Lecture Notes in Functional Analysis

by William L. Paschke

edition 0.9

Contents

Frontmatter

I. Hilbert Space

II. Bounded Operators

III. Compact Operators

IV. The Spectral Theorem

Index and References

## D. Non-Emptiness of the Spectrum

Proposition II.40

Let $$$T\in \mathcal{L}\mathopen{}\left( X\right)\mathclose{}$$$. Then

1. for $$${λ}_{0}\in \mathbb{C}\setminus \mathop{\sigma}\mathopen{}\left( T\right)\mathclose{}$$$ we have $$\lim_{λ\to{λ}_{0}}{} \frac{1}{λ-{λ}_{0}}\mathopen{}\left({ \mathopen{}\left(λ-T\right)\mathclose{} }^{-1}-\mathopen{}\left({λ}_{0}-{T}^{-1}\right)\mathclose{}\right)\mathclose{} = {-} {\mathopen{}\left({λ}_{0}-T\right)\mathclose{}}^{{-}2}$$
2. $$$\lim_{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}\to\infty}{} {\mathopen{}\left(λ-T\right)\mathclose{}}^{-1} = 0$$$.

Proof.

1. Note that $${\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}-{ \mathopen{}\left({λ}_{0}-T\right)\mathclose{} }^{-1}= {\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}\mathopen{}\left(\mathopen{}\left({λ}_{0}-T\right)\mathclose{}-\mathopen{}\left(λ-T\right)\mathclose{}\right)\mathclose{}{ \mathopen{}\left({λ}_{0}-T\right)\mathclose{} }^{-1}= \mathopen{}\left({λ}_{0}-λ\right)\mathclose{}{\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}{ \mathopen{}\left({λ}_{0}-T\right)\mathclose{} }^{-1}\text{,}$$ divide by $$$λ-{λ}_{0}$$$ and use continuity of inversion.
2. For $$$\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}\gt \mathopen{}\left\lVert{}T\right\rVert\mathclose{}$$$, $$${\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}= { \mathopen{}\left(λ\mathopen{}\left(1-\frac{1}{λ}T\right)\mathclose{}\right)\mathclose{} }^{-1}= \frac{1}{λ}{ \mathopen{}\left(1-\frac{1}{{λ}_{1}}T\right)\mathclose{} }^{-1}$$$. Then $$\mathopen{}\left\lVert{}{\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}\right\rVert\mathclose{}= \frac{1}{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}}\mathopen{}\left\lVert{}{ \mathopen{}\left(1-\frac{1}{λ}T\right)\mathclose{} }^{-1}\right\rVert\mathclose{}\leq \frac{1}{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}}\frac{1}{1-\frac{\mathopen{}\left\lVert{}T\right\rVert\mathclose{}}{\mathopen{}\left\lVert{}λ\right\rVert\mathclose{}}}= \frac{1}{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}-\mathopen{}\left\lVert{}T\right\rVert\mathclose{}} \text{.}$$

Our proof that the spectrum of a bounded operator is always non-empty uses Liouville's Theorem and the Hahn-Banach Theorem, specifically the consequence of the latter asserting that the intersection of the kernels of all the bounded linear functionals on a normed linear space is $$$\mathopen{}\left\{\, 0\,\right\}\mathclose{}$$$.

Theorem II.41

The spectrum $$$\mathop{\sigma}\mathopen{}\left( T\right)\mathclose{}$$$ is nonempty for every $$$T\in \mathcal{L}\mathopen{}\left( X\right)\mathclose{}$$$.

Proof. Suppose $$$T\in \mathcal{L}\mathopen{}\left( X\right)\mathclose{}$$$ has empty spectrum. That is, $$$λ-T\in { \mathcal{L}\mathopen{}\left( X\right)\mathclose{} }^{-1}$$$ for all $$$λ\in \mathbb{C}$$$. Take $$$φ\in {\mathcal{L}\mathopen{}\left( X\right)\mathclose{}}^{*}$$$ and consider $$$f\mathopen{}\left( λ\right)\mathclose{}= φ\mathopen{}\left( { \mathopen{}\left(λ-T\right)\mathclose{} }^{-1}\right)\mathclose{}$$$. It follows from Proposition II.40 and the continuity of $$$φ$$$ that $$$f$$$ is analytic on $$$\mathbb{C}$$$ (with derivative $$$f' \mathopen{}\left( λ\right)\mathclose{}= {-} φ\mathopen{}\left( {\mathopen{}\left(λ-T\right)\mathclose{}}^{{-}2}\right)\mathclose{}$$$). Furthermore, $$$\lim_{\mathopen{}\left\lvert{}λ\right\rvert\mathclose{}\to\infty}{}\mathopen{}\left\lvert{}f\mathopen{}\left( λ\right)\mathclose{}\right\rvert\mathclose{}= 0$$$ by Proposition II.40 — so $$$f$$$ is bounded on $$$\mathbb{C}$$$. Then, by Liouville's Theorem, $$$f$$$ must be constant — and so $$$f\equiv 0$$$. This means for all $$$λ\in \mathbb{C}$$$ and for all $$$φ\in \mathcal{L}\mathopen{}\left( X\right)\mathclose{}^{*}$$$, we have $$$φ\mathopen{}\left( {\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}\right)\mathclose{}= 0$$$.

It follows from the Hahn-Banach Theorem that $$${\mathopen{}\left(λ-T\right)\mathclose{}}^{-1}= 0$$$ for every $$$λ$$$, which is impossible.