Linear Operators

Prove Proposition 11.1.10.

Proposition 11.1.10: Let $X, Y$ be normed linear spaces. The operator norm on $\mathcal{L}(X, Y)$ is a norm.

Prove Proposition 11.1.11.

Proposition 11.1.11: Let $X, Y$ be normed linear spaces and let $T \in \mathcal{L}(X, Y)$. Then (a) $\left\| T\right\| = \sup \left\{ \left\| Tx\right\| : \left\| x\right\| \leq 1\right\} = \sup \left\{ \left\| Tx\right\| : \left\| x\right\| = 1\right\}$. (b) $\left\| Tx\right\| \leq \left\| T\right\| \left\| x\right\|$, for all $x \in X$.

Let $X, Y$ be normed linear spaces and $T \in \mathcal{L}(X, Y)$. Suppose that $M>0$ and $\left\| T\right\| \leq M$.

Let $X, Y$ be normed linear spaces, and let $T: X \rightarrow Y$ be a linear operator. Prove that $T$ is bounded if and only if it maps bounded sets to bounded sets.

Let $X$ be a normed linear space and $x_{0} \in X$. Prove that a linear operator on a normed space is continuous if and only if it is continuous at $x_{0}$.

Let $X$ be a normed linear space and let $T, S \in \mathcal{L}(X, X)$. Prove that the composition $S T$ belongs to $\mathcal{L}(X, X)$ and that $\left\| S T\right\| \leq \left\| S\right\| \left\| T\right\|$. (We write $S T$ for $S \circ T$ just like when $S$ and $T$ are matrices.)

Let $X$ be a Banach space and $\mathcal{F}$ a family of bounded linear operators from $X$ to a normed linear space $Y$. Prove that $\mathcal{F}$ is equicontinuous if and only if it is uniformly bounded.

Let $\mathcal{H}$ be Hilbert space and $T \in \mathcal{L}(\mathcal{H})$. Prove that

Let $X, Y$ be normed linear spaces and $T: X \rightarrow Y$ a linear map such that $\left\{ T x_{n}\right\}$ is bounded in $Y$ whenever the sequence $\left\{ x_{n}\right\}$ converges to 0 in $X$. Prove that $T \in \mathcal{L}(X, Y)$.

Let $X, Y$ be normed linear spaces, and let $T: X \rightarrow Y$ be a linear operator. Prove that $T$ is bounded if and only if it maps every Cauchy sequence in $X$ to a Cauchy sequence in $Y$.

Prove that the set of all linear transformations from a vector space $X$ to a vector space $Y$ is a vector space.

Let $Z$ be a subspace of a normed linear space $X$, let $Y$ be a Banach space, and let $A: Z \rightarrow Y$ be a continuous linear operator. Prove that there exists a unique continuous linear operator $\bar{A}: \bar{Z} \rightarrow Y$ such that $\bar{A} \mid Z=A$ and $\left\| \bar{A}\right\| =\left\| A\right\|$.

Let $X, Y$ be normed spaces, let $T \in \mathcal{L}(X, Y)$, and let $\tilde{T}: X / \operatorname {ker} T \rightarrow Y$ be defined by $\tilde{T}([x])=T x$ (see Exercise 10.1.13). Prove that $\tilde{T}$ is a welldefined, injective, bounded, linear operator for which $\left\| \tilde{T}\right\| =\left\| T\right\|$.

Let $X$ be a normed space and $V$ a closed subspace of $X$ (see Exercise 10.1.13). Prove that:

Let $\mathcal{H}$ be Hilbert space and let $T \in \mathcal{L}(\mathcal{H})$. Prove that $T=0$ if and only if $\left\langle T x, y \right\rangle =0$ for all $x, y \in \mathcal{H}$.

Let $X$ be a Banach space and $T \in \mathcal{L}(X, X)$ with $\left\| T\right\| <1$. Define $T^{0}=\operatorname {Id}$. Prove that $\sum_{n=0}^{\infty } T^{n}$ converges in $\mathcal{L}(X, X)$.

