Math Problems Vault

Problem 12.3

Which of the following sets can be the spectrum of a compact operator in $l_{2}$ :

?

(1)

$\left[0,1\right]$

(2)

$\left\{ 0,1\right\}$

(3)

$\left\{ 0\right\}$

(4)

$\left\{ 1\right\}$

(5)

$\left\{ \frac{1}{n}: n \geq 1\right\}$

(6)

$\left\{ 0\right\} \cup \left\{ \frac{1}{n}: n \geq 1\right\}$

(7)

$\left\{ 0\right\} \cup \left\{ \frac{i}{n}: n \geq 1\right\}$

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Problem 12.4

(1)(2)

Determine whether the operator $(A x)(t) = \int_{0}^{1} t s^{3} x(s) d s$ is compact, and find its spectrum:

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(1)

in the space $L_{2}(\left[0,1\right])$ ;

(2)

in the space $C(\left[0,1\right])$ .

Problem 12.5

(1)(2)

Let $u, v \in C(\left[a, b\right])$ . Determine whether the operator

(A x)(t) = \int _{a}^{b} u(t) v(s) x(s) d s

is compact, and find its spectrum:

?

(1)

in the space $L_{2}(\left[a, b\right])$ ;

(2)

in the space $C(\left[a, b\right])$ .

Problem 12.6

Find the spectrum of an operator $A$ in $C(\left[0,1\right])$ .

?

(1)

$(A x)(t) = \int_{0}^{1} e^{t-s} x(s) d s$ ;

(2)

$(A x)(t) = t^{3} \int_{0}^{1} s^{2} x(s) d s$ ;

(3)

$(A x)(t) = t \int_{0}^{1} \sin s x(s) d s$ ;

(4)

$(A x)(t) = \sin t \cdot x\left(\frac{\pi }{6}\right)$ ;

(5)

$(A x)(t) = \int_{0}^{1} s^{\alpha } x(s) d s, \alpha \geq 0 ;$

(6)

$(A x)(t) = e^{t} \int_{0}^{1} s x(s) d s$ ;

(7)

$(A x)(t) = 2 x(0)+t x(1)$ ;

(8)

$(A x)(t) = t \int_{0}^{1} \frac{x(s)}{1+s^{2}} d s$ .

Problem 12.7

Find the spectrum of an operator $A$ in $L_{2}(\left[0,1\right])$ .

?

(1)

$(A x)(t) = t^{\alpha } \int_{0}^{1} x(s) d s, \alpha > -\frac{1}{2}$ ;

(2)

$(A x)(t) = e^{2 t} \int_{0}^{1} s x(s) d s$ ;

(3)

$(A x)(t) = t^{5} \int_{0}^{1} s x(s) d s$ ;

(4)

$(A x)(t) = \int_{0}^{1} e^{t-3 s} x(s) d s$ ;

(5)

$(A x)(t) = t^{2} \int_{0}^{1} s^{2} x(s) d s$ ;

(6)

$(A x)(t) = t \int_{0}^{1} \frac{x(s)}{1+s} d s$ ;

(7)

$(A x)(t) = t^{\alpha } \int_{0}^{1} s^{\beta } x(s) d s, \alpha , \beta \geq 0 ;$

(8)

$(A x)(t) = \chi_{\left[\frac{1}{3}, \frac{2}{3}\right]}(t) \cdot \int_{0}^{1} e^{s} x(s) d s$ .

Problem 12.8

(1)(2)(3)(4)

Find the spectrum of an operator $A \in \mathcal{L}(X)$ .

?

(1)

$(A x)(t) = \int_{-1}^{1}\left(3 t^{2} s-2 t s^{3}\right) x(s) d s, X = C(\left[-1,1\right]) ;$

(2)

$(A x)(t) = \int_{0}^{1}\left(4 t^{4} s^{3}+t\right) x(s) d s, X = L_{p}(\left[0,1\right]), 1 \leq p \leq +\infty ;$

(3)

$(A x)(t) = \int_{0}^{1}\left(8 t^{3}+5 s^{4}\right) x(s) d s, X = C(\left[0,1\right]) ;$

(4)

$(A x)(t) = \int_{0}^{1}\left(s-3 t^{2} s^{3}\right) x(s) d s, X = L_{p}(\left[0,1\right]), 1 \leq p \leq +\infty$ .

Problem 12.9

Determine whether an operator $A$ is compact in $L_{2}(\left[a, b\right])$ , find its spectrum and norm.

?

