# Article

Full entry | PDF   (0.3 MB)
Keywords:
BK spaces; bases; matrix transformations; measure of noncompactness
Summary:
In this paper we investigate linear operators between arbitrary BK spaces $X$ and spaces $Y$ of sequences that are $(\bar{N},q)$ summable or bounded. We give necessary and sufficient conditions for infinite matrices $A$ to map $X$ into $Y$. Further, the Hausdorff measure of noncompactness is applied to give necessary and sufficient conditions for $A$ to be a compact operator.
References:
[1] R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina and B. N. Sadovskii: Measures of noncompactness and condensing operators. Oper. Theory Adv. Appl. 55 (1992), Birkhäuser Verlag, Basel. DOI 10.1007/978-3-0348-5727-7_1 | MR 1153247
[2] A. M. Aljarrah and E. Malkowsky: BK spaces, bases and linear operators. Suppl. Rend. Circ. Mat. Palermo (2) 52 (1998), 177–191. MR 1644548
[3] J. Banás and K. Goebl: Measures of noncompactness in Banach spaces. Lecture Notes in Pure and Appl. Math. 60 (1980), Marcel Dekker, New York and Basel. MR 0566245
[4] G. H. Hardy: Divergent Series. Oxford University Press, 1973. MR 0030620
[5] E. Malkowsky: Linear operators in certain BK spaces. Bolyai Soc. Math. Stud. 5 (1996), 259–273. MR 1432674 | Zbl 0861.40007
[6] E. Malkowsky and S. D. Parashar: Matrix transformations in spaces of bounded and convergent difference sequences of order $m$. Analysis 17 (1997), 87–97. DOI 10.1524/anly.1997.17.1.87 | MR 1451207
[7] E. Malkowsky and V. Rakočević: The measure of noncompactness of linear operators between certain sequence spaces. Acta Sci. Math. (Szeged) 64 (1998), 151–170. MR 1631981
[8] E. Malkowsky and V. Rakočević: The measure of noncompactness of linear operators between spaces of $m^{th}$-order difference sequences. Studia Sci. Math. Hungar. 35 (1999), 381–395. MR 1762251
[9] V. Rakočević: Funkcionalna analiza. Naučna knjiga. Beograd, 1994.
[10] A. Wilansky: Summability through functional analysis. North-Holland Math. Stud. 85 (1984). MR 0738632 | Zbl 0531.40008

Partner of