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 $H$.

The following two definitions of diagonalizable operator are equivalent:

By a Basis
$A$ is said to be diagonalizable iff there exist:


 * a basis $E$ for $H$
 * a collection $\left({\alpha_e}\right)_{e \in E} \subseteq \Bbb F$ of scalars (with $E$ as indexing set)

such that:


 * $\forall e \in E: Ae = \alpha_e e$

Value Set
The collection $\left({\alpha_e}\right)_{e \in E}$ may be called the value set of $A$ (with respect to the basis $E$).

By a Partition of Unity
$A$ is said to be diagonalizable iff there exist:


 * a partition of unity $\left({P_i}\right)_{i \in I}$ on $H$
 * a collection $\left({\alpha_i}\right)_{i \in I} \subseteq \Bbb F$ of scalars (with the same $I$ as indexing set)

such that:


 * $\forall i \in I: \forall h \in \operatorname{ran} P_i: Ah = \alpha_i h$

To express that $A$ is diagonalizable, one writes $A = \displaystyle \sum_{i \in I} \alpha_i P_i$ or $A = \displaystyle \bigoplus_{i \in I} \alpha_i P_i$.

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

Examples

 * Orthogonal Projection