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Let $H$ be a Hilbert space over $\Bbb F \in \set {\R, \C}$.

Let $A \in \map B H$ be a bounded linear operator.

Let $\alpha \in \Bbb F$ be an eigenvalue of $A$.

A nonzero vector $h \in H$ is said to be an eigenvector for $\alpha$ if and only if:

$h \in \map \ker {A - \alpha I}$

That is, if and only if $Ah = \alpha h$.


The eigenspace for an eigenvalue $\alpha$ is the set $\map \ker {A - \alpha I}$.

By Kernel of Linear Transformation is Closed Linear Subspace, it is a closed linear subspace of $H$.

Also see