Spectral Theory

Abstract

To a bounded operator A on normed space X over ${\mathbb K}$, we associate a subset of ${\mathbb K}$, known as the spectrum of A. It is intimately related to the invertibility of a specific linear combination of the operator A and the identity operator. Eigenvalues and approximate eigenvalues of A form a part of the spectrum of A. Determining the spectrum of a bounded operator is one of the central problems in functional analysis. In case X is a Banach space, we show that the spectrum of a bounded operator A on X is a closed and bounded subset of ${\mathbb K}$. We explore special properties of the spectrum of a compact operator on a normed space. We find relationships between the spectrum of a bounded operator A and the spectra of the transpose $A'$ and the adjoint $A^{*}$. They yield particularly interesting results when the operator A is ‘well behaved’ with respect to the adjoint operation. In the last section of this chapter, we show how a compact self-adjoint operator can be represented in terms of its eigenvalues and eigenvectors. This is used in obtaining explicit solutions of operator equations.

Exercises

• 5.1 Let X be a normed space, and let $A\in BL(X)$ be invertible. Then $\sigma (A^{-1}) = \{ \lambda ^{-1} : \lambda \in \sigma (A) \}.$ In fact, if $k\in {\mathbb K},\, k\ne 0$, and $A-kI$ is invertible, then $A^{-1}\!-k^{-1}I$ is invertible, and $-k(A-kI)^{-1}A$ is its inverse.

• 5.2 Let X be a normed space, $A\in BL(X)$, and let p be a polynomial. Then $\{ p(\lambda ) : \lambda \in \sigma (A) \}\subset \sigma (p(A))$, where equality holds if ${\mathbb K}:= {\mathbb C}$.

• 5.3 Let X be a normed space, and let $A,B\in BL(X)$. If $I-AB$ is invertible, then $I-BA$ is invertible, and $I+B(I-AB)^{-1}A$ is its inverse. Consequently, $\sigma (AB)\setminus \{0\}=\sigma (BA)\setminus \{0\}$.

• 5.4 (Gershgorin theorem) Let an $n\times n$ matrix $M := [k_{i,j}]$ define an operator $A\in BL({\mathbb K}^n)$. For $i=1,\ldots ,n$, let $r_i:=|k_{i,1}|+\cdots +|k_{i,i-1}|+|k_{i,i+1}|+\cdots +|k_{i,n}|$, and $D_i := \{k\in {\mathbb K}: | k - k_{i,i}| \le r_i\}$. Then $\sigma (A)\subset D_1\cup \cdots \cup D_n$. (Note: $D_1,\ldots , D_n$ are known as the Gershgorin disks of A. They localize the eigenvalues of A.)

• 5.5 Let X be a linear space over ${\mathbb K}$, and let $A:X\rightarrow X$ be of finite rank. Suppose $A(x):=g_1(x)x_1+\cdots +g_n(x)x_n$ for all $x\in X$, where $x_1,\ldots ,x_n$ are in X and $g_1,\ldots , g_n$ are linear functionals on X. Let M be the $n\times n$ matrix $[g_i(x_j)]$. Let $x\in X$, $u:=(u(1),\ldots ,u(n))\in {\mathbb K}^n$, and nonzero $\lambda \in {\mathbb K}$. Then $A(x)=\lambda x$ and $u=(g_1(x),\ldots ,g_n(x))$ if and only if $Mu^t=\lambda u^t$ and $x=\big (\sum _{j=1}^nu(j)x_j\big )/\lambda$. In this case, x is an eigenvector of A corresponding to $\lambda$ if and only if $u^t$ is an eigenvector of M corresponding to $\lambda$.

• 5.6 Let $X:=C([a,b])$ with the sup norm, and let A denote the multiplication operator considered in Example 5.6(ii). Then $\sigma _e(A)$ consists of all $\lambda \in {\mathbb K}$ such that $x_{0}(t) = \lambda$ for all t in a nontrivial subinterval of [a, b].

