Definition:Diagonalizable Operator
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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 (2nd ed.) ... (previous) ... (next) $\text {II}.4.6$
- 1990: John B. Conway: A Course in Functional Analysis (2nd ed.) ... (previous) ... (next) $\text {II}.7.3$