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 $\lambda \in ℂ$ 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. 2 strengthens this result for normal operators.

Title	eigenvalues of normal operators
Canonical name	EigenvaluesOfNormalOperators
Date of creation	2013-03-22 17:33:32
Last modified on	2013-03-22 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