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

单词	EigenvaluesOfNormalOperators
释义	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 $\\lambda\\in\\mathbb{C}$ 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. 2 strengthens this result for normal operators.
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