Let E and F be locally convex spaces and G their completed e-tensor product. It is shown that if S and T are weakly almost periodic equicontinuous semigroups of operators on E and F respectively, then, under mild restrictions on E or F, SoT is a weakly almost periodic equicontinuous semigroup of operators on G, and the almost periodic and flight vector subspaces of G are related in a natural way to the corresponding subspaces of E and Fvia the e-tensor product. Furthermore, if Eand Fboth decompose into a direct sum of these subspaces then so does G. 1. Weakly almost periodic semigroups. Let E be a locally convex (Hausdorff) linear topological space with topological dual E', and let L(E) denote the space of continuous linear operators on E. A semigroup of operators on E is a subset S of L(E) containing the identity operator and closed under composition. A vector x E E is said to be weakly (strongly) almost periodic under S if its orbit Sx= {ux: u E S} is relatively compact in the weak (strong) topology of E. The set of all weakly (strongly) almost periodic vectors in E shall be denoted by W(E, S) (A(E, S)). Occasionally we shall suppress the symbols E or S from the notation if they are understood from context. If W(E, S)=E (A(E, S)=E) we say that S is weakly (strongly) almost periodic. It is easily seen that multiplication in a semigroup of operators S on a locally convex space E is separately continuous with respect to the weak or strong operator topologies on L(E), that is to say S is a topological semigroup. Moreover if S is equicontinuous then multiplication is actually jointly continuous in the strong operator topology. The following lemma is at the heart of the theory of weak almost periodicity. A proof can be found in [1]. LEMMA 1.1. Let S be a weakly almost periodic equicontinuous semigroup of operators on a locally convex space E, and let 3 denote the closure Received by the editors March 26, 1973 and, in revised form June 21, 1973. AMS (MOS) subject classiflcations (1970). Primary 47D05, 46M05; Secondary 43A60, 22A15.