Symbolic denotation of the space of all eigenvectors

Question

There is a symbolic notation for the set of all eigenvalues $$\operatorname{spec} \varphi = \lbrace \lambda \in K \mid \lambda \textrm{ is an eigenvalue} \rbrace$$ There is also a notation for the eigenspace $$V_\lambda = \lbrace \alpha \in V \mid \varphi(\alpha) = \lambda \alpha \rbrace$$ Is there any standard notation for the set of all eigenvectors? So that instead of writing

Let $v$ be an eigenvector

we could write

Let $v \in \dots$

Answer 1

Not sure this is an answer, too long for a comment.

I'd be surprised if there were a standard notation for the set of eigenvectors, since that set doesn't have nice geometry. (Think about it for the eigenvectors of the matrix with 1, 2, 2, on the diagonal.)

Arguments about eigenvectors usually begin

Let $v$ be an eigenvector with eigenvalue $\lambda$ ...

Answer 2

I would just write

Let $v \in V_\lambda \backslash \{0\}$ for some $\lambda \in \text{spec}\; \phi$

This is convenient since anything useful you can say about $v$ is likely to involve the eigenvalue $\lambda$. If for some reason you're determined not to identify the particular eigenvalue, you could say

Let $v \in \bigcup_{\lambda \in \text{spec}\; \phi} V_\lambda \backslash \{0\}$