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Elementary Functional Analysis

Barbara MacCluer

Chapter 5

Banach and $C^{*}$-Algebras - all with Video Answers

Educators

Chapter Questions

01:32

Problem 1

Show that a Banach algebra $\mathscr{A}$ with an involution satisfying
$$
\left\|A^{*} A\right\| \geq\|A\|^{2}
$$
is a $C^{*}$-algebra, meaning that equality holds in this inequality.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:52

Problem 2

Suppose that $\mathscr{A}$ is a $\mathrm{C}^{*}$-algebra.
(a) Show that if $\mathscr{A}$ has a unit, it is unique (call it $I$ ); furthermore $I^{*}=I$ and $\|I\|=1$ (provided $\|A\| \neq 0$ for some $A \in \mathscr{A}$ ).
(b) Suppose $\mathscr{A}$ is unital. Show that if $A$ is invertible, so is $A^{*}$, with $\left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}$.
(c) Every $A \in \mathscr{A}$ can be written as $A=X+i Y$ where $X$ and $Y$ are self-adjoint.
(d) If $\mathscr{A}$ is unital and $U$ is unitary (meaning $U U^{*}=U^{*} U=I$ ), then $\|U\|=1$.

Nick Johnson
Nick Johnson
Numerade Educator
02:13

Problem 3

Let $\mathscr{G}$ denote the set of invertible elements in a unital Banach algebra. Show that the map of $\mathscr{G}$ into $\mathscr{G}$ defined by $A \rightarrow A^{-1}$ is continuous.

Nick Johnson
Nick Johnson
Numerade Educator
01:09

Problem 4

Suppose that $F: \Omega \rightarrow \mathscr{A}$ is a function defined on an open set $\Omega \subseteq \mathbb{C}$ and taking values in a Banach space $\mathscr{A}$. Show that if $f$ is strongly analytic in $\Omega$, then it is weakly analytic (as defined in Section 5.2).

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
02:04

Problem 5

Recall that for $T \in \mathscr{B}(\mathscr{H})$, the operator $T-\lambda I$ is invertible if and only if $T-\lambda I$ is bounded below and has dense range. So one way for $\lambda$ to get into the spectrum of $T$ is for $T-\lambda I$ to not be bounded below, meaning that there are unit vectors $h_{n}$ with $\left\|(T-\lambda I) h_{n}\right\| \rightarrow 0$. A point $\lambda$ with this property is said to be an approximate eigenvalue of $T$; the set of all approximate eigenvalues of $T$ is called the approximate point spectrum of $T$. Show
(a) Every eigenvalue of $T$ is in the approximate point spectrum of $T$.
(b) The approximate point spectrum of $T$ is a closed set (show its complement is open).
(c) Show that if $T_{n}$ is invertible for all $n$ and $T_{n} \rightarrow T$ where $T$ is not invertible, then 0 is an approximate eigenvalue of $T$. Hints: Explain why it suffices to show that if the range of $T$ is not dense, then there are unit vectors $h_{n}$ with $\left\|T h_{n}\right\| \rightarrow 0$. Then assume that the range of $T$ is not dense and find a nonzero vector $h$ with $h \perp \overline{\operatorname{ran} T}$ (why must such an $h$ exist?). Consider $h_{n}=T_{n}^{-1} h /\left\|T_{n}^{-1} h\right\| .$
(d) If $\lambda$ is in the boundary of $\sigma(T)$, then show that $\lambda$ is an approximate eigenvalue for $T$.
(e) Extend the result of (d) to the case that $T$ is a bounded linear operator on a Banach space $X$, with "approximate eigenvalue" defined in the analogous way.

Manik Pulyani
Manik Pulyani
Numerade Educator
01:49

Problem 6

Suppose that $T \in \mathscr{B}(\mathscr{H}) .$ Show that $\lambda$ is not an approximate eigenvalue of $T$ if and only if $T-\lambda I$ has a left inverse.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
02:17

Problem 7

Let $\sigma_{a p}(A)$ denote the approximate point spectrum for an operator $A \in \mathscr{B}(\mathscr{H})$.
(a) Show that $\Pi_{j=1}^{n}\left(A-\lambda_{j} I\right)$ is bounded below on $\mathscr{H}$ if and only if $A-\lambda_{j} I$ is bounded below for $1 \leq j \leq n .$
(b) Show that for any polynomial $p, \sigma_{a p}(p(A))=p\left(\sigma_{a p}(A)\right)$.
Does the analogous result, with "'approximate point spectrum" replaced by "point spectrum" hold? The point spectrum of $A$ is $\{\lambda: \operatorname{ker}(A-\lambda)$ is nontrivial $\}$, i.e., the set of eigenvalues of $A$.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:25

Problem 8

Suppose that $A$ is a bounded linear operator on a Hilbert space $\mathscr{H}$. Show that if $A-\lambda I$ does not have dense range in $\mathscr{H}$, then $\bar{\lambda}$ is an eigenvalue of $A^{*}$, and conversely, if $\mu$ is an eigenvalue of $A^{*}$, then $A-\bar{\mu} I$ does not have dense range. Thus the compression spectrum of $A$ can be described in terms of the eigenvalues of $A^{*}$.

