Math Problems Vault

Problem 10C.1

Prove that if $T$ is a compact operator on a Hilbert space $V$ and $e_{1}, e_{2}, \ldots$ is an orthonormal sequence in $V$ , then $\lim_{n \rightarrow \infty } T e_{n} = 0_V$ .

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Problem 10C.2

Prove that if $T$ is a compact operator on $L^{2}([0,1])$ , then $\lim_{n \rightarrow \infty } \sqrt{n}\left\| T\left(x^{n}\right)\right\|_{2} = 0$ , where $x^{n}$ means the element of $L^{2}([0,1])$ defined by $x \mapsto x^{n}$ .

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Problem 10C.3

Suppose $T$ is a compact operator on a Hilbert space $V$ and $f_{1}, f_{2}, \ldots$ is a sequence in $V$ such that $\lim_{n \rightarrow \infty } \left\langle f_{n}, g \right\rangle = 0$ for every $g \in V$ . Prove that $\lim_{n \rightarrow \infty }\left\| T f_{n}\right\| = 0$ .

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Problem 10C.4

Suppose $h \in L^{\infty }(\mathbb {R})$ . Define $M_{h} \in \mathcal{B}\left(L^{2}(\mathbb {R})\right)$ by $M_{h} f = f h$ . Prove that if $\left\| h\right\|_{\infty } > 0$ , then $M_{h}$ is not compact.

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Problem 10C.5

Suppose $\left(b_{1}, b_{2}, \ldots \right) \in \ell^{\infty }$ . Define $T: \ell^{2} \rightarrow \ell^{2}$ by

T\left(a_{1}, a_{2}, \ldots \right) = \left(a_{1} b_{1}, a_{2} b_{2}, \ldots \right)

Prove that $T$ is compact if and only if $\lim_{n \rightarrow \infty } b_{n} = 0$ .

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Problem 10C.6

Suppose $T$ is a bounded operator on a Hilbert space $V$ . Prove that if there exists an orthonormal basis $\left\{ e_{k}\right\}_{k \in \Gamma }$ of $V$ such that

\sum _{k \in \Gamma }\left\| T e_{k}\right\| ^{2} < \infty

then $T$ is compact.

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Problem 10C.7

Suppose $T$ is a bounded operator on a Hilbert space $V$ . Prove that if $\left\{ e_{k}\right\}_{k \in \Gamma }$ and $\left\{ f_{j}\right\}_{j \in \Omega }$ are orthonormal bases of $V$ , then

\sum _{k \in \Gamma }\left\| T e_{k}\right\| ^{2} = \sum _{j \in \Omega }\left\| T f_{j}\right\| ^{2}

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Problem 10C.8

Suppose $T$ is a bounded operator on a Hilbert space. Prove that $T$ is compact if and only if $T^{*} T$ is compact.

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Problem 10C.9

Prove that if $T$ is a compact operator on an infinite-dimensional Hilbert space, then $\left\| I-T\right\| \geq 1$ .

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Problem 10C.10

Show that if $T$ is a surjective but not injective operator on a vector space $V$ , then

\operatorname {null} T \varsubsetneqq \operatorname {null} T^{2} \varsubsetneqq \operatorname {null} T^{3} \varsubsetneqq \cdots

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Problem 10C.11

(a)(b)(c)

Suppose $T$ is a compact operator on a Hilbert space and $\alpha \in \mathbf{F} \backslash \left\{ 0\right\}$ .

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(a)

Prove that $\operatorname {range}(T-\alpha I)^{m-1} = \operatorname {range}(T-\alpha I)^{m}$ for some $m \in \mathbb {N}$ .

(b)

Prove that $\operatorname {null}(T-\alpha I)^{n-1} = \operatorname {null}(T-\alpha I)^{n}$ for some $n \in \mathbb {N}$ .

(c)

Show that the smallest positive integer $m$ that works in (a) equals the smallest positive integer $n$ that works in (b).

Problem 10C.12

Prove that if $f:[0,1] \rightarrow \mathbf{F}$ is a continuous function, then there exists a continuous function $g:[0,1] \rightarrow \mathbf{F}$ such that

f(x) = g(x)+\int _{0}^{x} g

for all $x \in [0,1]$ .

