Definition:Diagonal Operator
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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.
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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 $\map {CL} {X, Y}$