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You may apply your observation in the partial case to functions of operators. (For example: $T$ and $T^2$ are not similar since $T-1/2$ and $T^2-1/2$ have different index.) It allows to prove that degrees are equal, apply it to $aP$ and $aQ$ for small $a$. Next, considering $aP,aQ$ for $a$ close to 1 we get polynomials without roots on unit circle and apply it again. Then let $a$ tend to $1-0$, after that let $a$ tend to $1+0$.