  We use a ``weakly formulated'' Sylvester equation
$$A^{1/2}TM^{-1/2}-A^{-1/2}TM^{1/2}=F$$ to obtain new bounds for the rotation
of spectral subspaces of a nonnegative selfadjoint operator in a Hilbert space.
Our bound extends the known results of Davis and Kahan. Another application is
a bound for the square root of a positive selfadjoint operator which extends
the known rule: ``The relative error in the square root is bounded by the one
half of the relative error in the radicand''. Both bounds are illustrated on
differential operators which are defined via quadratic forms.
