Math Problems Vault

Problem 5.2.1

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.

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Problem 5.2.2

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.

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Problem 5.2.3

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 $\left\| C_{\phi }\right\| \leq 1$ .

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Problem 5.2.4

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

T(x)=\sum _{n=1}^{\infty } x_{n} a_{n}, \quad \text{ for all } x=\left\{ x_{n}\right\} _{n=1}^{\infty } \in \ell ^{1}

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

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Problem 5.2.5

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 $A$ .

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Problem 5.2.6

(a)(b)(c)(d)

Consider

A=\left(\begin{array}{ll} a & b \\ c & d \end{array}\right), \quad a, b, c, d \in \mathbb {C}

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 $\mathbb {C}^{2}$ .

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(a)

Endow $\mathbb {C}^{2}$ with the supremum norm, $\left\| \cdot \right\|_{\infty }$ . Show that in this case, the norm of $A$ is given by

\max (|a|+|b|,|c|+|d|)

(b)

Endow $\mathbb {C}^{2}$ with the 1-norm, $\left\| \cdot \right\|_{1}$ . Show that in this case, the norm of $A$ is given by

\max (|a|+|c|,|b|+|d|)

(c)

Endow $\mathbb {C}^{2}$ with the 2-norm, $\left\| \cdot \right\|_{2}$ . Consider the matrix

A^{*}=\left(\begin{array}{cc} \bar{a} & \bar{c} \\ \bar{b} & \bar{d} \end{array}\right)

and form the matrix $A A^{*}$ . Show that in this case, the norm of $A$ is given by

\left[\frac{1}{2}\left(\operatorname {tr}\left(A A^{*}\right)+\sqrt{\left(\operatorname {tr} A A^{*}\right)^{2}-4 \operatorname {det}\left(A A^{*}\right)}\right)\right]^{\frac{1}{2}}

Use this to write down a formula for the norm of $A$ as a function of its entries $a, b, c, d$ . Here "tr" denotes trace, and "det" denotes determinant.

(d)

Finally, compute these 3 operator norms of

A= \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}

Problem 5.2.7

(a)(b)

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

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(a)

Show that $C^{1}([0,1])$ , with the supremum norm, is a subspace of $C([0,1])$ but that it is not closed.

(b)

Prove that the differential operator

\frac{d}{d x}: C^{\prime }([0,1]) \rightarrow C([0,1])

is linear but not bounded.