**Exercise 6.1** Let \(1\leq q < \infty\) and let \(S:\ell_q\rightarrow\ell_q\) be the left shift operator given by

$$S(\eta_1,\eta_2,\eta_3,\ldots) = (\eta_2, \eta_3, \eta_4, \ldots),\quad (\eta_k)_{k=1}^\infty\in\ell_q.$$

Show that λ is an eigenvalue for *S* if and only if \(|\lambda|<1\). What are the eigenvalues if \(q=\infty\)?

**Exercise 6.2** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a bounded linear operator. If *X* is reflexive, show that \(T(B_X)\) is closed in *Y*.

**Exercise 6.3** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a bounded linear operator. Show that \(T^\ast:(Y^\ast,w^\ast)\rightarrow (X^\ast,\|\cdot\|)\) is continuous only if \(T^\ast\) has finite rank.

**Exercise 6.4** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a compact operator. If \((x_n)_{n=1}^\infty\) is a sequence in *X* such that \(x_n\rightarrow 0\) weakly, then show that \(\displaystyle\lim_{n\rightarrow\infty} \|T(x_n)\|= 0\).

**Exercise 6.5** Show that the converse to Exercise 6.4 is true if *X* is a reflexive Banach space. That is, if *X* and *Y* are Banach spaces such that *X* is reflexive, and \(\displaystyle\lim_{n\rightarrow\infty} \|T(x_n)\|= 0\) whenever \(x_n\rightarrow 0\) weakly, show that *T* is a compact operator.

**Exercise 6.6** Let *X* be a reflexive Banach space and let *E* be an infinite-dimensional closed proper subspace of *X*. Show there exists an \(x\in X\) with \(\|x\|=1\) such that \(d(x,E)=\inf\{\|x-e\|:e\in E\}=1\). (*Hint:* Find a weak limit point of the sequence \((e_n)_{n=1}^\infty\) in the proof of Lemma 6.34.)

**Exercise 6.7** Let *X* be a reflexive Banach space. Show that if \(T:X\rightarrow X\) is a compact operator, then there exists an element \(x\in X\) with \(\|x\|= 1\) such that \(\|Tx\|=\|T\|\). (*Hint:* Use Theorem 5.39.)

**Exercise 6.8** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a compact operator. If *T*(*X*) is dense in *Y*, then show that *Y* is separable.

**Exercise 6.9** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a compact operator. If \(T(X)=Y\), then show that *Y* has finite dimension.

**Exercise 6.10** Suppose *X* and *Y* are Banach spaces and \(T:X\rightarrow Y\) is a bounded linear operator. If *T*(*X*) is a closed infinite-dimensional subset of *Y*, is *T* a compact operator? Explain your answer.

**Exercise 6.11** Let *E* be a Banach space. Show that *E* is infinite-dimensional if and only if \(E^\ast\) is infinite-dimensional. Also, show that \(\mathrm{dim}(E) = \mathrm{dim}(E^\ast)\) if *E* is finite-dimensional. (*Hint:* Use the Hahn–Banach Theorem.)

**Exercise 6.12** Let

*X* be a Banach space and suppose

\(K:X\rightarrow X\) is a compact operator. Let

\(\lambda\neq 0\) and let

\(T=\lambda I - K\) be a compact perturbation of a scaling of the identity.

- (a)
Show there exists an \(N\in\mathbb{N}\) such that \(\mathrm{ker}\big((T^\ast)^m\big) = \mathrm{ker}\big((T^\ast)^{N}\big)\) for all \(m\geq N\).

- (b)
Use (a) to show that \(\mathrm{ran} (T^m) = \mathrm{ran} (T^{N})\) for all \(m\geq N\), where *N* is the natural number from (a). (*Hint:* Use Exercise 5.20.)

**Exercise 6.13** Suppose that \(T:C[0,1]\rightarrow C[0,1]\) is a compact operator. Show that there exists a sequence \((T_n)_{n=1}^\infty\) of finite rank operators such that \(\displaystyle \lim_{n\rightarrow\infty} \|T-T_n\|=0\).

**Exercise 6.14** Define a map \(T_K:C[0,1]\rightarrow C[0,1]\) by

$$T_K f(x) = \int_0^1 \sin\big((x-y)\pi\big) f(y)\, dy,\quad f\in C[0,1], \ x\in [0,1].$$

Show that *T* _{ K } is a compact operator. Compute \(\mathrm{ker}(T_K)\) and \(\mathrm{ran}(T_K)\).

**Exercise 6.15** Let *X* and *Y* be Banach spaces and suppose \(T:X\rightarrow Y\) is a bounded linear operator. The operator *T* is called *weakly compact* if the set \(T(B_X)\) is relatively compact in the weak topology on *Y*. Show that *T* is weakly compact if and only if \(T^{\ast\ast}(X^{\ast\ast})\subseteq Y\).

**Exercise 6.16 (Gantmacher’s Theorem)** Prove the following: If *T* is weakly compact, then \(T^\ast\) is weakly compact.

**Exercise 6.17** Let *X* and *Y* be Banach spaces. Show that if *S* and *T* in \(\mathcal{L}(X,Y)\) are weakly compact operators, then \(S+T\) is a weakly compact operator.

**Exercise 6.18** Assume that *W*, *X*, *Y*, and *Z* are Banach spaces and suppose that the map \(T:X\rightarrow Y\) is a weakly compact operator. Show that if \(A:W\rightarrow X\) and \(B:Y\rightarrow Z\) are bounded linear operators, then \(BTA:W\rightarrow Z\) is weakly compact.

**Exercise 6.19** Prove the following theorems of Pettis:

- (a)
If *X* is reflexive and \(T: X \rightarrow \ell_1\) is a bounded linear operator, then *T* is compact.

- (b)
If *X* is reflexive and \(S:c_0\rightarrow X\) is a bounded linear operator, then *S* is compact.