I have a self-adjoint compact operator $\Gamma : L^2[0,1] \to L^2[0,1]$ with positive eigenvalues $\lambda_j$, which of course tend to zero,and a general finite dimensional linear subspace $S \subset L^2[0,1]$, which could, for example, be the space of polynomials of order $q$. Could I then say something about the minimum of theRayleigh quotient
$$\inf_{\beta \in S, \beta \neq 0} \frac{\langle \Gamma \beta, \beta \rangle_{L^2}}{||\beta||^2_{L^2}}$$
possibly in terms of the eigenvalues of $\Gamma$? A lower bound would be most helpful here.
It seems to me that if $S$ is the span of the some eigenfunctions of $\Gamma$ then the infimum would be one of the corresponding eigenvalues, but, if not, it's extremely hard to make precise statements.
Thank you.
Notice, that $L^2[0,1]=S\oplus S^\perp$, so $\Gamma(\beta)=s+w$, where $s\in S$ and $w\in S^\perp$. Let $P_S$ be orthogonal projector on $S$, then
$$ \inf_{\beta \in S, \beta \neq 0} \frac{\langle \Gamma \beta, \beta \rangle_{L^2}}{\|\beta\|^2_{L^2}}=\inf_{\beta \in S, \beta \neq 0} \frac{\langle P_S\Gamma \beta, \beta \rangle_{L^2}}{\|\beta\|^2_{L^2}}. $$ As $P_S\Gamma |_S$ is an operator from $S$ to $S$ you can say nothing more about your $\inf$, then in case of an arbitary operator in S.