(1)

$(A x)(t) = \int_{0}^{2 \pi }(\cos t \cos s+1) x(s) d s,\left[a, b\right] = \left[0,2 \pi \right] ;$

(2)

$(A x)(t) = \int_{0}^{2 \pi } \sin (t-s) x(s) d s,\left[a, b\right] = \left[0,2 \pi \right] ;$

(3)

$(A x)(t) = \int_{0}^{\pi } \cos (t+s) x(s) d s,\left[a, b\right] = \left[0, \pi \right] ;$

(4)

$(A x)(t) = 2 x(t)+\int_{-1}^{1}\left(t s-2 t^{4} s^{4}\right) x(s) d s,\left[a, b\right] = \left[-1,1\right] ;$

(5)

$(A x)(t) = \int_{0}^{1}\left(3 t s+4 t^{2} s^{2}\right) x(s) d s,\left[a, b\right] = \left[0,1\right] ;$

(6)

$(A x)(t) = \int_{-1}^{1}\left(t^{2} s-t s^{2}\right) x(s) d s,\left[a, b\right] = \left[-1,1\right] ;$

(7)

$(A x)(t) = 3 i x(t)+\int_{0}^{\pi } 2(\cos t-\cos s) x(s) d s,\left[a, b\right] = \left[0, \pi \right] ;$

(8)

$(A x)(t) = \int_{0}^{1}\left(t^{4}+s^{4}\right) x(s) d s,\left[a, b\right] = \left[0,1\right] ;$

(9)

$(A x)(t) = x(t)+\int_{-1}^{1}\left(t s+3 t^{2} s^{2}\right) x(s) d s,\left[a, b\right] = \left[-1,1\right] ;$

(10)

$(A x)(t) = -x(t)+\int_{-\pi }^{\pi } \cos (t-s) x(s) d s,\left[a, b\right] = \left[-\pi , \pi \right]$ .

Problem 12.10

(1)(2)(3)(4)

Determine whether an operator $A$ in $l_{p}, 1 \leq p \leq +\infty$ , is compact, find its spectrum. Also find the norm of the operator in case $p = 2$ .

?

(1)

$A x = \left(x_{1}+3 x_{2}, 3 x_{1}+x_{2}, 2 x_{3}, 5 x_{4}, 5 x_{5}, \ldots \right)$ ;

(2)

$A x = \left(i x_{2},-i x_{1}, 2 i x_{4},-2 i x_{3}, x_{5}, 0,0, \ldots \right)$ ;

(3)

$A x = \left(x_{1}+x_{3}, x_{2}+2 x_{4}, x_{1}+x_{3}, 2 x_{2}+x_{4}, 0,0, \ldots \right)$ ;

(4)

$A x = \left(x_{2}+x_{3}, x_{1}+x_{3}, x_{1}+x_{2}, 2 x_{4}, 2 x_{5}, \ldots \right)$ .

Problem 12.11

Find the spectrum, the spectral radius, and the norm of the operator $A$ in $l_{2}$ , defined by the formula $A x = \left(2 x_{1}+\sqrt{2} x_{2}, 3 x_{2}, 0, \ldots \right)$ .

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Problem 12.12

Let $X$ be a Banach space, $A \in S_{\infty }(X)$ and $\lambda \neq 0$ . Prove that $R(A-\lambda I)$ is a closed subspace in $X$ .

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Problem 12.13

(1)(2)(3)

Provide an example of a compact operator in $l_{2}$ , for which

?

(1)

0 is not an eigenvalue;

(2)

0 is an eigenvalue of finite multiplicity;

(3)

0 is an eigenvalue of infinite multiplicity.

Problem 12.14

Let $A$ be a compact self-adjoint operator in an infinite-dimensional Hilbert space. Prove that if $A$ has finitely many eigenvalues, then 0 is an eigenvalue of $A$ of infinite multiplicity.

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Problem 12.15

Let $\left\{ \varphi_{n}: n \geq 1\right\}$ be an orthonormal basis in the Hilbert space $H, A \in \mathcal{L}(H)$ and $A \varphi_{n} = \lambda_{n} \varphi_{n}, n \geq 1$ , where $\left\{ \lambda_{n}: n \geq 1\right\}$ is a bounded sequence of numbers. Prove that $A x = \sum_{n = 1}^{\infty } \lambda_{n}\left(x, \varphi_{n}\right) \varphi_{n}$ for all $x \in H$ .

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Problem 12.16

Let $A \in \mathcal{L}\left(L_{2}(\left[0, \pi \right])\right),\left(A \varphi_{n}\right)(t) = \frac{1}{n^{2}} \varphi_{n}(t), t \in [0, \pi ]$ , where $\varphi_{n}(t) = \sin n t, n \geq 1$ . Prove that $A$ is a Hilbert-Schmidt integral operator. Find its kernel $K$ .

