# Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem

Does the following generalization of that fact also hold?

Theorem: Let $H$ be a complex Hilbert space, $T:D \to H$ be an unbounded operator defined on a dense domain $D \subset H$. Assume that $T$ is self-adjoint and has discrete spectrum contained in some interval $[c,\infty[$, $c > 0$. Enumerate the eigenvalues by $\lambda_1 \leq \lambda_2 \leq \ldots$ counting with multiplicities. Then for each $k \in \mathbb{N}$ $$\lambda_k = \min_{\substack{U \subset D, \\ \dim U=k }}{\max_{\substack{x \in U, \\\|x\|=1}}{\langle Tx, x \rangle}}$$

If this is true, can you give a good reference for this?

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I am trying to understand. What is the role of $U$ –  Seyhmus Güngören Sep 6 '12 at 14:52
$U$ is a $k$-dimensional subspace of $D$ –  Meneldur Sep 9 '12 at 19:34