Dharmendra Jain
Dharmendra Jain
Numerade Educator
01:25

Problem 9

(An Inversion Spectral Mapping Theorem.) Suppose that $A$ is an invertible operator in $\mathscr{B}(\mathscr{H})$. The goal of this problem is to show that
$$
\sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\}
$$
(a) Show that if $A-\lambda I$ is not bounded below, then $A^{-1}-\frac{1}{\lambda} I$ is not bounded below, and conversely that if $A^{-1}-\mu I$ is not bounded below, then $A-\frac{1}{\mu} I$ is not bounded below. Show that the eigenvalues of $A^{-1}$ are precisely the reciprocals of the eigenvalues of $A$.
(b) Show that $A-\lambda I$ fails to have dense range in $\mathscr{H}$ if and only if $A^{-1}-\frac{1}{\lambda} I$ fails to have dense range in $\mathscr{H}$. Exercise $5.8$ may be helpful here.
Conclude that
$$
\sigma\left(A^{-1}\right)=\left\{\frac{1}{\lambda}: \lambda \in \sigma(A)\right\}
$$
This result holds more generally for any invertible element in a unital Banach algebra; see for example, p. 204 in [8].

Dharmendra Jain
Dharmendra Jain
Numerade Educator
08:50

Problem 10

Consider the operator on $\ell^{\infty}$ defined by
$$
T\left(x_{1}, x_{2}, \ldots\right)=\left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots\right)
$$
where $\left(\lambda_{1}, \lambda_{2}, \ldots\right)$ is in $\ell^{\infty}$. Find $\sigma_{p}(T), \sigma(T)$ and show that $\sigma(T) \backslash \sigma_{p}(T)$ is the residual spectrum of $T$.

Ruirui Liu
Ruirui Liu
Numerade Educator
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Problem 11

Recall from Exercise $2.10$ in Chapter 2 that if $W$ is a weighted shift and $\lambda \in \mathbb{C}$ satisfies $|\lambda|=1$, then $\lambda W$ is a weighted shift which is unitarily equivalent to $W$. Show that weighted shifts have "circularly symmetric" spectra, that is, if $\mu \in \sigma(W)$ and $|\lambda|=1$, then $\lambda \mu \in \sigma(W)$.

Victor Salazar
Victor Salazar
Numerade Educator
01:27

Problem 12

Recall the Banach space
$$
C^{1}[0,1]=\{f: f \text { is continuously differentiable on }[0,1]\}
$$
with norm $\|f\|_{\infty}+\left\|f^{\prime}\right\|_{\infty}$.
(a) Show that under pointwise multiplication, $C^{1}[0,1]$ is a Banach algebra. Is it a $C^{*}$-algebra if we define $f^{*}=\bar{f} ?$
(b) Let $g(x)=x$ for $x \in[0,1]$. What is the norm of $g$ in $C^{1}[0,1]$ ? What is the spectral radius $r(g)$ ?
(c) Show that for each closed set $E \subseteq[0,1]$,
$$
\mathscr{J}_{E} \equiv\left\{f \in C^{1}[0,1]: f(x)=0 \text { for } x \in E\right\}
$$
is a closed, two-sided ideal in $C^{1}[0,1]$.
(d) Find a closed ideal in $C^{1}[0,1]$ which is not of the form $\mathscr{J}_{E}$ as in (c).

Carson Merrill
Carson Merrill
Numerade Educator

Problem 13

Suppose $\mathscr{A}$ is a Banach algebra and $\mathscr{J}$ is a proper, closed ideal. Show that
$$
(A+\mathscr{J})(B+\mathscr{J})=A B+\mathscr{J}
$$
is a well-defined multiplication on $\mathscr{A} / \mathscr{J}$ under which this quotient space becomes a complex algebra.

Check back soon!
06:58

Problem 14

Recall that an operator $T \in \mathscr{B}(X)$, where $X$ is a Banach space, is an isometry if $\|T x\|=\|x\|$ for all $x \in X$.
(a) Show that the spectrum of an isometry $T$ is contained in the unit circle $\partial \mathbb{D}$ if $T$ is invertible.
(b) Show that if $T$ is an isometry but is not invertible, then its spectrum is $\bar{D}$. Hint: By Exercise $5.5$, the boundary of the spectrum is contained in the set of approximate eigenvalues of $T$.
(c) Give an example of a continuous $\varphi:[0,1] \rightarrow[0,1]$ so that the composition operator $C_{\varphi}$ (see Exercise $2.3$ in Chapter 2 ) is an isometry on $C[0,1]$ and $\sigma\left(C_{\varphi}\right)=\overline{\mathbb{D}} .$