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Problem 12.17

(1)(2)(3)

Let $H$ be a Hilbert space, $\left\{ \varphi_{n}: n \geq 1\right\}$ be an orthonormal sequence in $H,\left\{ \lambda_{n}: n \geq 1\right\} \subset \mathbb {C} \backslash \left\{ 0\right\}$ be a bounded sequence, $A x = \sum_{n = 1}^{\infty } \lambda_{n}\left(x, \varphi_{n}\right) \varphi_{n}, x \in H$ .

Prove that

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(1)

$A \in \mathcal{L}(H)$ ;

(2)

$\sigma_{p}(A) \backslash \left\{ 0\right\} = \left\{ \lambda_{n}: n \geq 1\right\}$ ;

(3)

$0 \notin \sigma_{p}(A)$ if and only if $\left\{ \varphi_{n}: n \geq 1\right\}$ is a basis in $H$ .

Under what conditions on the sequence $\left\{ \lambda_{n}: n \geq 1\right\}$ is the operator $A$ selfadjoint? nonnegative? normal? compact? a Hilbert-Schmidt operator?

Problem 12.18

(1)(2)

Let $A$ be a self-adjoint compact operator in a Hilbert space $H, A = \sum_{n = 1}^{\infty } \lambda_{n}\left(\cdot , \varphi_{n}\right) \varphi_{n}$ be its Schmidt expansion. Prove that

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(1)

$\forall k \geq 1: A^{k} x = \sum_{n = 1}^{\infty } \lambda_{n}^{k}\left(x, \varphi_{n}\right) \varphi_{n}, x \in H$ ;

(2)

$\forall \lambda \in \mathbb {C} \backslash \left\{ 0, \lambda_{n}, n \geq 1\right\}$ :

R_{\lambda }(A) y = \sum _{n = 1}^{\infty } \frac{\left(y, \varphi _{n}\right)}{\lambda _{n}-\lambda } \varphi _{n}-\frac{1}{\lambda } P y = \frac{1}{\lambda }\left(\sum _{n = 1}^{\infty } \frac{\lambda _{n}\left(y, \varphi _{n}\right)}{\lambda _{n}-\lambda } \varphi _{n}-y\right), y \in H

where $P$ is the orthogonal projector onto $\operatorname {Ker} A$ .

Problem 12.19

(1)(2)

Let $H$ be a separable Hilbert space, $A$ be a compact self-adjoint operator in $H,\left\{ \varphi_{n}: n \geq 1\right\}$ be an orthonormal basis in $H$ , such that $A \varphi_{n} = \lambda_{n} \varphi_{n}, n \geq 1$ . For fixed $\lambda \in \mathbb {C} \backslash \left\{ 0\right\}$ and $y \in H$ consider the equation $(A-\lambda I) x = y$ . Prove that

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(1)

if $\lambda_{n} \neq \lambda$ for all $n \geq 1$ , then the solution is unique and has the form

x = \frac{1}{\lambda }\left(\sum _{n = 1}^{\infty } \frac{\lambda _{n}}{\lambda _{n}-\lambda }\left(y, \varphi _{n}\right) \varphi _{n}-y\right)

(2)

if $J = \left\{ n \geq 1 \mid \lambda_{n} = \lambda \right\} \neq \varnothing$ (the set $J$ is finite, because the operator $A$ is compact), then the solution exists if and only if $\left(y, \varphi_{n}\right) = 0$ for all $n \in J$ . In this case, the general solution has the form

x = \frac{1}{\lambda }\left(\sum _{n \in \mathbb {N} \backslash J} \frac{\lambda _{n}}{\lambda _{n}-\lambda }\left(y, \varphi _{n}\right) \varphi _{n}-y\right)+\sum _{n \in J} C_{n} \varphi _{n},

where $C_{n}, n \in J$ , are arbitrary constants.

Problem 12.20

Let $A$ be a nonnegative compact operator in a Hilbert space $H$ . Prove that there exists a unique nonnegative compact operator $B$ such that $B^{2} = A$ . In this case, if $A = \sum_{n = 1}^{\infty } \lambda_{n}\left(\cdot , \varphi_{n}\right) \varphi_{n}$ is the Schmidt expansion, then $B = \sum_{n = 1}^{\infty } \sqrt{\lambda_{n}}\left(\cdot , \varphi_{n}\right) \varphi_{n}$ .

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Remark.

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The operator $B$ is called square root of the operator $A$ . The square root of the operator $A$ is denoted as $B = \sqrt{A}$ or $B = A^{\frac{1}{2}}$ . In problem 9.52* the existence and uniqueness of the square root of an arbitrary nonnegative operator $A$ is proven by another method without assumption that the operator $A$ is compact.

