Isometric shifts on

Let c be the set of all convergent sequences with the sup norm and X a strictly convex Banach space. Let Q be the natural projection from ( c ⊕ X ) ∞ to X and let T be an isometric shift on ( c ⊕ X ) ∞ . We prove the following:(1) striction of Q ∘ T to X is a surjective isometry. c into c and the restriction of T to c is an isometric shift. so show that the space ( c ⊕ ℓ 1 ) ∞ admits an isometric shift.