Let $X, Y$ be normed linear spaces, let $\left\{ T_{n}\right\}$ converge to $T$ in $\mathcal{L}(X, Y)$, and let $\left\{ x_{n}\right\}$ converge to $x$ in $X$. Prove that $\left\{ T_{n} x_{n}\right\}$ converges to $T x$ in $Y$.

Let $X, Y$ be normed linear spaces and $\left\{ T_{n}\right\} , T \in \mathcal{L}(X, Y)$. Prove that $\lim_{n} T_{n}=T$ if and only if $\left\{ T_{n} x\right\}$ converges to $T x$ uniformly on the unit ball of $X$.

Let $g$ be a continuous function on $[0,1]$ and let $T: L^{p}\left({[0,1]}\right) \rightarrow L^{p}\left({[0,1]}\right)$ be defined by $T(f)=g f$. Prove that $T$ is a bounded operator and $\left\| T\right\| =\left\| g\right\|_{\max }$.

Let $p, q$ be real numbers such that $q>1$ and $1 / p+1 / q=1$, and let $\left(a_{i j}\right)$ be an infinite matrix of real numbers such that

Prove that

defines a bounded linear operator

from $\ell^{p}$ to $\ell^{q}$.

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{\infty }\right)$,

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a continuous function. Prove that

defines an operator in $\mathcal{L}(C[0,1])$,

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

where $\left\| K(s, t)\right\|_{t, \infty }$ means the $L^{\infty }$ norm of $K$ as a function of $t$ for a fixed $s$. Prove that

defines an operator in $\mathcal{L}\left(L^{1}\left({[0,1]}\right)\right)$,

Let $q>1$ and let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{p}\left({[0,1]}\right), L^{q}\left({[0,1]}\right)\right),(1 / p+1 / q=1)$

Let $K:[0,1] \times [0,1] \rightarrow \mathbb {R}$ be a measurable function such that

Prove that

defines an operator in $\mathcal{L}\left(L^{2}\left({[0,1]}\right)\right)$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space, let $h$ be a bounded measurable function on $X$, and let $T$ be the operator on $L^{p}\left(X, \mu \right)$ defined by $T f=h f$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a finite measure space and let $p \geq q$. Find the norm of the identity operator $\operatorname {Id}: L^{p}\left(X, \mu \right) \rightarrow L^{q}\left(X, \mu \right)$.

Let $1 \leq p \leq q<\infty$. Determine for what functions $h$ on $[0,1]$ is the multiplication by $h$ a bounded operator from $L^{p}[0,1]$ to $L^{q}[0,1]$.

Let $T: C[0,1] \rightarrow C[0,1]$ be defined by $T f(x)=x^{3} f(x)$, for all $f \in C[0,1]$ and each $x \in [0,1]$.

Let $T: C[0,1] \rightarrow C[0,1]$ be defined by $T f(x)=\int_{0}^{x} f(t) d t$, for all $f \in C[0,1]$ and each $x \in [0,1]$. Prove that $T$ is a bounded linear operator and find its norm.

Let $T: L^{1}[0,1] \rightarrow L^{1}[0,1]$ be defined by $T f(x)=f(\sqrt{x})$, for all $f \in C[0,1]$ and each $x \in [0,1]$. Prove that $T$ is a bounded linear operator and find its norm.

For $f \in L^{1}[0,1]$ let $x_{n}=\int_{0}^{1} f(t) t^{n} d t$, and let $T(f)=\left\{ x_{n}\right\}$. Prove that the operator $T$ is a bounded linear operator mapping $L^{1}[0,1]$ to $c_{0}$ and determine its norm. (See Exercise 10.1.15.)

Let ( $X, \mathfrak {M}, \mu$ ) and ( $Y, \mathfrak {N}, \nu$ ) be $\sigma$-finite measure spaces, let $K$ be an $\mathfrak {M} \otimes \mathfrak {N}$ measurable function on $X \times Y$ and let $C>0$. Suppose that

Prove that the operator $T$ defined on $L^{p}\left(X, \mu \right)$ by

is a bounded operator and that $\left\| T f\right\|_{p} \leq C\left\| f\right\|_{p}$.

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $1 \leq p<r \leq \infty$. Prove that:

Let ( $X, \mathfrak {M}, \mu$ ) be a measure space and let $1 \leq p<r \leq \infty$. Prove that: