Isometric multipliers of a vector valued Beurling algebra on a discrete semigroup

Let (S,ω) be a weighted abelian semigroup, let M _ω (S) be the semigroup of ω-bounded multipliers of S, and let (mathcal {A}) be a strictly convex commutative Banach algebra with identity. It is shown that T is an onto isometric multiplier of (ell ^{1}(S,omega , mathcal {A})) if and only if there exists an invertible σ ∈ M _ω (S), a unitary point (a in mathcal {A}), and a k>0 such that (T(f)= ka{sum }_{x in S} f(x)delta _{sigma (x)}) for each (f={sum }_{x in S}f(x)delta _{x} in ell ^{1}(S,omega ,mathcal {A})). It is also shown that an isomorphism from (ell ^{1}(S_{1},omega _{1},mathcal {A})) onto (ell ^{1}(S_{2},omega _{2}, mathcal {B})) induces an isomorphism from (M(ell ^{1}(S_{1},omega _{1},mathcal {A}))), the set of all multipliers of (ell ^{1}(S_{1},omega _{1},mathcal {A})), onto (M(ell ^{1}(S_{2},omega _{2},mathcal {B}))).