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[parent] eigenvalues of normal operators (Theorem)

Let $ H$ be a Hilbert space and $ B(H)$ the algebra of bounded operators in $ H$. Suppose $ T \in B(H)$ is a normal operator. Then

  1. - If $ \lambda \in \mathbb{C}$ is an eigenvalue of $ T$, then $ \overline{\lambda}$ is an eigenvalue of $ T^*$ (the adjoint operator of $ T$) for the same eigenvector.
  2. - Eigenvectors of $ T$ associated with distinct eigenvalues are orthogonal.

Remark - It is known that for any linear operator eigenvectors associated with distinct eigenvalues are linearly independent. Point 2 strengthens this result for normal operators.



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Cross-references: linearly independent, linear operator, orthogonal, eigenvectors, eigenvector, adjoint operator, eigenvalue, normal operator, bounded operators, algebra, Hilbert space
There is 1 reference to this entry.

This is version 7 of eigenvalues of normal operators, born on 2007-09-26, modified 2007-11-22.
Object id is 9965, canonical name is EigenvaluesOfNormalOperators.
Accessed 855 times total.

Classification:
AMS MSC: 15A18 (Linear and multilinear algebra; matrix theory :: Eigenvalues, singular values, and eigenvectors)
 47A10 (Operator theory :: General theory of linear operators :: Spectrum, resolvent)
 47A15 (Operator theory :: General theory of linear operators :: Invariant subspaces)
 47A75 (Operator theory :: General theory of linear operators :: Eigenvalue problems)
 47B15 (Operator theory :: Special classes of linear operators :: Hermitian and normal operators )
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