Notation: $X$ and $Y$ are vectors in $\ell^2$. Let $P_X$ denote the projection operator onto the vector space spanned by $\{L^jX\}_{j=0}^{\infty}$ where $L$ is the right-shift operator. $P_X$ is a Toeplitz operator.

I need to know how small changes to $X$ might affect $P_X Y$. More formally, is it possible to come up with a bound $M$ such that for an arbitrarily small $\varepsilon\in\ell^2$ $$ \frac{||P_{X+\varepsilon} Y-P_X Y||}{||\varepsilon||}\leq M$$

Thanks in advance for your help; I've been stuck on this for a while! I'm not a mathematician and will try to clarify any issues.

Edit: cleaned up some notation


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