Math Problems Vault

Problem 86

Let $P$ be an orthogonal projection defined on a Hilbert space $H$ . Show that $\left\| P\right\| = 1$ .

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Problem 87

Given a function $\phi : [0,1] \rightarrow \mathbb {C}$ , consider the operator

P: L^{2}[0,1] \rightarrow L^{2}[0,1] \text{ defined by } P f(x) = \phi (x) f(x).

Find necessary and sufficient conditions on the function $\phi$ for $P$ to be an orthogonal projection.

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Problem 88

Consider the right-shift on the Hilbert space $\ell^{2}$ :

S: \ell ^{2} \rightarrow \ell ^{2}, \quad S\left(\alpha _{1}, \alpha _{2}, \ldots \right) = \left(0, \alpha _{1}, \alpha _{2}, \ldots \right)

Define its adjoint operator.

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Problem 89

Let $A: \ell^{2} \rightarrow \ell^{2}$ be defined by

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

Prove $A$ is linear, continuous, self-adjoint and positive. Find $\sqrt{A}$ .

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Problem 90

(a)(b)(c)

Let $(X, \Omega , \mu )$ be a $\sigma$ -finite measure space. Consider the Hilbert space $H = L^{2}(X, \Omega , \mu ) = : L^{2}(\mu )$ . If $\phi \in L^{\infty }(\mu )$ , define

M_{\phi }: L^{2}(\mu ) \rightarrow L^{2}(\mu ) \quad \text{ by } \quad M_{\phi } f = \phi f.

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(a)

Show that

M_{\phi } \in \mathcal{B}(H) \text{ and }\left\| M_{\phi }\right\| = \left\| \phi \right\| _{\infty }.

Here $\left\| \phi \right\|_{\infty }$ is the $\mu$ -essential supremum norm.

(b)

Show that $M_{\phi }^{*} = M_{\bar{\phi }}$ .

(c)

Show that $M_{\phi }$ is normal. When $M_{\phi }$ is self adjoint? unitary?

Problem 91

(a)(b)

Let $H$ be a Hilbert space and $A \in \mathcal{B}(H)$ . Show that

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(a)

$\overline{\operatorname {Image} A} = \left(\operatorname {ker} A^{*}\right)^{\perp }$ .

(b)

$\operatorname {ker} A = \left(\operatorname {Image} A^{*}\right)^{\perp }$ .

Problem 92

(a)(b)

Let $(X, \Omega , \mu )$ be a measure space. Let $k: X \times X \rightarrow \mathbb {F}$ be an $\Omega \times \Omega$ -measurable function for which there are constants $c_{1}$ and $c_{2}$ such that

\begin{aligned} & \int _{X}\left|k(x, y)\right| d \mu (y) \leq c_{1} \quad \text{ a.e. }[\mu ], \\ & \int _{X}\left|k(x, y)\right| d \mu (x) \leq c_{2} \quad \text{ a.e. }[\mu ]. \end{aligned}

Consider the operator $K: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ defined by

(K f)(x) = \int k(x, y) f(y) d \mu (y)

The function $k$ is called the kernel of the operator $K$ .

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(a)

Show that $K$ is a bounded linear operator and $\left\| K\right\| \leq \sqrt{c_{1} c_{2}}$ .

(b)

Show that $K^{*}$ is the integral operator with kernel $k^{*}(x, y) = \overline{k(x, y)}$ .

Problem 93

Let $\mathcal{H} = H \oplus H$ where $H$ be a Hilbert space. Let $A \in \mathcal{B}(H)$ and $B$ be the operator defined on $\mathcal{H}$ by

B = \left[\begin{smallmatrix} 0 & i A \\ -i A^{*} & 0 \end{smallmatrix}\right]

Prove that $\left\| A\right\| = \left\| B\right\|$ and that $B$ is self-adjoint.

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Problem 94

Let $T$ be a self-adjoint operator on a Hilbert space $H$ . Show that its norm is given by

\left\| T\right\| = \sup _{\left\| x\right\| = 1}\left|\langle T x, x\rangle \right|.

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Problem 95

Let $H$ be a Hilbert space and $A$ a positive self-adjoint operator on $H$ . Prove that the following assertions are equivalent:

$A(H)$ is dense in $H$ .
$\operatorname {Ker} A = \left\{ 0\right\}$ .
$A$ is positive definite, i.e., $\langle A x, x\rangle > 0, \forall x \in H \backslash \left\{ 0\right\}$ .

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Problem 96

Let $H$ be a Hilbert space. If $A, B: H \rightarrow H$ are self-adjoint operators with $0 \leq A \leq B$ and $B$ is compact, prove that $A$ is compact.

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