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I am currently trying to understand "Analytic Pertubation Theory for Linear Operators" by T. Kato. In Chapter VII.§2.1 (p.375f) the following situation is considered:

Setup: Let $D \subset \mathbb{C}$ be open, $X$, $Y$, be complex Banach spaces, $C(X,Y)$ be the closed (but possibly unbounded) operators $X \to Y$ and let $T:D \to C(X,Y)$ be a holomorphic family of type (A). (This is defined earlier to be a map such that the domain $Z:=dom(D(z))$ is independent of $z \in D$ and such that for each $u \in Z$ the family $D(z)u$ is a holomorphic family $\Omega \to Y$). Kato shows that for each $z_0 \in D$ the norm $\|u\|_Z := \|u\|_X + \|T(z_0)u\|_Y$ turns $Z$ into a Banach space and $V:\Omega \to B(Z,Y)$, $V(z):=T(z)$, is a bounded-holomorphic family (i.e. just a holomorphic map).

Question: Now it is claimed that since $V$ is bounded-holomorphic, for any compact $K \subset D$ $ \sup_{z \in K}\sup_{u \in Z \setminus \{0\}} \frac{\|u\|_X + \|V(z_0)u\|_Y}{\|u\|_X + \|V(z)u\|_Y} < \infty$ Why is that? At a first glance this seems intuitively clear, but I can't find a formal reason for it.

Ideas: My first intuition was to note that the fraction is invariant under any rescaling of $u$. So it is sufficient to consider those $u$ that satisfy $\|u\|_Z=1$. Therefore it suffices to prove that $ \sup_{z \in K}\sup_{\|u\|_Z=1} \frac{1}{\|u\|_X + \|V(z)u\|_Y} < \infty$ by definition of $\| \_ \|_Z$. To see this it suffices to prove $ \inf_{z \in K}\inf_{\|u\|_Z=1} \|u\|_X + \|V(z)u\|_Y > 0 $ This could be proven by showing that $ z \mapsto \inf_{\|u\|_Z=1} \|u\|_X + \|V(z)u\|_Y $ is a continuous strictly positive function $\Omega \to \mathbf{R}$. But unfortunately I do not see why this function should be positive. :(

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    what does it mean by analytic operator in the Kato sens?2013-12-22

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