eigenvalues of normal operators

Let H be a Hilbert spaceMathworldPlanetmath and B(H) the algebra of bounded operatorsMathworldPlanetmathPlanetmath in H. Suppose TB(H) is a normal operator. Then

  1. 1.

    - If λ is an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath of T, then λ¯ is an eigenvalue of T* (the adjoint operator of T) for the same eigenvectorMathworldPlanetmathPlanetmathPlanetmath.

  2. 2.

    - Eigenvectors of T associated with distinct eigenvalues are orthogonalMathworldPlanetmath.

Remark - It is known that for any linear operatorMathworldPlanetmath eigenvectors associated with distinct eigenvalues are linearly independentMathworldPlanetmath. 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