• 5.7 (Multiplication operator) Let $X:=L^2([a,b]), x_0\in L^\infty ([a,b])$, and $A(x):=$ $x_0x$ for $x\in X$. Then $\sigma _a(A)$ and $\sigma (A)$ equal the essential range of $x_0$ consisting of $\lambda \in {\mathbb K}$ such that $m(\{t\in [a,b]:|x_0(t)-\lambda |<\epsilon \})>0$ for every $\epsilon >0$. Also, $\sigma _e(A)$ is the set of all $\lambda \in {\mathbb K}$ such that $m(\{t\in [a,b]:x_{0}(t) = \lambda \}>0$.

• 5.8 Let X be a normed space, and let $A\in BL(X)$. If $\lambda \in \sigma _a(A)$, and $n\in {\mathbb N}$, then $\lambda ^n\in \sigma _a(A^n)$, and so $|\lambda |\le \inf \big \{\Vert A^n\Vert ^{1/n}\!:n\in {\mathbb N}\big \}$.

• 5.9 Let X be a normed space, and let $A\in BL(X)$.

  • (i) If there is $\alpha >0$ such that $\Vert A(x)\Vert \le \alpha \Vert x\Vert$ for all $x\in X$, then $\sigma _a(A)$ is contained in $\{k\in {\mathbb K}:|k|\le \alpha \}$.

  • (ii) If there is $\beta >0$ such that $\Vert A(x)\Vert \ge \beta \Vert x\Vert$ for all $x\in X$, then $\sigma _a(A)$ is contained in $\{k\in {\mathbb K}:|k|\ge \beta \}$.

  • (iii) If A is an isometry, then $\sigma _a(A)$ is contained in $\{k\in {\mathbb K}:|k|=1\}$.

• 5.10 Let $X:=\ell ^1$, and let $A(x) := (0, x(1), 2x(2), x(3),2x(4),\ldots )$ for $x\in X$. Then $A\in BL(X), \Vert A\Vert =2$, but $|\lambda |\le \sqrt{2}$ for every $\lambda \in \sigma (A)$.

• 5.11 (Diagonal operator) Let $X:=\ell ^p$ with $p\in \{1,2,\infty \}$, and let $(\lambda _j)$ be a bounded sequence in ${\mathbb K}$. For $x:=(x(1),x(2),\ldots )$ in X, define $A(x) := (\lambda _1x(1), \lambda _2x(2), \ldots )$. Then $A\in BL(X),\, \Vert A\Vert =\sup \{|\lambda _j|:j\in {\mathbb N}\},\,\sigma _e(A)=\{\lambda _j:j\in {\mathbb N}\}$, while $\sigma _a(A)$ and $\sigma (A)$ equal the closure of $\{\lambda _j:j\in {\mathbb N}\}$.

• 5.12 Let E be a nonempty closed and bounded subset of ${\mathbb K}$. Then there is a diagonal operator $A\in BL(\ell ^2)$ such that $\sigma (A)=E$. Further, if $(\lambda _j)$ is a sequence in ${\mathbb K}$ such that $\lambda _j\rightarrow 0$, then there is a diagonal operator $A\in CL(\ell ^2)$ such that $\sigma _e(A)=\{\lambda _j:j\in {\mathbb N}\}$ and $\sigma (A)=\{\lambda _j:j\in {\mathbb N}\}\cup \{0\}$.

• 5.13 (Neumann expansion) Let X be a Banach space, and let $A\in BL(X)$. If $k\in {\mathbb K}$ and $|k|>\Vert A\Vert$, then $A-kI$ is invertible,

  $$ (A-kI)^{-1}=-\sum _{n=0}^\infty \frac{A^n}{k^{n+1}}\quad \text {and}\quad \Vert (A-kI)^{-1}\Vert \le \frac{1}{|k|-\Vert A\Vert }. $$

  In particular, if A denotes the right shift operator on $\ell ^p$ defined in Example 5.12(i), then $(A-kI)^{-1}(y)(j)=-y(j)/k-\cdots -y(1)/k^j$ for $y\in \ell ^p,\,j\in {\mathbb N}$.

