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?
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?
You need to define some things here friend. Presumably $\displaystyle S(\lambda)$ is the eigenspace of the eigenvalue $\displaystyle \lambda$? What does the $\displaystyle ^+$ mean?
3. Whoops my bad. $\displaystyle V(\lambda)^+$ is defined as the generalized eigenspace of the eigenvalue $\displaystyle \lambda$, i.e. $\displaystyle V(\lambda)^+ := \{v \in V | (\phi - \lambda .Id)^n (v)= 0$ for some $\displaystyle n \in \mathbb{N}\}$