Banach Algebras and Spectra; Compact Operators

Assume that $X$ is a normed linear space, and $S, T \in \mathcal{B}(X)$.

Assume that $X$ and $Y$ are normed linear spaces. Let $K$ be a compact operator in $\mathcal{B}(X, Y)$. Use Theorem 2.4 to prove that $K(B)$ has compact closure for each closed ball $B$ in $X$.

Theorem 2.4: A subset of a metric space is compact if and only if it is sequentially compact.

The situation discovered in Exercise 5.1.3 is somewhat redeemable. Let $X$ be a Banach space and $T \in \mathcal{B}(X)$. Assume that there are $U, V \in \mathcal{B}(X)$ satisfying

Prove that $U=V$. In other words, if $T \in \mathcal{B}(X)$ is both left and right invertible, then its left and right inverses must be equal.

In a unital normed algebra with multiplicative identity $e$, prove that $\left\| e\right\| \geq 1$.

In a normed algebra, prove that $\left\| a^{k}\right\| \leq \left\| a\right\|^{k}$ for each element $a$ in the algebra and each positive integer $k$.

Prove that $C([0,1])$, with the supremum norm, is a unital Banach algebra.

Assume that $X$ is a Banach space. In Theorem 5.6 we saw that any $S \in \mathcal{B}(X)$ that is sufficiently close to an invertible operator $T$ is also invertible. Given that $\left\| S-T\right\| <\left\| T^{-1}\right\|^{-1}$, prove that $S^{-1}$ satisfies

Assume that $X$ is a Banach space. Assume that $S, T \in \mathcal{B}(X)$ and $\lambda \in \mathbb {C} \backslash \sigma (S T), \lambda \neq 0$.

Consider the operator on $L^{2}((0,1))$ defined by

Prove that $I-T$ is an invertible operator.

Let $\ell^{1}(\mathbb {Z})$ denote the collection of all (doubly ended) sequences $\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ of complex numbers such that $\sum_{i=-\infty }^{\infty }\left|a_{i}\right|<\infty$. This space is very much like $\ell^{1}=\ell^{1}(\mathbb {N})$. With the norm of $a=\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ given by

$\ell^{1}(\mathbb {Z})$ becomes a Banach space. The point of this exercise is to see that we can define a multiplication to make $\ell^{1}(\mathbb {Z})$ into a Banach algebra. For $a=\left\{ a_{i}\right\}_{i=-\infty }^{\infty }$ and $b=\left\{ b_{i}\right\}_{i=-\infty }^{\infty }$ in $\ell^{1}(\mathbb {Z})$, define the product $a * b$ by

This multiplication is called convolution. Show that $\ell^{1}(\mathbb {Z})$ is a Banach algebra.

Prove that every compact operator is a bounded operator.

Consider the Volterra operator defined on $L^{2}([0,1])$ by

with $\left|k(s, t)\right| \leq C$ for some constant $C$ and every $s$ and $t$ in $[0,1]$.

In this exercise you are asked to fill in the details of Example 4.

Let $W$ be the weighted shift in Example 6.

Let $X$ be an infinite-dimensional Banach space. (Parts (a) and (b) are related only in that they are about compact operators.)

Assume that $X$ is a Banach space. Prove that the collection of all compact operators forms a closed, two-sided ideal in the Banach algebra $\mathcal{B}(X)$. Note: You may know what an "ideal" is from abstract algebra; if you do not, it is defined in the next chapter.

Learn enough about complex function theory to prove the results of this section.

Find a good reference (like [21] or [124]) and work through the proof of Theorem 5.11, which describes the spectrum of a compact operator.

Theorem 5.11: Let $X$ be an infinite-dimensional Banach space and $T \in \mathcal{B}(X)$ a compact operator. Then the spectrum of $T$ is either a finite set or a sequence converging to $0$. The point $0$ is in the spectrum, and each nonzero value of the spectrum is an eigenvalue. Further, for $\lambda \neq 0$, the eigenspace $\ker (\lambda I - T)$ is finite-dimensional.