**Exercise 8.1** Let *A* be a unital Banach algebra. Show that if \(a\in A\) has an inverse, then it must be unique. Show that if *a* has both a left-inverse and a right-inverse, then *a* is invertible.

**Exercise 8.2** Let *A* be a noncommutative unital Banach algebra. Suppose *x* and *y* are elements in *A* such that both *xy* and *yx* are invertible. Show that both *x* and *y* are invertible and show that \((xy)^{-1} = y^{-1}x^{-1}\).

**Exercise 8.3** Let *A* be a unital Banach algebra with identity element 1. Suppose \(a\in A\) is such that \(\sum_{n=0}^\infty \|a\|^n <\infty\). Show that 1 − *a* is an invertible element and that \((1-a)^{-1} = \sum_{n=0}^\infty a^n.\)

**Exercise 8.4** Let *X* be a Banach space and suppose \(T\in\mathcal{L}(X)\) is invertible. Show \(\lambda\in\mathrm{Sp}(T)\) if and only if \(1/\lambda\in\mathrm{Sp}\Big(T^{-1}\Big)\).

**Exercise 8.5** Let *A* be a unital Banach algebra. Show that the set of invertible elements of *A* is a group.

**Exercise 8.6** Let *A* be a commutative unital Banach algebra and let *G*(*A*) be the group of invertible elements of *A*. Show that \(x\in G(A)\) if and only if \(\phi(x)\neq 0\) for all multiplicative linear functionals *ϕ*.

**Exercise 8.7** Let *A* be a Banach algebra. An element \(x\in A\) is called a *topological divisor of zero* if there exists a sequence \((y_n)_{n=1}^\infty\) in *A* with \(\|y_n\|=1\) for all \(n\in\mathbb{N}\) such that \(\displaystyle \lim_{n\rightarrow\infty} xy_n = 0 = \lim_{n\rightarrow\infty} y_nx\). Find an example of a Banach algebra with a topological divisor of zero that is not a divisor of zero.

**Exercise 8.8** Let *A* be a unital Banach algebra and let *G*(*A*) be the group of invertible elements. Show that every \(x\in \partial G(A)\) is a topological divisor of zero, where \(\partial G(A)\) is the boundary of *G*(*A*).

**Exercise 8.9** Let *X* be a commutative Banach algebra. A proper ideal *I* of *X* is called *prime* if for any elements *a* and *b* in *X* the product \(ab\in I\) implies that either \(a\in I\) or \(b\in I\). Show that in a commutative unital Banach algebra any maximal ideal is prime.

**Exercise 8.10** Let *M* be the multiplier operator on \(L_2=L_2\big([0,2\pi), \frac{d\theta}{2\pi}\big)\) given by the formula \(Mf(\theta) = e^{i\theta} f(\theta)\) for all \(f\in L_2\) and \(\theta\in [0,2\pi)\). Show that the Fourier transform of the multiplier operator satisfies the equation \(\widehat{Mf}(n) = \hat{f}(n-1)\) for all \(f\in L_2\) and each \(n\in\mathbb{Z}\). Explicitly write out the relationship between *M* and the shift operator *R* on \(\ell_2(\mathbb{Z})\). (Recall that \(\hat{f}(n)=\int_0^{2\pi} f(\theta)\, e^{-in\theta}\, \frac{d\theta}{2\pi}\) for all \(n\in\mathbb{Z}\).)

**Exercise 8.11** If *N* is a subspace of a vector space *X*, then the *codimension* of *N* in *X* is the dimension of \(X/N\). Show that an ideal of a complex commutative unital Banach algebra *X* is maximal if and only if it has codimension 1. (*Hint:* See the proof of Theorem 8.40.)

**Exercise 8.12** Let *A* be a complex unital Banach algebra. Suppose there exists some \(M<\infty\) such that \(\|x\| \|y\| \leq M \|xy\|\) for all *x* and *y* in *A*. Show that \(A=\mathbb{C}\).

**Exercise 8.13** Let *A* be a Banach algebra and suppose \(a\in A\). Use Fekete’s Lemma (Exercise 3.10) to show that

$$\lim_{n\rightarrow\infty} \|a^n\|^{1/n} = \inf_{n\in\mathbb{N}} \|a^n\|^{1/n}.$$

(*Hint:* Let \(\theta_n = \log \|a^n\|\).)

**Exercise 8.14** Let \(A = \ell_1(\mathbb{Z}_+)\) be the convolution algebra from Example 8.6(c). Recall that the multiplication in *A* is given by the formula \((a\ast b)_k = \sum_{j=0}^k a_{k-j}\, b_j\) for all \(k\in\mathbb{Z}_+\), where \(a = (a_j)_{j=0}^\infty\) and \(b=(b_j)_{j=0}^\infty\) are sequences in *A*. Identify \(\Sigma = \Sigma(A)\), the spectrum of *A*, and show that \(C(\Sigma)\) is the collection of all functions having an absolutely convergent Taylor series in \(\overline{\mathbb{D}}\). (*Hint: A* is a subalgebra of *W*.)

**Exercise 8.15** If *f* has an absolutely convergent Taylor series in \(\overline{\mathbb{D}}\), and if \(f(z)\neq 0\) for any \(z\in \overline{\mathbb{D}}\), then show that \(1/f\) has an absolutely convergent Taylor series in \(\overline{\mathbb{D}}\). (*Hint:* Use the previous problem.)