Abhijit Das
Abhijit Das
Numerade Educator
06:58

Problem 15

Recall that an operator $T \in \mathscr{B}(X)$, where $X$ is a Banach space, is an isometry if $\|T x\|=\|x\|$ for all $x \in X$.
(a) Show that the spectrum of an isometry $T$ is contained in the unit circle $\partial \mathbb{D}$ if $T$ is invertible.
(b) Show that if $T$ is an isometry but is not invertible, then its spectrum is $\bar{D}$. Hint: By Exercise $5.5$, the boundary of the spectrum is contained in the set of approximate eigenvalues of $T$.
(c) Give an example of a continuous $\varphi:[0,1] \rightarrow[0,1]$ so that the composition operator $C_{\varphi}$ (see Exercise $2.3$ in Chapter 2 ) is an isometry on $C[0,1]$ and $\sigma\left(C_{\varphi}\right)=\overline{\mathbb{D}} .$

Abhijit Das
Abhijit Das
Numerade Educator
00:59

Problem 16

Consider the Volterra integral operator $V$ acting on $L^{2}([0,1], d x)$ defined by
$$
V f(x)=\int_{0}^{x} f(t) d t
$$
(a) Show that for any positive integer $n$,
$$
V^{n+1} f(x)=\frac{1}{n !} \int_{0}^{x}(x-t)^{n} f(t) d t .
$$
(b) Show that $\sigma(V)=\{0\}$.

Raj Bala
Raj Bala
Numerade Educator
01:09

Problem 17

Let $\mathscr{A}$ be the Banach algebra $C(T)$ in the supremum norm, where $T$ denotes the unit circle $\partial \mathbb{D}$. Let $\mathscr{B}$ be the subalgebra of $C(T)$ consisting of those $f \in C(T)$ for which there exist polynomials $p_{n}$ in $z$ with $p_{n}$ converging uniformly to $f$ on $T$.
(a) Show that $g(z)=\bar{z}$ is not in $\mathscr{B}$ (but of course it is in $\mathscr{A}$ ).
(b) Consider the function $f(z)=z$ which is in both $\mathscr{A}$ and $\mathscr{B}$. What is $\sigma_{\mathscr{\mathscr}}(f)$ ? Show that $\sigma_{\mathscr{B}}(f)=\overline{\mathbb{D}}$, the closed unit disk. Observe that although $\sigma_{\mathscr{A}}(f) \neq \sigma_{\mathscr{B}}(f)$, the spectral radius of the element $f$ doesn't change in passing from $\mathscr{A}$ to $\mathscr{B}$.

Raj Bala
Raj Bala
Numerade Educator
06:04

Problem 18

Suppose $\mathscr{A}$ and $\mathscr{B}$ are Banach algebras with common identity and $\mathscr{B} \subseteq \mathscr{A}$. Show $\sigma_{\mathscr{I}}(A) \subseteq \sigma_{\mathscr{B}}(A)$ and $\partial \sigma_{\mathscr{D}}(A) \subseteq \partial \sigma_{\mathscr{I}}(A)$, for any $A \in \mathscr{B}$. Hint for the second part: Since the first part implies that the interior of $\sigma_{\sigma f}(A)$ is contained in the interior of $\sigma_{\mathscr{B}}(A)$ for any $A$ in $\mathscr{B}$, argue first that it suffices to show that if $\lambda \in \partial \sigma_{\mathscr{D}}(A)$, then $\lambda \in \sigma_{\sigma d}(A)$.

Mengchun Cai
Mengchun Cai
Numerade Educator
07:44

Problem 19

Suppose that $\mathscr{A}$ is a $C^{*}$-algebra with unit $I_{\mathscr{A}}, \mathscr{B}$ is a $C^{*}$-algebra with unit $I_{\mathscr{D}}$ and $\rho: \mathscr{A} \rightarrow \mathscr{B}$ is a *-homomorphism with $\rho\left(I_{\mathscr{A}}\right)=I_{\mathscr{B}} .$ Prove the following:
(a) For every $A \in \mathscr{A}, \sigma(\rho(A)) \subseteq \sigma(A)$, and hence $r(\rho(A)) \leq r(A)$.
(b) For every $A \in \mathscr{A},\|\rho(A)\| \leq\|A\|$.
(c) If $\rho$ is a *-isomorphism, then $\rho$ is an isometry.

Anthony Ramos
Anthony Ramos
Numerade Educator
03:06

Problem 20

Find the norm of the operator on $\mathscr{B}(\mathscr{H})$ where $\mathscr{H}=\mathbb{C}^{2}$ which is given by the matrix
$$
\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
$$
Give your answer in terms of the numbers $S=|a|^{2}+|b|^{2}+|c|^{2}+|d|^{2}$ and $D=$ $a d-b c$.

Tamara Worner
Tamara Worner
Numerade Educator
08:01

Problem 21

Suppose that $\|\cdot\|_{1}$ and $\|\cdot\|_{2}$ are two norms on a $*$-algebra $\mathscr{A}$, each of which make $\mathscr{A}$ into a $C^{*}$-algebra. Show $\|\cdot\|_{1}=\|\cdot\|_{2}$.

