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Matrix summability of statistically convergent sequences. (English) Zbl 0801.40005
Let K be an index set and ϕ K :=(ϕ j K ) be defined by ϕ j K =1 (jk), ϕ j K =0 otherwise. For a non-negative regular (cc) matrix A, the A- density of K is δ A (K):=lim A ϕ K whenever ϕ K c A . For a sequence x=(x k ) k1 and a number x 0 , denote K ε :={k: |x k -x 0 |ε}; then x is said to be A-statistically convergent to x 0 (xst A ) if δ A (K ε )=0 for every ε>0. We also denote the K-section of x as x [K] , where x k [K] =x k (kK), x k [K] =0 otherwise, and a sequence space X is called section-closed if x [K] x for all xX and for every index set K. The K-column-section of a matrix B is denoted by B [K] :=(b nk [K] ), where b nk [K] =b nk (kK), b nk [K] =0 otherwise. For any two sequence spaces W, Y, we denote by (W,Y) the set of all matrices H which map W into Y (WY H ). The main Theorem 4.1 shows: Let X be a section- closed sequence space containing e:=(1,1,) and let Y be an arbitrary sequence space; then B(Xst A ,Y) if and only if B(Xc,Y) and B [K] (X,Y) (δ A (K)=0). There are some modifications and corollaries, and a final section which considers matrix maps of statistically convergent bounded sequences (where we take X= ).

MSC:
40C05Matrix methods in summability
40D25Inclusion theorems; equivalence theorems
40J05Summability in abstract structures
46A45Sequence spaces
40A05Convergence and divergence of series and sequences