Problem 12.21

(1)(2)(3)

Let $K(t, s) = \sum_{n = 1}^{\infty } \frac{1}{n^{2}} \sin n \pi t \sin n \pi s, t, s \in [-1,1]$ . Consider the operator $A$ in $L_{2}(\left[-1,1\right])$ , defined by the formula

(A x)(t) = \int _{-1}^{1} K(t, s) x(s) d s, t \in [-1,1]

?

(1)

Prove that $A$ is a self-adjoint Hilbert-Schmidt operator (in particular, $A$ is a compact operator).

(2)

Find $\sigma (A), r(A)$ and $\left\| A\right\|$ , as well as an orthonormal sequence of eigenfunctions of the operator $A$ .

(3)

Find the Schmidt expansion of $A$ .

Problem 12.22

(1)(2)(3)

Let $K$ be an even $2 \pi$ -periodic real-valued function that belongs to $L_{2}(\left[-\pi , \pi \right]), K(t) = \sum_{n \in \mathbb {Z}} c_{n} e^{i n t}, t \in \mathbb {R}$ be its Fourier series. Consider the operator $A$ in $L_{2}(\left[-\pi , \pi \right])$ , defined by the formula

(A x)(t) = \int _{-\pi }^{\pi } K(t-s) x(s) d s, t \in [-\pi , \pi ]

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(1)

Prove that $A$ is a self-adjoint Hilbert-Schmidt operator and the functions $\varphi_{n}(t) = \frac{1}{\sqrt{2 \pi }} e^{i n t}, t \in [-\pi , \pi ], n \in \mathbb {Z}$ , which form an orthonormal basis in the space $L_{2}(\left[-\pi , \pi \right])$ , are the eigenfunctions of the operator $A$ .

(2)

Find $\sigma_{p}(A), \sigma (A), r(A)$ and $\left\| A\right\|$ .

(3)

Find the Schmidt expansion of $A$ .

Problem 12.23

(1)(2)(3)

Let $K$ be an even $2 \pi$ -periodic real-valued function that belongs to $L_{2}(\left[-\pi , \pi \right]), K(t) = \frac{a_{0}}{2}+\sum_{n = 1}^{\infty } a_{n} \cos n t, t \in \mathbb {R}$ be its Fourier series. Consider the operator $A$ in $L_{2}(\left[-\pi , \pi \right])$ , defined by the formula

(A x)(t) = \int _{-\pi }^{\pi } K(t+s) x(s) d s, t \in [-\pi , \pi ]

?

(1)

Prove that $A$ is a self-adjoint Hilbert-Schmidt operator with the following eigenvalues and orthonormal eigenfunctions: $\lambda_{0} = \pi a_{0}, \varphi_{0}(t) = \frac{1}{\sqrt{2 \pi }} ; \lambda_{n}^{(1)} = \pi a_{n}, \varphi_{n}^{(1)}(t) = \frac{\cos n t}{\sqrt{\pi }} ; \lambda_{n}^{(2)} = -\pi a_{n}, \varphi_{n}^{(2)}(t) = \frac{\sin n t}{\sqrt{\pi }}, n \geq 1$ .

(2)

Find $\sigma (A), r(A)$ and $\left\| A\right\|$ .

(3)

Find the Schmidt expansion of $A$ .

Problem 12.24

Let $A$ be the integral operator in $L_{2}(\left[-\pi , \pi \right])$ with a kernel $K(t, s)$ . Find the norm and the spectrum of $A$ and the eigenfunctions that correspond to nonzero eigenvalues.

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(1)

$K(t, s) = \sum_{n = 1}^{N} \lambda_{n} \cos n(t-s), N \in \mathbb {N}, \lambda_{1}, \ldots , \lambda_{N} \in \mathbb {C} \backslash \left\{ 0\right\}$ ;

(2)

$K(t, s) = \cos^{4}(t-s)$ ;

(3)

$K(t, s) = \sum_{n = 1}^{\infty } \frac{1}{n^{2}} \cos n t \cos n s$ ;

(4)

$K(t, s) = \sum_{n = 1}^{\infty } \frac{1}{n^{2}} \sin (2 n+1) t \sin (2 n+1) s$ ;

(5)

$K(t, s) = \sum_{n = 0}^{\infty } \frac{1}{(2 n+1)^{2}} \cos (2 n+1) t \cos (2 n+1) s$ ;

(6)

$K(t, s) = \sum_{n = 0}^{\infty } \frac{1}{(2 n+1)^{2}} \cos (2 n+1)(t-s)$ .

