$. Let $X$ and $Y$ be subsets of $w$. Then $A$ defines a matrix mapping from $X$ into $Y$ if $A\left(x\right)$ 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 ${L}_{A}:X$ $\to$ $Y$ by ${L}_{A}\left(x\right)=A\left(x\right)$ for all $x\in X$. Let $\left(X,\parallel ·{\parallel }_{X}\right)$ be a $BK$-space, then the matrix domain ${X}_{T}=\left\{x\in w:Ax\in X\right\}$ is also a $BK$-space with the norm ${\parallel x\parallel }_{{X}_{T}}={\parallel Tx\parallel }_{X}$ 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 ${t}_{nn}\ne 0$ and ${t}_{nk}=0$ for all $k>n$ $\left(n\in ℕ\right)$. 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