Andrija Isakov
Andrija Isakov
Numerade Educator
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Problem 22

Show that the noncommutative unital Banach algebra $\mathbb{M}_{n}(\mathbb{C})$ of all $n \times n \mathrm{ma-}$ trices with complex entries has no nontrivial two-sided ideals. (Hints: Take any nonzero matrix $A$. Show that by multiplying $A$ on the left and right by the appropriate sequence of matrices you can isolate any entry of $A$ and move it anywhere you want. Recall that the elementary row and column operations of interchanging two rows or two columns of $A$ can be obtained by multiplying $A$ by the appropriate elementary matrix.)

Nick Johnson
Nick Johnson
Numerade Educator
09:35

Problem 23

Suppose that $\mathscr{J}$ is a closed two-sided ideal in $\mathscr{B}(\mathscr{H})$, for $\mathscr{H}$ a Hilbert space. The goal of this problem is to show that either $\mathscr{J}=\{0\}$, or $\mathscr{J}$ contains $\mathscr{K}(\mathscr{H})$, the ideal of compact operators on $\mathscr{H}$. (Compare this with the statement in Exercise 5.22.)
(a) Suppose $T$ is a nonzero operator in $\mathscr{J}$. Find vectors $f_{0}, f_{1}$ with $T f_{0}=f_{1}$ and $f_{1} \neq 0$. Show that if $g_{0}, g_{1}$ are any pair of nonzero vectors in $\mathscr{H}$, then the rank one operator $S$ defined by
$$
S f=\frac{\left\langle f, g_{0}\right\rangle g_{1}}{\left\|g_{0}\right\|^{2}}
$$
is in $\mathscr{J}$. By Exercise $2.6$ in Chapter 2 , this will show that $\mathscr{J}$ contains all rank 1 operators. Hint: Let
$$
A f=\frac{\left\langle f, g_{0}\right\rangle f_{0}}{\left\|g_{0}\right\|^{2}}
$$
and
$$
B f=\frac{\left\langle f, f_{1}\right\rangle g_{1}}{\left\|f_{1}\right\|^{2}}
$$
and compute $B T A$.
(b) Apply Exercise $4.9$ of Chapter 4 to show that $\mathscr{J}$ contains all finite rank operators.

Chris Trentman
Chris Trentman
Numerade Educator
01:35

Problem 24

Let $(X, \Omega, \mu)$ be a $\sigma$-finite measure space and suppose $\varphi \in L^{\infty}(\mu)$. Define $M_{\varphi}$ on $L^{2}(\mu)$ by $M_{\varphi}(f)=\varphi f$, so that $M_{\varphi}$ is the multiplication operator with symbol $\varphi$. Recall that $M_{\varphi}$ is a bounded linear operator on $L^{2}(\mu)$ with $\left\|M_{\varphi}\right\|=\|\varphi\|_{\infty}$.
(a) Show that $M_{\varphi}$ is normal, with $M_{\varphi}^{*}=M_{\bar{\varphi}}$.
(b) Show that $\varphi \rightarrow M_{\varphi}$ is a $*$ - homomorphism from $L^{\infty}(\mu)$ into $\mathscr{B}\left(L^{2}(\mu)\right)$.
(c) Show that the eigenvalues of $M_{\varphi}$ are the complex numbers $\lambda$ for which $\varphi^{-1}(\{\lambda\})$ has positive measure, and that $\sigma\left(M_{\varphi}\right)$ is the essential range of $\varphi$. The essential range of $\varphi$ is defined as:
$$
\{w \in \mathbb{C}: \mu\{x:|f(x)-w|<\varepsilon\}>0 \text { for all } \varepsilon>0\} .
$$
(d) Show directly that any closed set in $\mathbb{C}$ that contains the range of $\varphi$ must contain the essential range of $\varphi$.
(e) Suppose $f$ is a continuous function on $\sigma\left(M_{\varphi}\right)$. Identify $f\left(M_{\varphi}\right)$ in the continuous functional calculus. (Hint: Make a guess, and use the uniqueness statement for the functional calculus to prove your guess correct.)

Aman Gupta
Aman Gupta
Numerade Educator
02:07

Problem 25

Suppose that $A$ is a self-adjoint operator in $\mathscr{B}(\mathscr{H})$ for some Hilbert space $\mathscr{H}$. Show that if ker $(A-t I)=\{0\}$ and $A-t I$ has closed range for some real number $t$, then the range of $A-t I$ is $\mathscr{H}$. Conclude that a self-adjoint operator has no residual spectrum.

ET
Ed Tam
Numerade Educator
01:36

Problem 26

Suppose $S$ is a set and $\tau_{1}, \tau_{2}$ are topologies on $S$ with $\tau_{1}$ weaker than $\tau_{2}$. For an arbitrary set $A$ in $S$, how does the closure of $A$ relative to $\tau_{1}$ compare to the closure of $A$ relative to $\tau_{2} ?$ Is it easier for a set to be compact in the $\tau_{1}$-topology or the $\tau_{2}-$ topology? Is it easier for a sequence (or net) to converge in the $\tau_{1}$-topology or the $\tau_{2}$-topology?

