# How to bound marginal changes to Toeplitz projections onto different Hilbert spaces?

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