• 5.14 (Left shift operator) Let X denote one of the spaces $\ell ^1,\ell ^2,\ell ^\infty ,c_0, c$, and define $B(x) := (x(2), x(3), \ldots )$ for $x := (x(1), x(2), \ldots )\in X.$ If $X:=\ell ^1,\, \ell ^2 \,\text {or}\,c_0$, then $\sigma _{e}(B) = \{\lambda \in {\mathbb K}: |\lambda | < 1 \}$; if $X:= \ell ^\infty$, then $\sigma _{e}(B) = \{\lambda \in {\mathbb K}: |\lambda | \le 1 \}$; if $X:=c$, then $\sigma _{e}(B) = \{\lambda \in {\mathbb K}: |\lambda | < 1 \} \cup \{1\}$. In all cases, $\sigma _{a}(B) = \{\lambda \in {\mathbb K}: |\lambda | \le 1\} = \sigma (B).$

• 5.15 Let $p\in \{1,2,\infty \}$, and let A be the right shift operator on $L^p([0,\infty ))$ defined in Exercise 4.22. If $p\in \{1,2\}$, then $\sigma _e(A)=\{\lambda \in {\mathbb K}:|\lambda |<1\}$, and if $p:=\infty$, then $\sigma _e(A)=\{\lambda \in {\mathbb K}:|\lambda |\le 1\}$. In all cases, $\sigma _a(A)$ and $\sigma (A)$ equal $\{\lambda \in {\mathbb K}:|\lambda |\le 1\}$.

• 5.16 Let X be a nonzero inner product space, and let $A\in BL(X)$. For nonzero $x\in X$, let $q_A(x):=\langle {A(x)},{x} \rangle /\langle {x},{x} \rangle$ and $r_A(x):=A(x)-q_A(x)x$. The scalar $q_A(x)$ is called the Rayleigh quotient of A at x, and the element $r_A(x)$ of X is called the corresponding residual. Then $r_A(x)\perp x$, and $\Vert r_A(x)\Vert =\min \{\Vert A(x)-k\,x\Vert :k\in {\mathbb K}\}$. (Note: This is known as the minimum residual property of the Rayleigh quotient.)

• 5.17 Let X be a normed space, and let $A\in CL(X)$. Then $\sigma (A)$ is a closed and bounded subset of ${\mathbb K}$. If $\lambda \in \sigma (A)$, then $|\lambda |\le \inf \{\Vert A^n\Vert ^{1/n}:n\in {\mathbb N}\}$.

• 5.18 (Weighted-shift operators) Let $X:=\ell ^p$ with $p\in \{1,2,\infty \}$, and let $(w_n)$ be a sequence in ${\mathbb K}$ such that $w_n\rightarrow 0$. For $x:=(x(1),x(2),\ldots )\in X$, define $A(x):=(0,w_1x(1),w_2x(2),\ldots )$ and $B(x):=(w_2x(2),w_3x(3),\ldots )$. Then $A,\,B\in CL(X)$, and $\sigma (A)=\sigma (B)=\{0\}$. Further, $0\in \sigma _e(A)$ if and only if there is $j\in {\mathbb N}$ such that $w_j=0$, and then the dimension of the corresponding eigenspace of A is the number of times 0 occurs among $w_1,w_2,\ldots .$ Also, $0\in \sigma _e(B)$, and the dimension of the corresponding eigenspace of B is one plus the number of times 0 occurs among $w_2,w_3,\ldots .$

• 5.19 (Volterra integration operator) Let $X := L^{2} ([a,b])$. For $s,t\in [a,b]$, let $k(s,t):=1$ if $t\le s$ and $k(s,t):=0$ if $s<t$, and let A denote the Fredholm integral operator on X with kernel $k(\cdot \,,\cdot )$. Then $A\in CL(X)$, $\sigma _{e}(A) = \emptyset$, and $\sigma _a(A) =\sigma (A)= \{0\}$.

• 5.20 Let $X := C([0,1])$, and $Y:= L^{p}([0,1])$ with $p\in \{1,2,\infty \}$. For $s,t\in [0,1]$, let $k(s,t):=\min \{s,t\}$, and let A denote the Fredholm integral operator on X, and on Y, with kernel $k(\cdot \,,\cdot )$. Then

  $$\sigma _{e}(A) = \{ 4/(2n-1)^{2} \pi ^{2}:n\in {\mathbb N}\} \quad \text {and} \quad \sigma _{a}(A) = \sigma (A) = \sigma _e(A)\cup \{0\}.$$

  Further, the eigenspace corresponding to the eigenvalue $4/(2n-1)^{2} \pi ^{2}$ of A is $\mathrm{span\,}\{x_{n}\},$ where $x_{n}(s) := \sin (2n-1)\pi s/2,\, 0 \le s \le 1,$ for $n\in {\mathbb N}$.

