An Introduction to Abstract Linear Operator Theory

Let $H$ be a Hilbert space and $x_{0}$ be a fixed element of $H$. Prove that the map taking $x$ to $\left\langle x, x_{0} \right\rangle$ is a linear operator from $H$ to $\mathbb {C}$.

Consider the operator defined by

This is a Fredholm operator and is often referred to as the Cesaro operator.

In this section you encountered infinite matrices, perhaps for the first time. In general, matrix multiplication is not commutative. This is the case for finite as well as for infinite matrices. But multiplication for infinite matrices gets even worse. For finite matrices, there are certain situations in which multiplication is, in fact, commutative. For example, if $A B=I$ for two square matrices $A$ and $B$, then $A$ and $B$ commute. This shows that a finite matrix with a "left inverse" also has a "right inverse" (and they are equal). For infinite-dimensional matrices, this need not be the case. Give examples of infinite matrices $A$ and $B$ such that $A B=I$, yet $B A \neq I$. See Exercise 5.3.3 for more on this topic.

Prove that $\mathcal{B}(X, Y)$ is a normed linear space. That is, prove Theorem 5.3.

Theorem 5.3: Let $X$ and $Y$ be normed linear spaces. The collection $\mathcal{B}(X, Y)$ of all bounded linear operators $T: X \to Y$, endowed with the operator norm, forms a normed linear space.

Show that the identity operator from $\left(\mathcal{C}([0,1]),\left\| \cdot \right\|_{\infty }\right)$ to $\left(\mathcal{C}([0,1]),\left\| \cdot \right\|_{1}\right)$ is a bounded linear operator, but that the identity map from $\left(\mathcal{C}([0,1]),\left\| \cdot \right\|_{1}\right)$ to $\left(\mathcal{C}([0,1]),\left\| \cdot \right\|_{\infty }\right)$ is unbounded. This phenomenon cannot happen on a Banach space; see Exercise 6.3.2.

Fix a continuous function $\phi :[0,1] \rightarrow [0,1]$ and define the composition operator $C_{\phi }: C([0,1]) \rightarrow C([0,1])$ by $C_{\phi } f(x)=f(\phi (x))$. Prove that $C_{\phi }$ is a bounded linear operator with $\left\| C_{\phi }\right\| \leq 1$.

For $\left\{ a_{n}\right\}_{n=1}^{\infty } \in \ell^{\infty }$ define

Show that $T$ is a bounded linear operator from $\ell^{1}$ to $\mathbb {C}$, and compute $\left\| T\right\|$.

Consider an $n \times n$ matrix $A=\left(a_{i j}\right)$ that satisfies $a_{i j}=\overline{a_{j i}}$ for all pairs of indices $i$ and $j$. Such a matrix is called Hermitian. View $A$ as a linear operator $\mathbb {C}^{n} \rightarrow \mathbb {C}^{n}$. Assume that $\mathbb {C}^{n}$ is endowed with the usual norm. Prove that $\left\| A\right\| =\max_{k}\left\| \lambda_{k}\right\|$, where $\lambda_{k}$ are the eigenvalues of $A$.

Consider

as a linear operator from $\mathbb {C}^{2}$ to itself. The aim of this exercise is to show that the operator norm of $A$ depends on the choice of norm on $\mathbb {C}^{2}$.

Let $C^{1}([0,1])$ denote the collection of all continuous functions with continuous derivative (including one-sided derivatives at the endpoints).

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.

Verify that the Hermitian matrices (as defined in Exercise 5.2.5) are exactly the Hermitian operators on the finite-dimensional Hilbert space $\mathbb {C}^{n}$.

Describe the Fredholm integral operators of the first kind that are Hermitian.

Let $K$ be a subspace of a Hilbert space $H$.

Complete the proof of Theorem 5.15 by verifying the uniqueness assertion.

Theorem 5.15: Assume that $K$ is a closed subspace of a Hilbert space $H$ and that $x \in H$. Then there exists $y \in K$ such that $\left\| x - y\right\| = \inf \left\{ \left\| x - z\right\| \mid z \in K\right\}$. Further, the element $y$ is the unique element of $K$ with this property.

Prove Theorem 5.17.

Theorem 5.17: Let $H$ be a Hilbert space. If $T \in \mathcal{B}(H)$ is Hermitian, then $\ker (T)$ and $\operatorname {range}(T)$ are subspaces of $H$ and satisfy $\ker (T)^{\perp } = \operatorname {range}(T)$ and $\operatorname {range}(T)^{\perp } = \ker (T)$ (here $\ker (T)$ and $\operatorname {range}(T)$ denote, respectively, the kernel and range of the map $T$).

It follows from Theorem 5.18 that any eigenvalue of a Hermitian operator is real-valued. Give a more direct proof of this fact.

Theorem 5.18: Let $H$ be a Hilbert space. If $T \in \mathcal{B}(H)$ is Hermitian, then $\sigma (T) \subseteq \mathbb {R}$.

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

For $0 \leq \alpha \leq 1$ set

Verify each of the assertions made in the paragraph that follows Theorem 5.23. That is, prove that $P$ is bounded, linear, and satisfies $P^{2}=P$.

Theorem 5.23 (The Spectral Theorem for Compact Hermitian Operators): Let $T$ be a nonzero, compact, and Hermitian operator on a Hilbert space $H$. There is a sequence of nonzero real-valued eigenvalues $\left\{ \lambda_{k}\right\}$ and a corresponding sequence of orthonormal eigenvectors $\left\{ x_{k}\right\}$ such that: if the sequence of eigenvalues contains infinitely many distinct values, it tends to zero; each nonzero eigenvalue of $T$ appears in the sequence $\left\{ \lambda_{k}\right\}$, and each eigenspace $\ker (\lambda_{k} I - T)$ is finite-dimensional, with dimension equal to the number of times $\lambda_{k}$ appears in the sequence; and for each $x \in H$, $Tx = \sum_{k} \lambda_{k} \left\langle x, x_{k} \right\rangle x_{k}$. In this context, $P$ denotes the orthogonal projection onto an eigenspace arising from this decomposition.

In each of the following examples, explain why $T$ is compact and Hermitian. Then give the spectral decomposition guaranteed by Theorem 5.23. That is, find the eigenvalues, the corresponding eigenspaces, and projections.

Theorem 5.23 (The Spectral Theorem for Compact Hermitian Operators): Let $T$ be a nonzero, compact, and Hermitian operator on a Hilbert space $H$. There is a sequence of nonzero real-valued eigenvalues $\left\{ \lambda_{k}\right\}$ and a corresponding sequence of orthonormal eigenvectors $\left\{ x_{k}\right\}$ such that: if the sequence of eigenvalues contains infinitely many distinct values, it tends to zero; each nonzero eigenvalue of $T$ appears in the sequence $\left\{ \lambda_{k}\right\}$, and each eigenspace $\ker (\lambda_{k} I - T)$ is finite-dimensional, with dimension equal to the number of times $\lambda_{k}$ appears in the sequence; and for each $x \in H$, $Tx = \sum_{k} \lambda_{k} \left\langle x, x_{k} \right\rangle x_{k}$.