Let $\displaystyle V = W_1 \oplus W_2, \phi: V \rightarrow V $a linear map. Prove that $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.

$ Injectivity is easy enough, but I can't seem to prove $\displaystyle V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+ $ without assuming that $\displaystyle W_1, W_2 $ are $\displaystyle \phi-$stable. Am I missing something here, or is this assumption required for the implication to hold?