eigenvalues of normal operators
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 $\lambda \in \u2102$ 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^{}. 2 strengthens this result for normal operators.
Title  eigenvalues of normal operators 

Canonical name  EigenvaluesOfNormalOperators 
Date of creation  20130322 17:33:32 
Last modified on  20130322 17:33:32 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  10 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 47B15 
Classification  msc 47A75 
Classification  msc 47A15 
Classification  msc 47A10 
Classification  msc 15A18 