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 iff 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 $A = \displaystyle \sum_{i \in I} \alpha_i P_i$ or $A = \displaystyle \bigoplus_{i \in I} \alpha_i P_i$.


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


Examples


Sources