Math Problems Vault

Problem 10A.1

Define $T: \ell^{2} \rightarrow \ell^{2}$ by $T\left(a_{1}, a_{2}, \ldots \right) = \left(0, a_{1}, a_{2}, \ldots \right)$ . Find a formula for $T^{*}$ .

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Problem 10A.2

Suppose $V$ is a Hilbert space, $U$ is a closed subspace of $V$ , and $T: U \rightarrow V$ is defined by $T f = f$ . Describe the linear operator $T^{*}: V \rightarrow U$ .

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Problem 10A.3

Suppose $V$ and $W$ are Hilbert spaces and $g \in V, h \in W$ . Define $T \in \mathcal{B}(V, W)$ by $T f = \left\langle f, g \right\rangle h$ . Find a formula for $T^{*}$ .

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Problem 10A.4

Suppose $V$ and $W$ are Hilbert spaces and $T \in \mathcal{B}(V, W)$ has finite-dimensional range. Prove that $T^{*}$ also has finite-dimensional range.

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Problem 10A.5

Prove or give a counterexample: If $V$ is a Hilbert space and $T: V \rightarrow V$ is a bounded linear map such that $\operatorname {dim}\operatorname {null}(T)< \infty$ , then $\operatorname {dim}\operatorname {null}(T^{*}) < \infty$ .

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Problem 10A.6

Suppose $T$ is a bounded linear map from a Hilbert space $V$ to a Hilbert space $W$ . Prove that $\left\| T^{*} T\right\| = \left\| T\right\|^{2}$ .

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Remark.

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This formula for $\left\| T^{*} T\right\|$ leads to the important subject of $C^{*}$ -algebras.

Problem 10A.7

Suppose $V$ is a Hilbert space and $\operatorname {Inv}(V)$ is the set of invertible bounded operators on $V$ . Think of $\operatorname {Inv}(V)$ as a metric space with the metric it inherits as a subset of $\mathcal{B}(V)$ . Show that $T \mapsto T^{-1}$ is a continuous function from $\operatorname {Inv}(V)$ to $\operatorname {Inv}(V)$ .

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Problem 10A.8

(a)(b)

Suppose $T$ is a bounded operator on a Hilbert space.

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(a)

Prove that $T$ is left invertible if and only if $T^{*}$ is right invertible.

(b)

Prove that $T$ is invertible if and only if $T$ is both left and right invertible.

Problem 10A.9

Suppose $b_{1}, b_{2}, \ldots$ is a bounded sequence in $\mathbf{F}$ . Define a bounded linear map $T: \ell^{2} \rightarrow \ell^{2}$ by

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

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(a)

Find a formula for $T^{*}$ .

(b)

Show that $T$ is injective if and only if $b_{k} \neq 0$ for every $k \in \mathbb {N}$ .

(c)

Show that $T$ has dense range if and only if $b_{k} \neq 0$ for every $k \in \mathbb {N}$ .

(d)

Show that $T$ has closed range if and only if

\inf \left\{ \left|b_{k}\right|: k \in \mathbb {N} \text{and } b_{k} \neq 0\right\} > 0.

(e)

Show that $T$ is invertible if and only if

\inf \left\{ \left|b_{k}\right|: k \in \mathbb {N}\right\} > 0

Problem 10A.10

(a)(b)(c)(d)

Suppose $h \in \mathcal{L}^{\infty }(\mathbb {R})$ and $M_{h}: L^{2}(\mathbb {R}) \rightarrow L^{2}(\mathbb {R})$ is the bounded operator defined by $M_{h} f = f h$ .

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(a)

Show that $M_{h}$ is injective if and only if $\left|\left\{ x \in \mathbb {R}: h(x) = 0\right\} \right| = 0$ .

(b)

Find a necessary and sufficient condition (in terms of $h$ ) for $M_{h}$ to have dense range.

(c)

Find a necessary and sufficient condition (in terms of $h$ ) for $M_{h}$ to have closed range.

(d)

Find a necessary and sufficient condition (in terms of $h$ ) for $M_{h}$ to be invertible.

Problem 10A.11

(a)(b)

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(a)

Prove or give a counterexample: If $T$ is a bounded operator on a Hilbert space such that $T$ and $T^{*}$ are both injective, then $T$ is invertible.

(b)

Prove or give a counterexample: If $T$ is a bounded operator on a Hilbert space such that $T$ and $T^{*}$ are both surjective, then $T$ is invertible.

Problem 10A.12

(a)(b)(c)(d)

Define $T: \ell^{2} \rightarrow \ell^{2}$ by $T\left(a_{1}, a_{2}, a_{3}, \ldots \right) = \left(a_{2}, a_{3}, a_{4}, \ldots \right)$ . Suppose $\alpha \in \mathbf{F}$ .

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(a)

Prove that $T-\alpha I$ is injective if and only if $\left|\alpha \right| \geq 1$ .

(b)

Prove that $T-\alpha I$ is invertible if and only if $\left|\alpha \right| > 1$ .

(c)

Prove that $T-\alpha I$ is surjective if and only if $\left|\alpha \right| \neq 1$ .

(d)

Prove that $T-\alpha I$ is left invertible if and only if $\left|\alpha \right| > 1$ .

Problem 10A.13

(a)(b)

Suppose $V$ is a Hilbert space.

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(a)

Show that $\operatorname {LeftInv}(V) := \left\{ T \in \mathcal{B}(V): T \text{ is left invertible}\right\}$ is an open subset of $\mathcal{B}(V)$ .

(b)

Show that $\operatorname {RightInv}(V) := \left\{ T \in \mathcal{B}(V): T \text{ is right invertible}\right\}$ is an open subset of $\mathcal{B}(V)$ .

Problem 10A.14

(a)(b)

Suppose $T$ is a bounded operator on a Hilbert space $V$ .

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(a)

Prove that $T$ is invertible if and only if $T$ has a unique left inverse. In other words, prove that $T$ is invertible if and only if there exists a unique $S \in \mathcal{B}(V)$ such that $S T = I$ .

(b)

Prove that $T$ is invertible if and only if $T$ has a unique right inverse. In other words, prove that $T$ is invertible if and only if there exists a unique $S \in \mathcal{B}(V)$ such that $T S = I$ .