محمد رضا امیدی

For the Frechet algebras (A, (p_k)) and (B, (q_k)) and n in N, n >=2, a linear map T from A in to B is called almost n-multiplicative, with respect to (p_k) and (q_k), if there exists "epsilon >= 0 such that q_k(T(a_1)(a_2)...( a_n)- Ta1Ta2 ... Tan) <= p_k(a1)p_k(a2) .... p_k(an); for each k 2 N and a_1; a_2; ...; a_n in A. The linear map T is called weakly almost n-multiplicative, if there exists " epsilon >= 0 such that for every k in N there exists n(k) in N with qk(Ta1a2 ....an - Ta1Ta2 ...Tan) <=p_n(k)(a1)p_n(k)(a2) .... p_n(k)(an); for each k 2 N and a_1; a_2;... ; a_n in A. The linear map T is called n-multiplicative if Ta1a2 .... an = Ta1Ta2 ... Tan; for every a1; a2; ... ; an in A. In this paper, we investigate automatic continuity of (weakly) almost n-multiplicative maps be- tween certain classes of Frechet algebras, including Banach algebras. We show that if (A; (pk)) is a Frechet algebra and T : A ! C is a weakly almost n-multiplicative linear functional, then either T is n-multiplicative, or it is continuous. Moreover, if (A; (pk)) and (B; (qk)) are Frechet algebras and T from A in to B is a continuous linear map, then under certain conditions T is weakly almost n- multiplicative for each n>=2. In particular, every continuous linear functional on A is weakly almost n-multiplicative for each n>=2.