Definition:Diagonalizable Operator

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

Let $A:H \to H$ be a linear operator on $H$.

Then $A$ is said to be diagonalizable iff there exist:


 * a basis $\left({e_i}\right)_{i \in I}$ for $H$
 * a collection $\left({\alpha_i}\right)_{i \in I}$ of scalars (with the same indexing set $I$)

such that:


 * $\forall i \in I: Ae_i = \alpha_ie_i$

Value Set
The collection $\left({\alpha_i}\right)_{i \in I}$ may be called the value set of $A$.

Examples

 * Orthogonal Projection