Numerical range of a symmetric positive matrix as an operator acting on Hilbert space

Question

Let $A$ be a $3\times 3$ real constant symmetric positive matrix, $\Omega\subset\mathbb{R}^3$ a bounded Lipschitz domain and $(L^2(\Omega))^3$ the space of square integrable functions on $\Omega$.
I already know that the numerical range of $A$ is the interval $[\lambda_{min},\lambda_{max}]$ where $\lambda_{min}$ (resp. $\lambda_{max}$) is the smaller (resp. greatest) eigenvalue of $A$. But, if we consider $A$ as an operator acting on the space $(L^2(\Omega))^3$, did then the numerical range still the same as above ? For the definition of numerical range, one could consult for example math.wsu.edu/faculty/tsat/files/short.pdf

Answer 1

When you consider $A$ as an operator, it is still a selfajoint operator with three-point spectrum. For any selfajoint operator, the numerical range is $[\lambda_\min,\lambda_\max]$.