A note on compact operators on matrix domains

Abstract

We establish some identities or estimates for the operator norms and Hausdorff measures of noncompactness of linear operators given by infinite matrices that map the matrix domains of triangles in arbitrary BK spaces with AK, or in the spaces of all convergent or bounded sequences, into the spaces of all null, convergent or bounded sequences, or of all absolutely convergent series. Furthermore, we apply these results to the characterizations of compact operators on the matrix domains of triangles in the classical sequence spaces, and on the sequence spaces studied in [I. Djolović, Compact operators on the spaces a r 0 ( ) and a r c ( ),

On compact operators on some spaces related to matrix B(r, s)

“…(4.9) Further, applying (Lemma 1, [7]), we obtain the following: [4,Theorem 3.7] or use the result [5,Theorem 2.8].…”

On compact operators and some EulerB(m)-difference sequence spaces

“…In this section, we will show that L A is a compact operator for every matrix A ∈ (( ∞ ) T , c 0 ) or A ∈ (( ∞ ) T , c) by applying the Hausdorff measure of noncompactness and using some results in [11,13,14], where L A (x) = Ax for all x ∈ ( ∞ ) T . First we give some notations, definitions and well-known results.…”

“…In this paper, we go on defining some new Euler sequence spaces by using generalized difference matrix order m and give the some topological properties of these spaces. Also, we characterize some compact matrix classes between these spaces by applying the Hausdorff measure of noncompactness and using some results in [11][12][13][14].…”

Compact operators into the spaces of strongly summable and bounded sequences

“…An important application of the Hausdorff measure of noncompactness of bounded linear operators between Banach spaces is the characterisation of compact matrix transformations between BK spaces [11][12][13][14][15][16][17][18][19][20][21][22]. The characterisations of compact matrix operators between the classical sequence spaces in almost all cases can be found in [23].…”

“…Results characterising compact operators L ∈ B(X, Y ) can be applied in the theory of Fredholm operators, for instance, to establish sufficient conditions for L to be a Fredholm operator, as in [16,21]. One more application is in studies on the invertibility of operators and the solvability of infinite systems of linear equations, for instance, in [25,26].…”