Problem 12.25

Find the norm, the spectrum, the eigenvalues and the eigenfunctions of an operator $A$ in $L_{2}(\left[a, b\right])$ , defined by the formula $(A x)(t) = \int_{a}^{b} K(t, s) x(s) d s, t \in \left[a, b\right]$ , in the following cases:

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(1)

$K(t, s) = \begin{cases} t(1-s), & t \leq s, \\ s(1-t), & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(2)

$K(t, s) = \begin{cases} \sin t \cos s, & t \leq s, \\ \cos t \sin s, & s \leq t, \end{cases} \quad a = 0, b = \pi$ ;

(3)

$K(t, s) = \begin{cases} \cos t \sin s, & t \leq s, \\ \sin t \cos s, & s \leq t, \end{cases} \quad a = 0, b = \pi$ ;

(4)

$K(t, s) = \begin{cases} (t+1) s, & t \leq s, \\ t(s+1), & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(5)

$K(t, s) = \begin{cases} \sin t \sin (1-s), & t \leq s, \\ \sin (1-t) \sin s, & s \leq t, \end{cases} \quad a = 0, b = 1$ ;

(6)

$K(t, s) = \begin{cases} (1+t)(1-s), & t \leq s, \\ (1-t)(1+s), & s \leq t, \end{cases} \quad a = -1, b = 1$ .

Problem 12.26

Find the norm of the operator $A$ in $L_{2}(\left[0,1\right])$ , defined by the formula $(A x)(t) = \int_{0}^{t} x(s) d s$ .

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Problem 12.27

Let $H$ be a Hilbert space, $A \in \mathcal{L}(H)$ and $A^{*} = A$ . Prove that if $\sigma (A)$ is a countable set which has no nonzero limit points, and every nonzero point of its spectrum is an eigenvalue of finite multiplicity, then the operator $A$ is compact.

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Problem 12.28

(1)(2)(3)

(Nikolsky) Prove that Theorem 12.1 remains valid for the operator $A \in \mathcal{L}(X)$ , where $X$ is a Banach space, if the condition $A \in S_{\infty }(X)$ is replaced with the condition $A^{k} \in S_{\infty }(X)$ for some $k \in \mathbb {N}$ .

Theorem 12.1: Let $A$ be a compact operator in an infinite-dimensional Banach space. Then

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(1)

$0 \in \sigma (A) ;$

(2)

$\sigma (A)$ is a countable set, the only possible limit point of which is 0 ;

(3)

any nonzero point of the spectrum is an eigenvalue of finite multiplicity, i.e., if $\lambda \in \sigma (A) \backslash \left\{ 0\right\}$ , then $0 < \operatorname {dimKer}(A-\lambda I) < +\infty$ .

Problem 12.29

(1)(2)

(Sudakov)

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(1)

Let $X$ be a Banach space, $M \subset X, A \in \mathcal{L}(X), 1 \notin \sigma_{p}(A), A^{k} \in S_{\infty }(X)$ for some $k \in \mathbb {N}$ , and there exists $C > 0$ such that $\left\| A x-x\right\| < C$ for all $x \in M$ . Prove that the set $M$ is bounded.

(2)

Prove that in the compactness criteria in $L_{p}(\left[a, b\right])$ by Kolmogorov and M. Riesz (see problems 10.81 and 10.82) condition (a) follows from condition (b).

Problem 12.30

(1)(2)(3)

Let $H$ be a separable Hilbert space, $\left\{ \varphi_{n}: n \geq 1\right\}$ and $\left\{ \psi_{n}: n \geq 1\right\}$ be orthonormal sequences in $H,\left\{ s_{n}: n \geq 1\right\}$ be a sequence of numbers such that $\lim_{n \rightarrow \infty } s_{n} = 0$ . For any $x \in H$ put $A_{n} x = \sum_{k = 1}^{n} s_{k}\left(x, \varphi_{k}\right) \psi_{k}, n \geq 1$ , and $A x = \sum_{n = 1}^{\infty } s_{n}\left(x, \varphi_{n}\right) \psi_{n}$ . Prove that

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(1)

$A_{n} \rightrightarrows A, n \rightarrow \infty$ ;

(2)

$A$ is a compact operator in $H$ ;

(3)

$A$ is a Hilbert-Schmidt operator if and only if $\sum_{n = 1}^{\infty }\left|s_{n}\right|^{2} < \infty$ .

Problem 12.31

Let $A$ be the operator from problem $12.30^{\circ }$ . Find $A^{*}, A A^{*}, A^{*} A$ .

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Problem 12.32

Prove that any compact operator $A$ in a Hilbert space $H$ can be represented as

A = \sum _{n = 1}^{N} s_{n}\left(\cdot , \varphi _{n}\right) \psi _{n},

where $N \leq +\infty ,\left\{ \varphi_{n}: 1 \leq n \leq N\right\}$ and $\left\{ \psi_{n}: 1 \leq n \leq N\right\}$ are orthonormal sequences and $\left\{ s_{n}: 1 \leq n \leq N\right\}$ are positive eigenvalues of the operator $\left|A\right| = \sqrt{A^{*} A}$ (square root is defined in problem 12.20). In this case, if $N = +\infty$ , then the series converges in operator norm. The numbers $s_{n}, 1 \leq n \leq N$ , are called the singular values of the operator $A$ .

