Definition:Spectrum (Spectral Theory)/Bounded Linear Operator

Definition

Let $\struct {X, \norm \cdot_X}$ be a Banach space over $\C$.

Let $A : X \to X$ be a bounded linear operator.

Let $\map \rho A$ be the resolvent set of $A$.

Let:

$\map \sigma A = \C \setminus \map \rho A$

We say that $\map \sigma A$ is the spectrum of $A$.

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Also see

• Definition:Point Spectrum of Linear Operator

• Results about spectra in the context of bounded linear operators can be found here.

Sources

• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $14.1$: The Resolvent and Spectrum