• 5.21 (Perturbation by a compact operator) Let X be a normed space, $A\in BL(X)$, and let $B\in CL(X)$. Then $\sigma (A)\setminus \sigma _e(A)\subset \sigma (A+B)$.

• 5.22 Let X be a normed space, $A\in CL(X)$, and let $k\in {\mathbb K}$ be nonzero. Then $A-kI$ is one-one if and only if $A-kI$ is onto.

• 5.23 Let X be a normed space, $A\in CL(X)$, and let $k\in {\mathbb K}$ be nonzero. Then $R(A-kI)$ is a closed subspace of X. (Compare Lemma 5.19.)

• 5.24 (Fredholm alternative) Let X be a normed space, and let $A\in CL(X)$. Exactly one of the following alternatives holds.

  • (i) For every $y\in X$, there is a unique $x\in X$ such that $x-A(x)=y$.

  • (ii) There is nonzero $x\in X$ such that $x-A(x)=0$.

  If the alternative (i) holds, then the unique solution x of the operator equation $x-A(x)=y$ depends continuously on the ‘free term’ $y\in X$. If the alternative (ii) holds, then the solution space of the homogeneous equation $x-A(x)=0$ is finite dimensional.

• 5.25 Let X be a normed space, and let $A\in CL(X)$.

  • (i) The homogeneous equation $x - A(x) = 0$ has a nonzero solution in X if and only if the transposed homogeneous equation $x' - A'(x') = 0$ has a nonzero solution in $X'.$ (Note: The solution spaces of the two homogeneous equations have the same dimension, that is, $\dim Z(I-A')=\dim Z(I-A)$. See [28, Theorem V.7.14 (a)].)

  • (ii) Let $y\in X$. There is $x\in X$ such that $x - A(x) = y$ if and only if $x'_{j}(y) = 0$ for $j = 1, \ldots , m,$ where $\{x'_{1}, \ldots , x'_{m}\}$ is a basis for the solution space of the transposed homogeneous equation $x' - A'(x') = 0.$ Further, if $x_{0}$ is a particular solution of the equation $x - A(x) =y$, then its general solution is given by $x := x_{0} + k_{1} x_{1} + \cdots + k_{m}x_{m},$ where $k_{1}, \ldots , k_{m}$ are in ${\mathbb K}$, and $\{x_{1}, \ldots , x_{m}\}$ is a basis for the solution space of the homogeneous equation $x-A(x) = 0.$

• 5.26 Let X be a normed space, and let $A\in BL(X)$. Then $\sigma _e(A)\subset \sigma _e(A'')$ and $\sigma _a(A)\subset \sigma _a(A'')$, while $\sigma (A'')=\sigma (A')\subset \sigma (A)$.

• 5.27 Let X be a normed space, $A\in BL(X)$, and $\lambda \in \sigma _e(A)$. A nonzero $x\in X$ is called a generalized eigenvector of A corresponding to $\lambda$ if there is $n\in {\mathbb N}$ such that $(A-\lambda I)^n(x)=0$. If H is a Hilbert space, and $A\in BL(H)$ is normal, then every generalized eigenvector of A is an eigenvector of A.

• 5.28 Let A be a normal operator on a separable Hilbert space H. Then $\sigma _e(A)$ is a countable set.

• 5.29 Let $A\in BL(\ell ^2)$ be defined by the infinite matrix $M:=[k_{i,j}]$. Suppose either M is upper triangular (that is, $k_{i,j}=0$ for all $i>j$) or M is lower triangular (that is, $k_{i,j}=0$ for all $i<j$). Then A is normal if and only if M is diagonal (that is, $k_{i,j}=0$ for all $i\ne j$).

• 5.30 Let H be a Hilbert space, and let $A\in BL(H)$ be unitary. Then $\sigma (A)$ is contained in $\{k\in {\mathbb K}:|k|=1\}$. Further, if $k\in {\mathbb K}$ and $|k|\ne 1$, then $\Vert (A-kI)^{-1}\Vert \le 1/|\,|k|-1 |$.