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Problem 12.33

Let $H$ be a Hilbert space, $A \in S_{\infty }(H),\left\| A^{*} x\right\| \leq \left\| A x\right\| , x \in H$ . Prove that $A^{*} A = A A^{*}$ .

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Problem 12.34

Let $H$ be a Hilbert space, $A \in S_{\infty }(H), A^{*} = A,\left\{ \lambda_{n}^{+}: 1 \leq n \leq N^{+}\right\} ,\left\{ \lambda_{n}^{-}: 1 \leq n \leq N^{-}\right\}$ , where $N^{+}, N^{-} \leq +\infty$ , are the sequences of positive and negative eigenvalues of the operator $A$ , numbered counting their multiplicities, and $\lambda_{1}^{-} \leq \lambda_{2}^{-} \leq \ldots < 0 < \ldots \leq \lambda_{2}^{+} \leq \lambda_{1}^{+}$ (one or both of the sequences $\left\{ \lambda_{n}^{+}\right\} ,\left\{ \lambda_{n}^{-}\right\}$ may be empty). Also, let $\varphi_{n}^{ \pm }$ be the unit eigenvectors corresponding to the eigenvalues $\lambda_{n}^{ \pm }$ . Prove that

\begin{aligned} & \lambda _{1}^{+} = \max \left\{ (A x, x) \mid \left\| x\right\| = 1\right\} = \left(A \varphi _{1}^{+}, \varphi _{1}^{+}\right) \\ & \lambda _{n}^{+} = \max \left\{ (A x, x) \mid \left\| x\right\| = 1,\left(x, \varphi _{k}^{+}\right) = 0,1 \leq k \leq n-1\right\} = \\ & = \left(A \varphi _{n}^{+}, \varphi _{n}^{+}\right), 2 \leq n \leq N^{+}. \end{aligned}

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Remark.

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For the numbers $\lambda_{n}^{-}$ , similar formulas are true with max replaced with $\min$ , because $\left\{ -\lambda_{n}^{-}: n \geq 1\right\}$ are the positive eigenvalues of the operator $-A$ .

Problem 12.35

(1)(2)

Let $\mathcal{L}_{n}$ be the family of all $n$ dimensional subspaces of $H, n \geq 1$ . In notations from problem 12.34, prove that for all $1 \leq n \leq N^{+}$

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(1)

$\lambda_{n}^{+} = \min_{L \in \mathcal{L}_{n-1}} \max_{x \in L^{\perp },\left\| x\right\| = 1}(A x, x)$ ;

(2)

$\lambda_{n}^{+} = \max_{L \in \mathcal{L}_{n}} \min_{x \in L,\left\| x\right\| = 1}(A x, x)$ .

Problem 12.36

(Cauchy interlacing theorem) Let $H$ be a Hilbert space, $A$ be a compact self-adjoint operator in $H, G \subset H$ be a subspace of codimension 1 (i.e., $\operatorname {dim} G^{\perp } =$ 1), $P$ be the orthogonal projector onto $G$ . Prove that the eigenvalues $\lambda_{n}^{ \pm }$ of the operator $A$ and the eigenvalues $\mu_{n}^{ \pm }$ of the operator $P A P$ , numbered in the same way as in problem 12.34, satisfy the following inequalities: $\lambda_{1}^{+} \geq \mu_{1}^{+} \geq \lambda_{2}^{+} \geq \mu_{2}^{+} \geq \ldots > 0$ , if $A$ has positive eigenvalues, $\lambda_{1}^{-} \leq \mu_{1}^{-} \leq \lambda_{2}^{-} \leq \mu_{2}^{-} \leq \ldots < 0$ , if $A$ has negative eigenvalues.

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Problem 12.37

Let $H$ be a Hilbert space, $A \in S_{\infty }(H)$ . Denote by $s_{n}(A), 1 \leq n \leq N_{A}, N_{A} \leq +\infty$ , the singular values of the operator $A$ (see problem 12.32), written in the order of decreasing taking into account their multiplicities. In case $N_{A} < +\infty$ put $s_{n}(A) = 0, n > N_{A}$ . Prove that

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(1)

$s_{n}\left(A^{*}\right) = s_{n}(A), n \geq 1 ;$

(2)

$s_{n}(A) = \min_{L: \operatorname {dim} L \leq n-1} \max_{x \in L^{\perp },\left\| x\right\| = 1}\left\| A x\right\| , 1 \leq n \leq \operatorname {dim} H$ , in particular, $s_{1}(A) = \left\| A\right\|$ ;

