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Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f.

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

1 Answer 1

I know this post is is maybe two years too late, but I have a reference for you to check. There is in fact a a corresponding theorem for self-adjoint operators on Hilbert spaces, but the conditions are altered slightly since operators have more than one kind of spectrum. An operator's spectrum can be divided into its discrete spectrum (eigenvalues) and its essential spectrum, which is defined in terms of projection-valued measures of an operator (using the functional calculus for self-adjoint operators you can define $\chi_B(A)$ where $A$ is self-adjoint, $\chi_B$ is the characteristic function on $B$, and $B$ is any Borel measurable set. The essential spectrum $\sigma_{ess}(H)$ are those $\lambda \in \mathbb{C}$ such that $\chi_{(\lambda - \epsilon, \lambda+\epsilon)}(A)$ has infinite-dimensional range for all $\epsilon>0$). Anyway the following statement of the theorem is taken out of Reed and Simon's Book on Operators.

Let $H$ be a self-adjoint operator bounded below (meaning $H \geq cI$ for some $c$). Define $$U_H(\varphi_1, \ldots, \varphi_m) \;\; =\;\; \inf_{\begin{subarray}{c} \psi\in D(H), \; ||\psi||=1 \\ \psi \in [ \varphi_1, \ldots, \varphi_m]^\perp \end{subarray} } \langle \psi, H\psi \rangle $$ and $$\mu_n(H) \;\; =\;\; \sup_{\varphi_1, \ldots, \varphi_{n-1}} U_H(\varphi_1, \ldots, \varphi_{n-1}) $$ where $[\varphi_1, \ldots, \varphi_{m-1}]^\perp = \{\psi \; : \; \langle \psi, \varphi_i\rangle =0, \; i= 1\ldots, m-1\}$ where the $\varphi_i$ are not necessarily independent. Then for each fixed $n$ one of the following occurs:

  1. There are $n$ eigenvalues (counting degenerate eigenvalues a number of times equal to their multiplicity) below the bottom of the essential spectrum, and $\mu_n(H)$ is the $n$th eigenvalue counting multiplicity.

  2. $\mu_n = \inf \sigma_{ess}(H)$ and in that case $\mu_n = \mu_{n+1} = \mu_{n+2} = \ldots$ and there are at most $n-1$ eigenvalues (counting multiplicities) below $\mu_n$.

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