Take $2n$-tuples of bounded positive operators $x_1,\dots x_n$ and $a_1,\dots a_n$ on a Hilbert space $H$ which have zero kernel and dense image and which satisfy the condition that (1) $$ x_1^* x_1+\dots+x_n^* x_n+a_1^* a_1+\dots+a_n^* a_n $$ is invertible and (2) that the image of $x_i^*x_i+a_i^*a_i$ and $x_i^*$ have dense intersection for each $1\le i\le n$ and (3) that we have the equality (used to define a function $T$ of the $2n$-tuple) $$ T:=x_1(x_1^*x_1+a_1^*a_1)^{-1}x_1^*=\dots=x_n(x_n^*x_n+a_n^*a_n)^{-1}x_n^*\in B(H). $$ The problem is to show that $T$ varies continuously as the $2n$-tuple (subject to the conditions) varies continuously in the norm topology.
[Edit: Each formula for $T$ is a densely defined operator of norm $\le 1$, and so it extends to a bounded operator on $H$. The problem is that as far as I can see no individual formula can show the continuity of $T$ as the $2n$-tuple varies...]
This question may seem strange, but it comes from representing rings of fractions in a Hilbert space. If anyone can say anything about it I would be very grateful!