(3)

if $B \in \mathcal{L}(H)$ , then

s_{n}(B A) \leq \left\| B\right\| s_{n}(A), s_{n}(A B) \leq \left\| B\right\| s_{n}(A), n \geq 1 ;

(4)

if $B \in S_{\infty }(H), A \geq B \geq 0$ , then $s_{n}(A) \geq s_{n}(B), n \geq 1$ ;

(5)

$s_{n}(A) = \min \left\| A-T\right\|$ , where the minimum is taken over all operators $T \in S_{0}(H)$ , for which $\operatorname {rank} T = \operatorname {dim} R(T) \leq n-1, n \geq 1 ;$

(6)

if $B \in S_{\infty }(H)$ , then $s_{m+n-1}(A+B) \leq s_{m}(A)+s_{n}(B), n, m \geq 1$ ;

(7)

if $B \in S_{\infty }(H)$ , then $\left|s_{n}(A)-s_{n}(B)\right| \leq \left\| A-B\right\| , n \geq 1$ .

Problem 12.38

Let $H$ be a separable Hilbert space, $A \in S_{\infty }(H),\left\{ s_{n}(A): n \geq 1\right\}$ be a sequence of singular values of the operator $A$ , numbered with respect to their multiplicities and, possibly, supplemented by zeros. Prove that $A$ is a Hilbert-Schmidt operator if and only if $\sum_{n = 1}^{\infty } s_{n}^{2}(A) < +\infty$ . Moreover $\left\| A\right\|_{2} = \left(\sum_{n = 1}^{\infty } s_{n}^{2}(A)\right)^{1 / 2}$ .

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Problem 12.39

Let $H$ be a separable Hilbert space, $A \in S_{\infty }(H),\left\{ s_{n}(A): n \geq 1\right\}$ be a sequence of singular values of the operator $A$ . It is said that the operator $A$ belongs to the Schatten class $S_{p}(H)$ , where $1 \leq p < +\infty$ , if $\left\| A\right\|_{p}: = \left(\sum_{n = 1}^{\infty } s_{n}^{p}(A)\right)^{1 / p} < +\infty$ . Prove that

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(1)

$A \in S_{p}(H)$ if and only if $A^{*} \in S_{p}(H)$ ;

(2)

$\left\| A\right\|_{p} \geq \left\| A\right\| , A \in S_{p}(H)$ ;

(3)

if $A \in S_{p}(H)$ and $B \in \mathcal{L}(H)$ , then $A B, B A \in S_{p}(H)$ , and

\left\| A B\right\| _{p} \leq \left\| A\right\| _{p} \cdot \left\| B\right\| , \quad \left\| B A\right\| _{p} \leq \left\| A\right\| _{p} \cdot \left\| B\right\| ;

(4)

$\left\| A\right\|_{p} = \max \left(\sum_{n = 1}^{\infty }\left|\left(A e_{n}, g_{n}\right)\right|^{p}\right)^{1 / p}, A \in S_{\infty }(H)$ , where the maximum is taken over all pairs of orthonormal bases $\left\{ e_{n}: n \geq 1\right\} ,\left\{ g_{n}: n \geq 1\right\}$ ;

(5)

if $A, B \in S_{p}(H)$ , then $A+B \in S_{p}(H)$ and

\left\| A+B\right\| _{p} \leq \left\| A\right\| _{p}+\left\| B\right\| _{p}

(6)

$S_{p}(H)$ endowed with the norm $\left\| \cdot \right\|_{p}$ is a Banach space;

(7)

the set of finite-dimensional operators $S_{0}(H)$ is dense in $S_{p}(H)$ .

Remark.

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In view of problem 12.38, the definition of the Schatten class $S_{p}(H)$ for $p = 2$ aligns with the definition of the set of Hilbert-Schmidt operators $S_{2}(H)$ .

Problem 12.40

Let $H$ be a separable Hilbert space. An operator $A$ that belongs to the Schatten class $S_{1}(H)$ is called nuclear (or trace-class operator), and the number $\left\| A\right\|_{1}$ is called the nuclear norm. Prove the following statements:

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(1)

If $A \in S_{1}(H)$ and $\left\{ e_{n}: n \geq 1\right\}$ is a complete orthonormal basis in $H$ , then the series $\sum_{n = 1}^{\infty }\left(A e_{n}, e_{n}\right)$ converges absolutely and its sum does not depend on the choice of basis $\left\{ e_{n}: n \geq 1\right\}$ . This sum is denoted by $\operatorname {tr} A$ and is called the trace of the operator $A$ .

