  In noncommutative geometry one is interested in invariants such as the
Fredholm index or spectral flow and their calculation using cyclic cocycles. A
variety of formulae have been established under side conditions called
summability constraints. These can be formulated in two ways, either for
spectral triples or for bounded Fredholm modules. We study the relationship
between these by proving various properties of the map on unbounded self
adjoint operators $D$ given by $f(D)=D(1+D^2)^{-1/2}$. In particular we prove
commutator estimates which are needed for the bounded case. In fact our methods
work in the setting of semifinite noncommutative geometry where one has $D$ as
an unbounded self adjoint linear operator affiliated with a semi-finite von
Neumann algebra $\aM$. More precisely we show that for a pair $D,D_0$ of such
operators with $D-D_0$ a bounded self-adjoint linear operator from $\aM$ and $
({\bf 1}+D_0^2)^{-1/2}\in \sE$, where $\sE$ is a noncommutative symmetric space
associated with $\aM$, then $$ \Vert f(D) - f (D_0) \Vert_{\sE} \leq C\cdot
\Vert D-D_0\Vert_{\aM}. $$ This result is further used to show continuous
differentiability of the mapping between an odd $\sE$-summable spectral triple
and its bounded counterpart.