Adriano Chikande
Adriano Chikande
Numerade Educator
00:26

Problem 27

Prove Theorem $5.40$ by modifying the proof of Proposition $5.32 .$

Vikash Ranjan
Vikash Ranjan
Numerade Educator
03:45

Problem 28

This problem explains why we require $Y$ to be a vector space in defining the $Y$-weak topology.
(a) Suppose that $X$ is a vector space and $\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n}$ are linear maps from $X$ into $\mathbb{C}$. Let $\varphi$ be a linear map from $X$ into $\mathbb{C}$. Show that $\varphi$ is in the linear span of $\left\{\varphi_{1}, \varphi_{2}, \ldots, \varphi_{n}\right\}$ if and only if
ker $\varphi_{1} \cap \operatorname{ker} \varphi_{2} \cap \cdots \cap \operatorname{ker} \varphi_{n} \subseteq \operatorname{ker} \varphi$.
(b) Suppose that $Y$ is a vector space of linear functionals on $X$ that separates the points of $X$. Show that a linear functional $\varphi$ on $X$ is continuous with respect to the $Y$-weak topology if and only if $\varphi$ is in $Y$.

Narayan Hari
Narayan Hari
Numerade Educator
01:36

Problem 29

Suppose $\mathscr{A}$ is an infinite-dimensional Banach space and $\mathscr{A}^{*}$ is its dual space.
(a) Show that a neighborhood basis of 0 in the weak $^{*}$ topology is given by the collection of sets
$$
\mathscr{O}_{A_{1}, \ldots, A_{n}} \equiv\left\{\varphi \in \mathscr{A}^{*}:\left|\varphi\left(A_{j}\right)\right|<1,1 \leq j \leq n\right\}
$$
where $n$ is a positive integer, and $A_{1}, A_{2}, \ldots, A_{n}$ are in $\mathscr{A} .$ To do this, you need to show that $\mathscr{O}_{A_{1}, \ldots, A_{n}}$ is a weak $^{*}$ open set containing 0, and for any weak $^{*}$ open set $\mathscr{O}$ containing 0 , there is some positive integer $n$, and points $A_{j} \in \mathscr{A}$ with $0 \in \mathscr{O}_{A_{1}, \ldots, A_{n}} \subseteq \mathscr{O} .$
(b) Let $\varphi_{0}$ be in $\mathscr{A}^{*}$. In the weak* topology, a sub-basic neighborhood of $\varphi_{0}$ has the form
$$
\left\{\varphi \in \mathscr{A}^{*}:\left|\varphi(A)-\varphi_{0}(A)\right|<\varepsilon\right\},
$$
where $\varepsilon>0$ and $A$ is fixed in $\mathscr{A}$. Basic neighborhoods of $\varphi_{0}$ are finite intersections of sub-basic neighborhoods:

Stanley Enemuo
Stanley Enemuo
Numerade Educator
02:27

Problem 30

Let $X=[0,1]$ and put a topology on $X$ by declaring the open sets to be the empty set and those subsets of $X$ whose complement is at most countable. Consider the set $A=[0,1)$. Show that the closure of $A$ is $[0,1]$, so that in particular 1 lies in the closure of $A$. Show that there is no sequence $\left\{x_{n}\right\}$ of points in $[0,1)$ that converges to 1 .

Angelo Rendina
Angelo Rendina
Numerade Educator
07:36

Problem 31

(a) Suppose $S$ is a Hausdorff topological space. Show that a net in $S$ converges to at most one point; i.e., if $x_{\alpha} \rightarrow x$ and $x_{\alpha} \rightarrow y$ then $x=y$.
(b) Give a proof of the "if" direction of Theorem $5.39$.

Muhammad Saleem
Muhammad Saleem
Numerade Educator
03:29

Problem 32

Let $\mathscr{H}$ be a Hilbert space with orthonormal basis $\left\{e_{n}\right\}_{0}^{\infty}$ and consider the set $E=\left\{e_{m}+m e_{n}: 0 \leq m<n, n=1,2,3, \ldots\right\} .$ Note that $E$ is countable. Show that 0 is in the weak closure of $E$, but there is no sequence $x_{n} \in E$ with $x_{n} \rightarrow 0$ weakly.

Joy Carpio
Joy Carpio
Numerade Educator
03:58

Problem 33

Prove the following statement used in Theorem 5.46: If $X$ and $Y$ are homeomorphic compact Hausdorff spaces, then $C(X)$ and $C(Y)$ are $*$-isomorphic unital $C^{*}$-algebras in a natural way.