• 5.31 Let H be a nonzero Hilbert space, and let $A\in BL(H)$. If $k\in {\mathbb K}\setminus \overline{\omega (A)}$, and $\beta :=d\big (k,\overline{\omega (A)}\big )$, then $A-kI$ is invertible, and $\Vert (A-kI)^{-1}\Vert \le 1/\beta$. In particular, if A is self-adjoint and $k\in {\mathbb K}\setminus [m_A,M_A]$, then $\beta$ is equal to $|\Im k|$ if $\Re k\in [m_A,M_A]$, to $|k-m_A|$ if $\Re k<m_A$ and to $|k-M_A|$ if $\Re k>M_A$. If in fact ${\mathbb K}:={\mathbb R}$, A is self-adjoint, and $k\in {\mathbb R}\setminus \sigma (A)$, then $\Vert (A-kI)^{-1}\Vert =1/d$, where $d:=d(k,\sigma (A))$. (Note: This result also holds if ${\mathbb K}:={\mathbb C}$ and A is normal.)

• 5.32 Let H be a Hilbert space over ${\mathbb C}$. If $A\in BL(H)$ is self-adjoint, then its Cayley transform$T(A) := (A-iI)(A+iI)^{-1}$ is unitary and $1 \not \in \sigma (T(A)).$ Conversely, if $B\in BL(H)$ is unitary and $1 \not \in \sigma (B),$ then its inverse Cayley transform $S(B) := i(I+B) (I-B)^{-1}$ is self-adjoint. Further, $S(T(A)) = A$ and $T(S(B)) = B.$ (Note: The function $z\longmapsto (z-i)(z+i)^{-1}$ maps ${\mathbb R}$ onto $E:=\{z\in {\mathbb C}:|z|=1\,\text {and}\,z\ne 1\}$, and its inverse function $w\longmapsto i(1+w)(1-w)^{-1}$ maps E onto ${\mathbb R}$.)

• 5.33 Let A be a self-adjoint operator on a nonzero Hilbert space. Then $A\ge 0$ if and only if $\sigma (A)\subset [0,\infty )$. In this case, $0 \in \omega (A)$ if and only if $0 \in \sigma _e (A).$

• 5.34 Let ${\mathbb K}:={\mathbb R}$, and let $\theta \in [0,2\pi )$. For $x:=(x(1),x(2))\in {\mathbb R}^2$, define

  $$A(x):=(x(1)\cos \theta -x(2)\sin \theta , x(1)\sin \theta +x(2)\cos \theta ).$$

  Then $A^{*}(x):=(x(1)\cos \theta +x(2)\sin \theta , -x(1)\sin \theta +x(2)\cos \theta )$ for $x\in {\mathbb R}^2$, and A is a unitary operator. Also, $\sigma (A)=\{1\}$ if $\theta :=0$, $\sigma (A)=\{-1\}$ if $\theta :=\pi$, and $\sigma (A)=\emptyset$ otherwise.

• 5.35 Let H be a separable Hilbert space over ${\mathbb K}$, and let A be a normal Hilbert–Schmidt operator on H. Let $(\lambda _n)$ be the sequence of nonzero eigenvalues of A, each such eigenvalue being repeated as many times as the dimension of the corresponding eigenspace. Then $\sum _n|\lambda _n|^2<\infty$.

• 5.36 Let H be a finite dimensional nonzero Hilbert space over ${\mathbb C}$, and let A be in BL(H). Then A is normal if and only if there are orthogonal projection operators $P_1,\ldots ,P_k$ on H, and distinct $\mu _1,\ldots ,\mu _k$ in ${\mathbb C}$ such that $I=P_1+\cdots +P_k$ and $A=\mu _1P_1+\cdots +\mu _kP_k$, where $P_iP_j=0$ for all $i\ne j$.

• 5.37 Let H be a Hilbert space over ${\mathbb K}$, and let $A\in BL(H)$ be nonzero. Then A is compact and self-adjoint if and only if there are orthogonal projection operators $P_0,P_1,P_2,\ldots$ on H and distinct $\mu _1,\mu _2,\ldots$ in ${\mathbb R}\setminus \{0\}$ such that $x=P_0(x)+P_1(x)+P_2(x)+\cdots$ for all $x\in H$ and $A=\mu _1P_1+\mu _2P_2+\cdots$, where $P_iP_j=0$ for all $i\ne j$, $P_1,\,P_2,\ldots$ are of finite rank, and either the set $\{\mu _1, \mu _2,\ldots \}$ is finite or $\mu _n\rightarrow 0$.

