If $A$ is a self-adjoint and nonnegative operator on a Hilbert space $H$, then there is an orthonormal basis of $H$ consisting of eigenvectors of $A$

Question

Let

• $H$ be a $\mathbb R$-Hilbert space
• $A$ be a compact and self-adjoint bounded linear operator on $H$
• $I:=\left\{n\in\mathbb N:n\le\operatorname{rank}A\right\}$

By the Hilbert-Schmidt theorem, there is a $(\lambda_i)_{i\in I}\subseteq\mathbb R\setminus\left\{0\right\}$ with $$Ae_i=\lambda_ie_i\;\;\;\text{for all }i\in I$$ for some orthonormal basis $(e_i)_{i\in I}$ of $\overline{AH}$.

Now, if $A$ is nonnegative (i.e. $\langle Ax,x\rangle_H\ge0$ for all $x\in H$) and has finite trace, why can we conclude that $\overline{AH}=H$?

Answer 1

Your claim is incorrect, even for finite dimensional spaces: let $H=\mathbb{R}^2$ and pick $A$ to be the operator corresponding to the matrix $\left(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}\right)$. Then $A$ is selfadjoint (since the matrix is symmetrical), nonnegative (since the eigenvalues $0$, $1$ are both nonnegative), and, trivially, has finite trace. However, $\overline{AH}=AH=\text{span}\{e_1\}\neq\mathbb{R}^2=H$.

Even simpler (but perhaps a little too trivial) is the operator $Ax:=0_H$ for any Hilbert space $H$.