Is self-adjointness stable under norm equivalence?

Question

Suppose that $H$ is a self-adjoint operator in some (let's say) Hilbert space ${\cal H}_{1}$. Now, consider that the norm on ${\cal H}_{1}$ is equivalent to the norm of another Hilbert space ${\cal H}_{2}$. Can we state that $H$ is self-adjoint in ${\cal H}_{2}$?

Answer 1

The anwer is negative. Consider a finite dimensional Hilbert space and, on it, define a second scalar product $(x|y) := \langle x,S y\rangle$, where $S$ is strictly positive and self adjoint with respect to the first scalar product $\langle,\rangle$. As the spaces are finite dimensional, their normes are equivalent. However, a selfadjoint operator for the first scalar product which does not commute with $S$ is not selfadjoint for the second scalar product.