Anthony Ramos
Anthony Ramos
Numerade Educator
01:32

Problem 34

Let $\mathscr{A}$ be a unital commutative Banach algebra, and suppose $A, B \in \mathscr{A}$. Show that $r(A+B) \leq r(A)+r(B)$ and $r(A B) \leq r(A) r(B)$. (Hint: Use the Gelfand transform.) Show the same result holds if $\mathscr{A}$ is not assumed to be commutative, provided $A B=B A$. Show the result fails in general (look in $\left.M_{2}(\mathbb{C})\right)$.

Angelo Rendina
Angelo Rendina
Numerade Educator
01:16

Problem 35

Suppose $\mathscr{A}$ is a unital $C^{*}$-algebra, and $A \in \mathscr{A}$ is a normal element. Show that $A$ is unitary if and only if $\sigma(A) \subseteq \partial \mathbb{D}$, the unit circle in the complex plane. Show that $A^{2}=A$ if and only if $\sigma(A) \subseteq\{0,1\}$.

Nick Johnson
Nick Johnson
Numerade Educator
02:24

Problem 36

Suppose that $N$ is a normal operator in $\mathscr{B}(\mathscr{H})$ for some Hilbert space $\mathscr{H}$. If $\lambda$ is in $\mathbb{C} \backslash \sigma(N)$, show that
$$
\left\|(N-\lambda I)^{-1}\right\|=\operatorname{dist}(\sigma(N), \lambda)^{-1}
$$
where dist $(\sigma(N), \lambda)$ is the distance from $\sigma(N)$ to $\lambda$.

Victor Salazar
Victor Salazar
Numerade Educator
01:27

Problem 37

Let $W$ denote the Wiener algebra. Show that the Gelfand transform $\Gamma: W \rightarrow$ $C\left(\mathscr{M}_{W}\right)$ is not isometric, and indeed is not even bounded below. Thus $W$ cannot be made into a $C^{*}$-algebra (for example, by defining $f^{*}=\bar{f}$ ).

Harshita Goel
Harshita Goel
Numerade Educator
19:24

Problem 38

The "one-sided Wiener algebra" $W_{+}$is defined to be the set of all $f$ in the Wiener algebra $W$ of the form
$$
f\left(e^{i \theta}\right)=\sum_{n=0}^{\infty} a_{n} e^{i n \theta} .
$$
(a) Show that $W_{+}$is a closed subalgebra of $W$.
(b) For $\lambda$ in the closed unit disk $\overline{\mathbb{D}}$, show that $\varphi_{\lambda}$ taking $\sum_{n=0}^{\infty} a_{n} e^{i n \theta} \in W_{+}$to $\sum_{n=0}^{\infty} a_{n} \lambda^{n}$ is a multiplicative linear functional on $W_{+}$.
(c) Show that
$$
\mathscr{M}_{W_{+}}=\left\{\varphi_{\lambda}: \lambda \in \overline{\mathbb{D}}\right\}
$$
and the map $\lambda \rightarrow \varphi_{\lambda}$ is a homeomorphism of $\overline{\mathbb{D}}$ and $\mathscr{M}_{W_{+}}$, the latter being equipped with the weak* topology.
(d) If $f \in W_{+}$, describe the spectra $\sigma_{+}(f)$ and $\sigma(f)$ of $f$ as an element of $W_{+}$and $W$, respectively.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
03:29

Problem 39

Consider the Banach space $\ell^{1}\left(\mathbb{N}_{0}\right)$ of sequences $\left\{x_{n}\right\}_{n=0}^{\infty}$ in the norm
$$
\left\|\left\{x_{n}\right\}\right\|=\sum_{n=0}^{\infty}\left|x_{n}\right|
$$
For $\left\{x_{n}\right\}$ and $\left\{y_{n}\right\}$ in $\ell^{1}\left(\mathbb{N}_{0}\right)$, define the convolution $\left\{x_{n}\right\} *\left\{y_{n}\right\}$ to be the sequence $\left\{z_{n}\right\}$ defined by
$$
z_{n}=\sum_{k=0}^{n} x_{k} y_{n-k}
$$
for $n=0,1,2, \ldots$
(a) Show that $\ell^{1}\left(\mathbb{N}_{0}\right)$ becomes a commutative unital Banach algebra under the convolution product.
(b) Show that as a Banach algebra, $\ell^{1}\left(\mathbb{N}_{0}\right)$ is isometrically isomorphic to the onesided Wiener algebra $W_{+}$defined in Exercise $5.38$.
(c) For $0<a<1$, set $x=\left\{a^{n}\right\}=\left(1, a, a^{2}, a^{3}, \ldots\right)$. Find $\sigma(x)$, the spectrum of $x$ in $\ell^{1}\left(\mathbb{N}_{0}\right)$

