Definition:Diagonal Operator

Definition

Let $X$ be a vector space.

Let $\dim X$ be the dimension of $X$.

Let $\set {e_i}_{1 \mathop \le i \mathop \le \dim X}$ be the basis of $X$.

Let $\Lambda : X \to X$ be a mapping such that:

$\forall i \in \N_{> 0} : i \le \dim X : \exists \lambda_i \in \C : \map \Lambda {e_i} = \lambda_i e_i$

Then $\Lambda$ is called the diagonal operator.

Further research is required in order to fill out the details. In particular: This is a placeholder for a more investigated definition. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by finding out more. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Research}} from the code.

Sources

• 1982: Paul R. Halmos: A Hilbert Space Problem Book (2nd ed.): Chapter $\S 7$: Multiplication Operators. $61$. Diagonal Operators
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis: Chapter $\S 2.3$: The normed space $\map {CL} {X, Y}$