Essential spectrum

The essential spectrum or essential spectrum is an object from the mathematical sub-area of functional analysis . It is not defined uniformly in the literature. What all definitions have in common, however, is that the essential spectrum is a subset of the spectrum of a linear operator in which points that are considered "benign" have been removed.

definition

One possible definition is: Let a linear operator on a Hilbert space , then there is a significant range of from all the no Fredholm operator is. It is thus a generalization of the concept of eigenvalue . ${\displaystyle A}$ ${\displaystyle \sigma _{ess}(A)}$ ${\displaystyle A}$ ${\displaystyle \lambda \in \mathbb {C} ,}$ ${\displaystyle A-\lambda I}$

properties

The essential spectrum from the above definition is invariant under perturbations with a compact operator It is therefore true . ${\displaystyle K.}$ ${\displaystyle \sigma _{ess}(A)=\sigma _{ess}(A+K)}$

For a normal operator on a Hilbert space belongs to if and only if there is no isolated eigenvalue of finite multiplicity. Alternatively, the essential spectrum can also be defined as the usual spectrum of the operator's image in Calkin's algebra . ${\displaystyle A}$ ${\displaystyle \lambda }$ ${\displaystyle \sigma _{ess}(A)}$ ${\displaystyle \lambda }$ ${\displaystyle A}$

literature

• Harro Heuser : Functional Analysis: Theory and Application . 3rd edition, BG Teubner, Stuttgart 1992. ISBN 3-519-22206-X .