Arpit Gupta
Arpit Gupta
Numerade Educator
19:24

Problem 40

Consider the Banach space $c$ of convergent sequences, as defined in Exercise $3.20$ of Chapter 3 . Define a product and involution respectively on $c$ by
$$
\left\{x_{n}\right\} \cdot\left\{y_{n}\right\}=\left\{x_{n} y_{n}\right\}
$$
and
$$
\left\{x_{n}\right\}^{*}=\left\{\overline{x_{n}}\right\} .
$$
This makes $c$ a commutative unital $C^{*}$-algebra, with unit $I=(1,1, \ldots)$. Show that its maximal ideal space is
$$
\mathscr{A}_{c}=\left\{\varphi_{k}: k \in \mathbb{N}\right\} \cup\left\{\varphi_{\infty}\right\}
$$
where $\varphi_{k}\left(\left\{x_{n}\right\}\right)=x_{k}$ for $k \in \mathbb{N}$ and $\varphi_{\infty}\left(\left\{x_{n}\right\}\right)=\lim _{n \rightarrow \infty} x_{n}$.
Note that the map $k \rightarrow \varphi_{k}$ is a homeomorphism of $\mathbb{N}$ onto its range in $\mathscr{M}_{c}$, and clearly $\varphi_{k} \rightarrow \varphi_{\infty}$ weak* as $k \rightarrow \infty$. Thus $\mathscr{M}_{c}$ is naturally homeomorphic to the onepoint compactification (see [33], p.183) of $\mathbb{N}$.

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
19:24

Problem 41

Consider $\ell^{\infty}$ as a commutative unital $C^{*}$-algebra with product $\left\{x_{n}\right\} \cdot\left\{y_{n}\right\}=$ $\left\{x_{n} y_{n}\right\}$ and involution $\left\{x_{n}\right\}^{*}=\left\{\overline{x_{n}}\right\}$. The cluster set at infinity of $x=\left\{x_{n}\right\} \in \ell^{\infty}$ is the set
$$
\begin{aligned}
C l_{\infty}(x) \equiv &\{\lambda \in \mathbb{C}: \text { for every } \varepsilon>0 \text { and } N \in \mathbb{N}\\
&\text { there exists } \left.n \geq N \text { with }\left|x_{n}-\lambda\right|<\varepsilon\right\} \\
=&\left\{\lambda \in \mathbb{C}: \text { there exists a subsequence }\left\{x_{n_{k}}\right\}\right. \text { of }\\
&\left.\left\{x_{n}\right\} \text { with } x_{n_{k}} \rightarrow \lambda \text { as } k \rightarrow \infty\right\} .
\end{aligned}
$$
Show that for any $x=\left\{x_{n}\right\} \in \ell^{\infty}$,
$\sigma(x)=\left\{x_{n}: n \in \mathbb{N}\right\} \cup C l_{\infty}(x)=\overline{\left\{x_{n}: n \in \mathbb{N}\right\}}$

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
19:24

Problem 42

Consider $\ell^{\infty}$ as a unital $C^{*}$-algebra as in the previous exercise. The fiber at infinity is the subset $X_{\infty} \subseteq \mathscr{M}_{e^{\infty}}$ defined by
$$
X_{\infty}=\left\{\varphi \in \mathscr{M}_{\ell}: \varphi(x)=\lim _{n \rightarrow \infty} x_{n} \text { for every } x=\left\{x_{n}\right\} \in c\right\}
$$
(a) For $n \in \mathbb{N}$, let $e_{n}=(0, \ldots, 0,1,0 \ldots)$ with a 1 in the $n$th position and 0 's elsewhere. Show that
$$
\begin{aligned}
X_{\infty} &=\left\{\varphi \in \mathscr{M}_{\ell^{\infty}}: \varphi\left(e_{n}\right)=0 \text { for all } n \in \mathbb{N}\right\} \\
&=\left\{\varphi \in \mathscr{M}_{\ell^{\infty}}: c_{0} \subseteq \operatorname{ker} \varphi\right\} .
\end{aligned}
$$
(b) Suppose that for each $k \in \mathbb{N}, \varphi_{k} \in \mathscr{M}_{\ell} \infty$ is defined by $\varphi_{k}(x)=x_{k}$ for $x=\left\{x_{n}\right\} \in$ $\ell^{\infty}$. Show that
$$
\mathscr{M}_{\ell^{\infty}}=\left\{\varphi_{k}: k \in \mathbb{N}\right\} \cup X_{\infty} .
$$
(c) With the terminology of the last exercise, argue that for any $x \in \ell^{\infty}$,
$$
C l_{\infty}(x)=\left\{\hat{x}(\varphi): \varphi \in X_{\infty}\right\} .
$$
(d) Let $x \in \ell^{\infty}$. Show that $x \in c$ if and only if $\hat{x}$ is constant on $X_{\infty}$.
(e) Show that $\left\{\varphi_{k}: k \in \mathbb{N}\right\}$ is dense in $\mathscr{M}_{\ell}^{\infty}$ in the weak* topology.
(f) Let $\varphi \in X_{\infty}$. Show that (the assertion of (e) notwithstanding) there is no subsequence $\left\{\varphi_{n_{k}}\right\}$ of $\left\{\varphi_{n}\right\}$ with $\varphi_{n_{k}} \rightarrow \varphi$ weak*

Oswaldo Jiménez
Oswaldo Jiménez
Numerade Educator
02:11

Problem 43

Can a Banach limit on $\ell^{\infty}$ (see Exercise $3.9$ ) be multiplicative?

