Resolvent Mapping is Analytic/Bounded Linear Operator

From ProofWiki
Jump to navigation Jump to search


This article needs to be linked to other articles.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding these links.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{MissingLinks}} from the code.


Theorem

Let $B$ be a Banach space.

Let $\map \LL {B, B}$ be the set of bounded linear operators from $B$ to itself.

Let $T \in \map \LL {B, B}$.

Let $\map \rho T$ be the resolvent set of $T$ in the complex plane.


Then the resolvent mapping $f : \map \rho T \to \map \LL {B, B}$ given by $\map f z = \paren {T - z I}^{-1}$ is analytic, and:

$\map {f'} z = \paren {T - z I}^{-2}$

where $f'$ denotes the derivative of $f$ with respect to $z$.


Proof 1

For $a \in \map \rho T$, define:

$R_a = \paren {T - a I}^{-1}$

Then we have:

\(\ds \frac {\norm {\map f {z + h} - \map f z - \paren {T - z I}^{-2} h }_*} {\size h}\) \(=\) \(\ds \frac {\norm {R_{z + h} - R_z - R_z^2 h}_*} {\size h}\)
\(\ds \) \(=\) \(\ds \frac {\norm {h R_{z + h} R_z - R_z^2 h }_*} {\size h}\) Resolvent Identity
\(\ds \) \(=\) \(\ds \frac {\size h \norm {R_{z + h} R_z - R_z^2 }_*} {\size h}\) Operator Norm is Norm
\(\ds \) \(=\) \(\ds \norm {R_{z + h} R_z - R_z^2 }_*\)

From Resolvent Mapping is Continuous we have:

$R_{z + h} \to R_z$ as $h \to 0$

Taking limits of both sides and using Norm is Continuous, we get:

$\ds \lim_{h \mathop \to 0} \dfrac {\norm {\map f {z + h} - \map f z - \paren {T - z I}^{-2} h }_*} {\size h} = \norm {R_z^2 - R_z^2}_* = 0$

which is the result.

$\blacksquare$


Proof 2

If $B=\set {\mathbf 0_B}$, the statement is trivial, since $\map f z = \mathbf 0_{\map \LL {B, B}}$ for all $z \in \C$.

We assume that $B \ne \set {\mathbf 0_B}$.

Especially, for all $z \in \map \rho T$:

$\map f z \ne \mathbf 0_{\map \LL {B, B}}$

since:

$\paren {T - z I} \map f z = I \ne \mathbf 0_{\map \LL {B, B}}$


Let $a \in \map \rho T$.

Then for each $z \in \C$:

\(\ds T - z I\) \(=\) \(\ds \paren {T - a I - \paren {z - a} I}\)
\(\ds \) \(=\) \(\ds \paren {T - a I} \paren {I - \paren {z - a} \paren {T - a I}^{-1} }\)
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds \paren {T - a I} \paren {I - \paren {z - a} \map f a}\)

Recall that $\map B {c, r}$ denotes the open disc of center $c \in \C$ and radius $r>0$.

For all $z \in \map B {a, \frac{1}{\norm {\map f a} } }$, we have:

\(\ds \paren {I - \paren {z - a} \map f a}^{-1}\) \(=\) \(\ds \sum_{n \mathop \ge 0} \paren {\paren {z - a} \map f a}^n\) Neumann Series, as $\norm {\paren {z - a} \map f a} < 1$
\(\text {(2)}: \quad\) \(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \paren {\map f a}^n \paren {z - a}^n\)

Therefore:

\(\ds \paren {T - zI}^{-1}\) \(=\) \(\ds \paren {\paren {T - a I} \paren {I - \paren {z - a} \map f a} }^{-1}\) by $(1)$
\(\ds \) \(=\) \(\ds \paren {I - \paren {z - a} \map f a}^{-1} \paren {T - a I}^{-1}\) Inverse of Product
\(\ds \) \(=\) \(\ds \paren {I - \paren {z - a} \map f a}^{-1} \map f a\)
\(\ds \) \(=\) \(\ds \paren {\sum_{n \mathop \ge 0} \paren {\map f a}^n \paren {z - a}^n} \map f a\) by $(2)$
\(\ds \) \(=\) \(\ds \sum_{n \mathop \ge 0} \paren {\map f a}^{n+1} \paren {z - a}^n\)

That is, in a neighborhood of each $a \in \map \rho T$, $f$ can be written as:

$\ds \map f z = \sum_{n \mathop \ge 0} \paren {\map f a}^{n+1} \paren {z - a}^n$

This means that $f$ is analytic on $\map \rho T$.


Finally, by Derivative of Power Series, in the neighborhood of each $a \in \map \rho T$:

$\ds \map {f'} z = \sum_{n \mathop \ge 1} \paren {\map f a}^{n+1} n \paren {z - a}^{n-1}$

Choosing $z = a$, we obtain:

$\map {f'} a = \paren {\map f a}^2 = \paren {T - aI}^{-2}$

$\blacksquare$

Retrieved from "https://proofwiki.org/w/index.php?title=Resolvent_Mapping_is_Analytic/Bounded_Linear_Operator&oldid=642607"