# Functional Analysis- Convergence

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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Nonnegative is that $\langle Tx,x\rangle\ge 0$ for all $x$. Over $\mathbb C$, in finite dimension, you'll have diagonalizable matrices with diagonal forms with positive or zero entries. Positive is then when none in the diagonal is zero. – plm May 26 '12 at 10:30
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. – plm May 26 '12 at 10:53