# 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. 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. 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 EigenvaluesOfNormalOperators 2013-03-22 17:33:32 2013-03-22 17:33:32 asteroid (17536) asteroid (17536) 10 asteroid (17536) Theorem msc 47B15 msc 47A75 msc 47A15 msc 47A10 msc 15A18