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Math Help - Generalized Eigenspaces

  1. #1
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    Generalized Eigenspaces

    Let V = W_1 \oplus W_2, \phi: V \rightarrow V a linear map. Prove that V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.<br /> <br />
Injectivity is easy enough, but I can't seem to prove V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+ without assuming that  W_1, W_2 are \phi-stable. Am I missing something here, or is this assumption required for the implication to hold?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by h2osprey View Post
    Let V = W_1 \oplus W_2, \phi: V \rightarrow V a linear map. Prove that V(\lambda)^+ = W_1(\lambda)^+ \oplus W_2(\lambda)^+.<br /> <br />
Injectivity is easy enough, but I can't seem to prove V(\lambda)^+ = W_1(\lambda)^+ + W_2(\lambda)^+ without assuming that  W_1, W_2 are \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 S(\lambda) is the eigenspace of the eigenvalue \lambda? What does the ^+ mean?
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    Whoops my bad. V(\lambda)^+ is defined as the generalized eigenspace of the eigenvalue \lambda, i.e. V(\lambda)^+ := \{v \in V | (\phi - \lambda .Id)^n (v)= 0 for some  n \in \mathbb{N}\}
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