Parallel sums and norm convergence (Q921523)

For positive operators A, B in a Hilbert space, the parallel sum A:B is defined by \(A:B=(A^{-1}+B^{-1})^{-1}\) for A and B invertible, and by \(A:B=\lim_{\epsilon \searrow 0}(A+\epsilon I):(B+\epsilon I)\) otherwise, where the convergence is understood in the strong topology. When A and B represent the impedance operator of two resistive parts, A:B represents the impedance of the pair, connected in parallel. This paper surveys some results about the norm convergence of the limit in the definition of A:B, and presents two examples disproving several conjectures about this norm convergence.