# Finding approximate eigenvectors: quantitative results

Let $$X$$ be a complex Banach space and $$T \colon X \to X$$ be a bounded operator. For every $$x \in X \setminus \{0\}$$, denote by $$Y_x$$ the smallest closed $$T$$-invariant subspace of $$X$$ containing $$x$$. By basic spectral theory, the restriction of $$T$$ to $$Y_x$$ has an approximate eigenvalue. So we have the following:

For every $$x \in X$$, and every $$\varepsilon > 0$$, there exists a polynomial $$P_{x, \varepsilon} \in \mathbb{C}[X]$$ such that, letting $$y = P_{x, \varepsilon}(T)(x)$$, we have $$\|y\| = 1$$ and $$dist(T(y), \mathbb{C}y) \leqslant \varepsilon.$$

General question: Are there known bounds on the degree of the polynomial $$P_{x, \varepsilon}$$, depending on $$X$$, $$T$$, $$x$$ and $$\varepsilon$$?

I would be interested in any reference of a work in this direction. But I am actually interested in a special case of this:

Specific question: Suppose $$T$$ is an isomorphism between $$X$$ and a proper subspace $$Y$$, and fix $$\varepsilon > 0$$. Does there exists a $$d \in \mathbb{N}$$ (depending on $$X$$, $$T$$, and $$\varepsilon$$) such that for every $$x \in X \setminus Y$$, one can find a polynomial $$P_{x, \varepsilon}$$ as above, with degree at most $$d$$?