Proposition 19

Let E be a complete normable space over $\mathbf{C}$, $v \in \mathscr{L}(E)$ and $g = \exp v$. Suppose that every $z \in \mathrm{Sp}\, v$ satisfies $-\pi < \mathscr{J}(z) < \pi$. Then, for every closed vector subspace $E'$ of $E$, the conditions $v(E') \subset E'$ and $g(E') = E'$ are equivalent.

The condition $v(E') \subset E'$ implies $g(E') \subset E'$ and $g^{-1}(E') \subset E'$ and hence $g(E') = E'$. Suppose that $g(E') = E'$. We use the notation $\Delta, \Delta'$ of Proposition 17. Since $\mathrm{Sp}\, v$ is a compact subset of $\Delta$, there exists a compact rectangle $Q = (a, b) \times (a', b')$ such that $\mathrm{Sp}\, v \subset Q \subset \Delta$. The set $\Delta - Q$ is connected. Hence $\mathrm{Sp}\, g \subset \exp Q \subset \Delta'$, the set $\exp Q$ is compact and the set $\Delta' - \exp Q$ is connected. The closure of the latter contains $]-\infty, 0]$ and hence

is connected. Then $\exp Q$ is polynomially convex (Spectral Theories, Chapter I, § 3, Corollary 2 to Proposition 9) and hence the function $\log$, defined on $\Delta'$, is a limit in $\mathscr{O}(\exp Q)$ of polynomial functions (Spectral Theories, Chapter I, § 4, Proposition 3). Hence $v = \log g$ is the limit in $\mathscr{L}(E)$ of elements of the form $P(g)$, where $P$ is a polynomial (Spectral Theories, Chapter I, § 4, Theorem 3). As $P(g)(E') \subset E'$, it follows that $v(E') \subset E'$.