Necessary and sufficient conditions have been established on an infinite matris A = (ap]^) to transform the generalized segnence spaces 1 (p), Cp(p) and 1„ (p) into, q, the spa,ce of semipe­ riodic seguences. The special case when pjj = 1 /t gives Cp(p) and Ijo (p) es T and ,T* the spaces introdueed by V. Ganapathy lyer [2].