140 $T$ stable implies BIBO stable

We deduce that $Y$ has at most a simple pole on the imaginary axis, so $T$ has no poles in the imaginary axis. Hence $T(s)$ has all its poles in LHP. Hence $T$ is stable.

Conversely, suppose that $T$ is stable. Then by Proposition 57, there exists a $SISO$ $(A,B,C,D)$ such that the transfer function is $T$ and the eigenvalues $\lambda$ of $A$ are the poles of $T$, hence satisfy . Then by Theorem 119, $(A,B,C,D)$ is $BIBO$ stable.