zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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 (n), where is the complex field and p n (x)=x n for all x=(x k )X. Let A be an infinite matrix with complex entries a nk (n,k) and let A n =(a nk ) k=0 be the sequence in the nth row of A for every n. If x=(x k )w, then the A-transform of x is the sequence Ax=(A n (x)) n=0 , where A n (x)= k=0 a nk x k (n), provided that the series on the right converges for each 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 xX. Let φ be the set of all finite complex sequences that terminate in zeros. If Xφ and Y are BK-spaces, then every infinite matrix A that maps X into Y defines a continuous linear operator L A :X Y by L A (x)=A(x) for all xX. Let (X,· X ) be a BK-space, then the matrix domain X T ={xw:AxX} is also a BK-space with the norm x X T =Tx X for all xX 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 Xφ into the BK-spaces c 0 ,c,l and l 1 , and into the matrix domains c 0 T ,c T ,l T l 1 T of infinite triangles matrices T, i.e., such that the complex entries of T satisfy t nn 0 and t nk =0 for all k>n (n). Further, the authors give necessary and sufficient (or only sufficient) conditions for such operators to be compact.

MSC:
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