My idea is as follows: LetKs consider the following Cauchy problem to the linear operator differen$ tial equation: dX/dt*$(AX+XA), X\t! $ X . As the operator A>0 is ...
My idea is as follows: Let’s consider the following Cauchy problem to the linear operator di¤erential equation: dX/dt=-(AX+XA), Xjt=0 = X0 : As the operator A>0 is positive de…nite, the semigroup exp( tA) : H ! H is well de…ned in the Hilbert space H and for the operator function X(t) : H ! H one …nds easily the following operator representation: X(t)=exp(-At)X0 exp(-At), t2 R: Assume then, by de…nition, that the operator X(t)=X0 =constant for all t>0. Thus we get that exp(-At)X0 exp(-At)=X0 for all t>0, in particular for t! 1: As the norm jof the operator semigroup jjexp(-At)jj! 0 as t! 1; one obtains easily that the operator norm limit limt!1 jjX(t)jj=jjexp(-At)X0 exp(-At)jj! 0 = jjX0 jj; meaning that X0 = 0 under condition that the norm jjX0 jj < 1: In the case jjX0 jj = 1 the analysis should be much more subtle and is still not found.
