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 $d$?
