Stein's (1952) theorem states:
"A necessary and sufficient condition that B is convergent, is that there exists a positive definite Hermitian matrix H for which H-B*HB is positive definite"
where B* denotes the conjugate transpose of B.
However, take: $$B=\left[ \begin{matrix} 0.9 & 0 \\ 0 & 0.3 \\ \end{matrix} \right] $$ Which is convergent. And take: $$H=\left[ \begin{matrix} 0.8 & 0.3 \\ 0.3 & 0.2 \\ \end{matrix} \right] $$ Which is positive semidefinite. Then, $$H-B^*HB=\left[ \begin{matrix} 0.152 & 0.219 \\ 0.219 & 0.182 \\ \end{matrix} \right] $$ But this matrix is not positive definite. The theorem seems to say that the positive definiteness of H-B*HB is a necessary condition for B to be convergent. I know I am making a logical mistake, but I'd appreciate your help in pointing it out. Thanks.
The theorem only says that such an $H$ exists, not that $H - B^*HB$ will be positive definite for all Hermitian matrices $H$.
For an example of an Hermitian $H$ for which $H - B^*HB$ is positive definite, we may simply take $$ H = \pmatrix{1&0\\0&1} $$