Estimates of solutions of certain classes of second-order differential equations in a Hilbert space

Linear second-order differential equations of the form
$$ u''(t)+(B+iD)u'(t)+(T+iS)u(t)=0 $$
in a Hilbert space are studied. Under certain conditions on the (generally speaking, unbounded) operators $T$, $S$, $B$ and $D$ the correct solubility of the equation in the “energy” space is proved and best possible (in the general case) estimates of the solutions on the half-axis are obtained.