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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 !

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    I mean, we assume the matrix is self-adjoint, hermitian. These correspond to hermitian inner products for positive matrices, or "semi-inner product" for nonnegative ones.2012-05-26

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