• 5.38 Let $H := L^2([0,1])$. For $s,t\in [0,1]$, let $k(s,t):=\min \{1-s,1-t\}$, and let A denote the Fredholm integral operator on H with kernel $k(\cdot \,,\cdot )$. Then $A(x)=\sum _{n=1}^\infty \lambda _n\langle {x},{u_n} \rangle u_n$ for $x\in H$, where $\lambda _n:= 4/(2n-1)^{2} \pi ^{2}$ and $u_{n}(s) :=\sqrt{2} \cos (2n-1)\pi s/2,\, s\in [0,1]$ and $n\in {\mathbb N}$.

• 5.39 Let A denote the Fredholm integral operator on $L^2([0,1])$ with kernel $k(s,t) := \min \{s,t\}, 0 \le s, t \le 1.$ For $x \in L^2([0,1])$ and $n \in {\mathbb N},$ let

  $$ s_n(x) := {\displaystyle {\int _{0}^{1}\!{x(t)\sin (2n - 1)\frac{\pi t}{2}}\,d{m(t)}}}. $$

  Then for every $x \in L^2([0,1])$,

  $$A(x)(s) = \frac{8}{\pi ^2} \sum ^\infty _{n=1} \frac{s_n(x)}{(2n-1)^2} \sin (2n -1) \frac{\pi s}{2}, \quad s \in [0,1],$$

  where the series on the right side converges in $L^2([0,1]).$ Let $y \in L^2([0,1])$.

  • (i) Suppose $\mu \in {\mathbb K},\,\mu \ne 0$ and $\mu \ne (2n-1)^2\pi ^2 /4$ for any $n\in {\mathbb N}$. Then there is a unique $x \in L^2([0,1])$ satisfying $x - \mu A(x) = y.$ In fact,

    $$ x(s) = y(s) + 8\mu \sum ^\infty _{n=1} \frac{s_n(y)}{(2n-1)^2\pi ^2 - 4\mu } \sin (2n-1)\frac{\pi s}{2},\quad s\in [0,1]. $$

    Further, $\Vert x\Vert \le \alpha \Vert y\Vert$, where $\alpha :=1+4|\mu |/\min _{n\in {\mathbb N}}\{|(2n-1)^2\pi ^2-4\mu |\}$.

  • (ii) Suppose $\mu := (2n_1 - 1)^2\pi ^2/4$, where $n_1\in {\mathbb N}$. Then there is x in $L^2([0,1])$ satisfying $x - \mu A(x) = y$ if and only if $s_{n_1}(y) = 0$. In this case,

    $$\begin{aligned} x(s) = y(s)+ & {} \frac{(2n_1-1)^2}{2} \sum _{n \ne n_1} \frac{s_n(y)}{(n-n_1)(n+n_1-1)} \sin (2n-1)\frac{\pi s}{2}\\+ & {} k_1\sin (2n_1 - 1)\frac{\pi s}{2}\quad \text {for}\,s\in [0,1],\,\text {where}\,k_1\in {\mathbb K}. \end{aligned}$$

• 5.40 Let H be a Hilbert space over ${\mathbb K}$, and let $A\in BL(H)$ be nonzero. Then A is compact if and only if there are countable orthonormal subsets $\{u_1,u_2,\ldots \}$ and $\{v_1,v_2,\ldots \}$ of H, and there are positive real numbers $s_1,s_2,\ldots$ such that $A(x)= \sum _ns_n\langle {x},{u_n} \rangle v_n$ for all $x\in H$ with $s_n\rightarrow 0$ if the set $\{v_1,v_2,\ldots \}$ is denumerable. In this case, $A^{*}A(x)= \sum _ns_n^2\langle {x},{u_n} \rangle u_n$ for $x\in H$, and $\sum _ns_n^2=\sum _n\Vert A(u_n)\Vert ^2$. In particular, if H is a separable Hilbert space, then A is a Hilbert–Schmidt operator if and only if $\sum _ns_n^2<\infty$. (Note: The positive real numbers $s_1,s_2,\ldots$ are called the nonzero singular values of A.)