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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6210 • Real Analysis

Fall 2007 • Question 3

Question

Math 6210 • Real Analysis Fall 2007 • Question 3

Let \(H\) be a complex Hilbert space, and let \(T:H\to H\) be a bounded self adjoint operator (so \(T=T^*\)).

a. Prove that if \(\lambda \) is an eigenvalue of \(T\), then \(\lambda \) is real.

b. Show that eigenvectors corresponding to different eigenvalues are orthogonal.

c. Assume in addition that \(T\) is compact. Show that if \(\{\lambda _n\}\) is a sequence of distinct eigenvalues with \(\lambda _n\to \lambda \), then \(\lambda =0\).

Raw LaTeX

Let \(H\) be a complex Hilbert space, and let \(T:H\to H\) be a bounded
self adjoint operator (so \(T=T^*\)).

a. Prove that if \(\lambda\) is an eigenvalue of \(T\), then \(\lambda\) is real.

b. Show that eigenvectors corresponding to different eigenvalues are
orthogonal.

c. Assume in addition that \(T\) is compact. Show that if \(\{\lambda_n\}\) is a
sequence of distinct eigenvalues with \(\lambda_n\to\lambda\), then \(\lambda=0\).

Topic Tags

Compactness in Metric and Normed Spaces Hilbert Spaces and Orthogonality Lp Spaces and Sequence Spaces