129 Solution of Sylvester’s equation

(i) First observe that by Lemma 118 there exist $M_1,M_2,{\delta }_1,{\delta }_2>0$ such that $\parallel \exp (tA)\parallel \le M_1{e}^{-{\delta }_{1}t}$ and $\parallel \exp (tB)\parallel \le M_2{e}^{-{\delta }_{2}t}$, so the integral is convergent. Also

$$AY+YB={\int }_0^{\mathrm{\infty }}\left(A\exp (tA)C\exp (tB)+\exp (tA)C\exp (tB)B\right)𝑑t$$

$$AY+YB={\int }_0^{\mathrm{\infty }}\frac{d}{dt}\left(\exp (tA)C\exp (tB)\right)𝑑t$$

$$={\left[\exp (tA)C\exp (tB)\right]}_0^{\mathrm{\infty }}=-C.$$

Uniqueness follows from the Proposition 127 since $A$ and $-B$ have no common eigenvalues.