Monatshefte f'fir Mathematik 79, 307--315 (1975) 9 by Springer-Verlag 1975 Summability Factors for Lower-Semi-Matrix Transformations By R. N. Mohapatra, Beirut, Lebanon, and (I. Das, Sambalpur, India (Received 17 January 1973; in revised ]orm 5 February 1974) Abstract In view of Kogbetliantz's identity [7] the absolute Cess summabflity of order k (k>--l) of an infinite series Xan is the same as the absolute is the n-th Ces~ro mean of order k of convergence of 27 (v~)n -1 where T~ sequence {nan}. ]:)As [5] has shown that similar dependence is true for certain classes of N6rlund means. The object of this paper is to establish two theorems on absolute summability factors involving two lower-semi- matrix transformations and thereby to genera]ise a result of CHOW [3] on absolute CesKro summability factors and a result of BOSAI~TQUET and DAS [1] on absolute Harmonic summability factors. 1. Definitions and Notations. Let 11 be the space of all complex u ~ suchthat~f_0 sequences u = { n}n = 0 ]un I < oc. For given sequences u and v, we let uv denote the componentwise product. Let A--= (a~) and B--= (b~) be two normal infinite matrices i. e. lower-semi- matrices with non-vanishing diagonals. Concerning normal matrices it is known that (see COOKE [2]) for a normal matrix A its two sided inverse A'--= (a~)exists and is also a normal matrix. For a matrix A, we denote by Au, the A-transform of the sequence u i. e. (A u)~ = Y~a~7~u~. Let ~~ be a given infinite series with s=(sn} for its n-th partial sum. The given series is said to be absolutely summable by method A or summable [A[ if ((As)n}eBV i.e. ~](As)n--(As)n_ll<oo. n=l In what follows, we associate with the given matrix A, two other lower-semi-matrices A--=(5"~) and d--(dn~)