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

• Definition:Orthogonal Projection

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) $\text {II}.7.3$