(2)

If $A \in S_{1}(H)$ and $A \geq 0$ , then $\operatorname {tr} A = \sum_{n = 1}^{\infty } \lambda_{n}$ , where $\lambda_{n}, n \geq 1$ , are the eigenvalues of the operator $A$ , numbered with respect to their multiplicities.

(3)

If $A, B \in S_{2}(H)$ , then $A B, B A \in S_{1}(H)$ ,

\left\| A B\right\| _{1} \leq \left\| A\right\| _{2} \cdot \left\| B\right\| _{2}, \quad \left\| B A\right\| _{1} \leq \left\| A\right\| _{2} \cdot \left\| B\right\| _{2}.

(4)

If $A, B \in S_{2}(H)$ , then $\operatorname {tr} A B = \operatorname {tr} B A$ .

(5)

For every $A \in S_{1}(H)$ there exist $B, C \in S_{2}(H)$ such that $A = B C$ .

(6)

The norm $\left\| A\right\|_{2}$ is defined by the scalar product $(A, B): = \operatorname {tr} B^{*} A$ .

(7)

$S_{2}(H)$ is a separable Hilbert space with an orthonormal basis, which consists of one-dimensional operators $A_{m, n} = \left(\cdot , e_{n}\right) g_{m}, m, n \in \mathbb {N}$ , where $\left\{ e_{n}: n \geq 1\right\}$ and $\left\{ g_{n}: n \geq 1\right\}$ are orthonormal bases in $H$ .

Remark.

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(2)

Lidskii's theorem states that the equality $\operatorname {tr} A = \sum_{n = 1}^{\infty } \lambda_{n}$ is valid for all $A \in S_{1}(H)$ . In particular, if an operator $A$ has no eigenvalues, then $\operatorname {tr} A = 0$ .

Problem 12.41

Let $H = L_{2}(T, \mu )$ be a separable Hilbert space. Prove that $A \in S_{2}(H)$ if and only if $A$ is an integral operator of the form $(A x)(t) = \int_{T} K(t, s) x(s) d \mu (s), t \in T, x \in H$ , where $K \in L_{2}(T \times T, \mu \times \mu )$ . Moreover $\left\| A\right\|_{2}^{2} = \int_{T} \int_{T}\left|K(t, s)\right|^{2} d \mu (t) d \mu (s)$ .

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Problem 12.42

(1)(2)(3)

Let $H_{1}, H_{2}$ be separable Hilbert spaces, $\left\{ e_{n}: n \geq 1\right\} ,\left\{ f_{n}: n \geq 1\right\}$ be orthonormal bases in $H_{1}$ and $H_{2}$ , respectively. An operator $A \in \mathcal{L}\left(H_{1}, H_{2}\right)$ is called a Hilbert-Schmidt operator if its Hilbert-Schmidt norm $\left\| A\right\|_{2} = \left(\sum_{j, k = 1}^{\infty }\left|\left(A e_{k}, f_{j}\right)\right|^{2}\right)^{\frac{1}{2}}$ is finite. Prove that

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(1)

Hilbert-Schmidt norm does not depend on the choice of orthonormal bases in spaces $H_{1}$ and $H_{2}$ ;

(2)

functions $\sinh t, e^{i k t}, t \in [-\pi , \pi ], k \in \mathbb {Z}$ , form a complete orthogonal system in the space $W_{2}^{1}(\left[-\pi , \pi \right])$ (see problems $2.32^{\circ }, 2.33^{*})$ ;

(3)

the embedding operator of the space $W_{2}^{1}(\left[-\pi , \pi \right])$ into $L_{2}(\left[-\pi , \pi \right])$ is a Hilbert-Schmidt operator.

Problem 12.43

(1)(2)(3)

Let $\alpha = \left(\alpha_{1}, \alpha_{2}, \ldots \right), \alpha_{k} \geq 1$ for all $k \geq 1, l_{2, \alpha }$ be the Hilbert space of all sequences $x = \left(x_{1}, x_{2}, \ldots \right)$ , which satisfy the condition

\left\| x\right\| _{\alpha } = \left(\sum _{n = 1}^{\infty } \alpha _{n} x_{n}^{2}\right)^{\frac{1}{2}} < \infty

endowed with the corresponding scalar product. Prove the following statements:

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(1)

$l_{2, \alpha } \subset l_{2}$ , and the norm of the embedding operator $A: l_{2, \alpha } \rightarrow l_{2}, A x = x$ , does not exceed one;

(2)

the operator $A$ is compact if and only if $\lim_{n \rightarrow \infty } \alpha_{n} = \infty$ ;

(3)

$A: l_{2, \alpha } \rightarrow l_{2}$ is a Hilbert-Schmidt operator if and only if $\sum_{n = 1}^{\infty } \frac{1}{\alpha_{n}} < \infty$ .