# On sequence spaces defined by a sequence of moduli and an extension of Kuttner’s theorem

### Abstract

Let *A *= (*a _{nk}*) be an infinite matrix with lim

*∑*

_{n}

_{k}*a*≠ 0, and let

_{nk }*p*=(

*p*) be a sequence of positive real numbers. If 1≤

_{k}*p*<

_{k}*H*<∞, then the strong

*A*-summability field (with the exponent

*p*) is included in the summability field of

*A*. But the result known as Kuttner’s theorem asserts that if 0<

*p*=

_{k}*˜p*<1 and

*A*is regular, then there is a sequence which is strongly (

*C*,1)-summable but which is not

*A*-summable. This result was extended by Thorpe (cf. Theorem 6) and Maddox (cf. [8] and [9]). The purpose of the present paper is to extend the results of Thorpe and Maddox to a lacunary strong summability with respect to a sequence of modulus functions.