On the similarity of dual operators

Let k be a field with an involution σ and 〈,〉:V×W→k a non-degenerate sesquilinear form, where V,W are n-dimensional k-spaces. Assume that Λ∈End(V) and Λ∗∈End(W) are dual operators. We show that if Λ and Λ∗ are similar, then Λ∗=ϕΛϕ-1, where ϕ:V→W is Hermitian.