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Problem 10C.13

Suppose $S$ is a bounded invertible operator on a Hilbert space $V$ and $T$ is a compact operator on $V$ .

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(a)

Prove that $S+T$ has closed range.

(b)

Prove that $S+T$ is injective if and only if $S+T$ is surjective.

(c)

Prove that $\operatorname {null}(S+T)$ and $\operatorname {null}\left(S^{*}+T^{*}\right)$ are finite-dimensional.

(d)

Prove that $\operatorname {dim} \operatorname {null}(S+T) = \operatorname {dim} \operatorname {null}\left(S^{*}+T^{*}\right)$ .

(e)

Prove that there exists $R \in \mathcal{B}(V)$ such that range $R$ is finite-dimensional and $S+T+R$ is invertible.

Problem 10C.14

Suppose $T$ is a compact operator on a Hilbert space $V$ . Prove that range $T$ is a separable subspace of $V$ .

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Problem 10C.15

(a)(b)

Suppose $T$ is a compact operator on a Hilbert space $V$ and $e_{1}, e_{2}, \ldots$ is an orthonormal basis of range $T$ . Let $P_{n}$ denote the orthogonal projection of $V$ onto $\operatorname {span}\left\{ e_{1}, \ldots , e_{n}\right\}$ .

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(a)

Prove that $\lim_{n \rightarrow \infty }\left\| T-P_{n} T\right\| = 0$ .

(b)

Prove that a bounded operator on a Hilbert space $V$ is compact if and only if it is the limit in $\mathcal{B}(V)$ of a sequence of bounded operators with finite-dimensional range.

Problem 10C.16

Prove that if $T$ is a compact operator on a Hilbert space $V$ , then there exists a sequence $S_{1}, S_{2}, \ldots$ of invertible operators on $V$ such that $\lim_{n \rightarrow \infty }\left\| T-S_{n}\right\| = 0$ .

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Problem 10C.17

Suppose $T$ is a bounded operator on a Hilbert space such that $p(T)$ is compact for some nonzero polynomial $p$ with coefficients in $\mathbf{F}$ . Prove that $\operatorname {sp}(T)$ is a countable set.

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Problem 10C.18

(a)(b)

Suppose $T \in \mathcal{B}\left(\ell^{2}\right)$ is defined by $T\left(a_{1}, a_{2}, a_{3}, \ldots \right) = \left(a_{2}, a_{3}, a_{4}, \ldots \right)$ . Suppose also that $\alpha \in \mathbf{F}$ and $\left|\alpha \right| < 1$ .

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(a)

Show that the geometric multiplicity of $\alpha$ as an eigenvalue of $T$ equals 1.

(b)

Show that the algebraic multiplicity of $\alpha$ as an eigenvalue of $T$ equals $\infty$ .

Remark.

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Suppose $T$ is a bounded operator on a Hilbert space. The algebraic multiplicity of an eigenvalue $\alpha$ of $T$ is defined to be the dimension of the subspace

\bigcup _{n = 1}^{\infty } \operatorname {null}(T-\alpha I)^{n}

As an easy example, if $T$ is the left shift as defined in the next exercise, then the eigenvalue $0$ of $T$ has geometric multiplicity 1 but algebraic multiplicity $\infty$ . The definition above of algebraic multiplicity is equivalent on finite-dimensional spaces to the common definition involving the multiplicity of a root of the characteristic polynomial. However, the definition used here is cleaner (no determinants needed) and has the advantage of working on infinite-dimensional Hilbert spaces.

Problem 10C.19

Prove that the geometric multiplicity of an eigenvalue of a normal operator on a Hilbert space equals the algebraic multiplicity of that eigenvalue.

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Problem 10C.20

Prove that every nonzero eigenvalue of a compact operator on a Hilbert space has finite algebraic multiplicity.

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Problem 10C.21

Prove that if $T$ is a compact operator on a Hilbert space and $\alpha$ is a nonzero eigenvalue of $T$ , then the algebraic multiplicity of $\alpha$ as an eigenvalue of $T$ equals the algebraic multiplicity of $\overline{\alpha }$ as an eigenvalue of $T^{*}$ .

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Problem 10C.22

Prove that if $V$ is a separable Hilbert space, then $\mathcal{C}(V)$ , the Banach space of compact operators on $V$ , is separable.

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