On weakly formulated Sylvester equations and applications

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.