Given an operator $H$ , and a sequence $\{ H_n \} _{n\geq 1 } $ in an arbitrary Hilbert Space , such that both $H$ and $ H_n $ are self-adjoint and non-negative.
How can I prove that $||(H_n+1)^{-1} - (H+1) ^ {-1} || \to 0 $ is equivalent to $||(H_n+i)^{-1} - (H+i) ^ {-1} || \to 0 $ ?
BTW- What is the meaning of a non-negative operator?
Thanks !