# Definition:Diagonalizable Operator

## Definition

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

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** if and only if there exist:

- a basis $E$ for $H$
- an indexed set $\family {\alpha_e}_{e \mathop \in E} \subseteq \Bbb F$ of scalars (with $E$ as indexing set)

such that:

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

#### Value Set

The indexed set $\family {\alpha_e}_{e \mathop \in E}$ 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** if and only if there exist:

- a partition of unity $\family {P_i}_{i \mathop \in I}$ on $H$
- an indexed set $\family {\alpha_i}_{i \mathop \in I} \subseteq \Bbb F$ of scalars (with the same $I$ as indexing set)

such that:

- $\forall i \in I: \forall h \in \Rng {P_i}: A h = \alpha_i h$

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

#### Value Set

The indexed set $\family {\alpha_i}_{i \mathop \in I}$ may be called the **value set of $A$ (with respect to the partition of unity $\family {P_i}_{i \mathop \in I}$)**.

## Also see

- Diagonalizable Operator Bounded iff Value Set Bounded
- Diagonalizable Operator Compact iff Value Set Converges to Zero

### Examples

## Sources

- 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $\text {II}.4.6$ - 1990: John B. Conway:
*A Course in Functional Analysis*... (previous) ... (next) $\text {II}.7.3$