Uma Kumari
Uma Kumari
Numerade Educator
02:07

Problem 44

Recall that in a unital $C^{*}$-algebra $\mathscr{A}$, the positive elements of $\mathscr{A}$ are defined to be those self-adjoint $A \in \mathscr{A}$ with $\sigma(A) \subseteq[0, \infty)$. Denote the collection of positive elements $\mathscr{A}_{+}$and set $\mathscr{A}_{-}=\left\{A:-A \in \mathscr{A}_{+}\right\}$.
(a) Show that $\mathscr{A}_{+} \cap \mathscr{A}_{-}=\{0\} .$
(b) Follow the outline below to show that every self-adjoint $A$ in a $C^{*}$-algebra $\mathscr{A}$ can be written in the form $A=A_{+}-A_{-}$, where $A_{+}$and $A_{-}$are both positive elements of $\mathscr{A}$ and $A_{+} A_{-}=A_{-} A_{+}=0$.
Outline: Note that the identity function $h(t)=t$ on the real line can be written as $f-g$, where $f(t)=\max (0, t)$ and $g(t)=-\min (0, t)$. Observe that $f, g$ are continuous nonnegative functions on the real line with $f g=0$. For $A$ self-adjoint, set $A_{+}=f(A)$ and $A_{-}=g(A)$ as given by the functional calculus. Check that $A_{+}$and $A_{-}$have the desired properties.

ET
Ed Tam
Numerade Educator
38:29

Problem 45

Suppose that $0 \leq A \leq B$ for self-adjoint elements $A, B$ in a $C^{*}$ algebra.
(a) Show that $B \leq\|B\| I$. Hint: Consider $C^{*}(B) \cong C(\sigma(B))$ where $B$ corresponds to the identity function on the spectrum of $B$. Use the functional calculus, with the function $f(x)=\|B\|-x$ on $\sigma(B)$.
(b) Show $\|A\| \leq\|B\| .$ Hint: Consider $C^{*}(A) \cong C(\sigma(A))$ with $A$ corresponding to the identity function on the spectrum of $A$. Use the functional calculus with the function $f(x)=\|B\|-x$.
(c) Show that $0 \leq A \leq B$ need not imply $A^{2} \leq B^{2}$ by considering
$$
X=\left[\begin{array}{ll}
1 & 0 \\
0 & 0
\end{array}\right]
$$
and
$$
Y=\left[\begin{array}{cc}
\frac{1}{2} & \frac{1}{2} \\
\frac{1}{2} & \frac{1}{2}
\end{array}\right]
$$
Show $0 \leq X \leq X+Y$. Is $X^{2} \leq(X+Y)^{2}$ ?
(d) Show that if $0 \leq A \leq B$ and $A$ and $B$ commute, then $A^{n} \leq B^{n}$ for every positive integer $n$. More generally, show that if there are positive elements $C_{j}, 1 \leq j \leq k$, with
$$
0 \leq A \leq C_{1} \leq C_{2} \leq \cdots \leq C_{k} \leq B
$$
so that any two neighbors in this list commute, then $A^{n} \leq B^{n}$ for any positive integer $n$.

Donald Albin
Donald Albin
Numerade Educator
01:35

Problem 46

Suppose that $P$ and $Q$ are orthogonal projections onto closed subspaces $M$ and $N$ in $\mathscr{H}$, respectively. Show that $P \geq Q$ if and only if $N \subseteq M$.

Victor Salazar
Victor Salazar
Numerade Educator
04:52

Problem 47

Let $\mathscr{H}$ be a Hilbert space. An operator $T$ in $\mathscr{B}(\mathscr{H})$ is said to be a contraction if $\|T\| \leq 1$.
(a) Show that $T$ is a contraction if and only if $I-T^{*} T \geq 0$.
(b) Suppose that $A$ and $B$ are bounded linear operators on $\mathscr{H}$ with $B$ invertible.
Show that $A B^{-1}$ is a contraction if and only if $A^{*} A \leq B^{*} B$.

Diogo Caetano
Diogo Caetano
Numerade Educator
02:24

Problem 48

What's wrong with the following "proof" that for an arbitrary element $B$ of a unital $C^{*}$-algebra $\mathscr{A}$, the element $B^{*} B$ is positive: Let $A=B^{*} B$. Clearly $A$ is selfadjoint. Using the Gelfand transform we have
$$
\Gamma(A)=\Gamma\left(B^{*} B\right)=\Gamma\left(B^{*}\right) \Gamma(B)=\overline{\Gamma(B)} \Gamma(B)=|\Gamma(B)|^{2} \geq 0
$$
so that $\sigma(A)=\sigma(\Gamma(A))=$ range $|\Gamma(B)|^{2} \subseteq[0, \infty)$.

Umar Sohail Qureshi
Umar Sohail Qureshi
Numerade Educator