Adjoints and operator classes of multiplication operators

Prove that the operator from problem 6.19 is a projection operator in each of the spaces а)-в) if and only if $\lambda_{n} \in \left\{ 0,1\right\}$ for every $n \in \mathbb {N}$.

Find the adjoint operator $A^{\prime }$ for the operator $A$ in the spaces $l_{p}, p \in [1, \infty )$, and $c_{0}$:

Find the adjoint operator $A^{\prime }$ for the following operators $A$:

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ self-adjoint in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(B x)(t) = a(t) x(t)$ self-adjoint in the space $L_{2}[0,1]$?

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ unitary in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ unitary in the space $L_{2}[0,1]$?

Under what conditions on the sequence $\left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ is the operator $A x = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \lambda_{3} x_{3}, \ldots \right)$ a partial isometry in the space $l_{2}$? Under what conditions on the function $a \in L_{\infty }[0,1]$ is the operator $(A x)(t) = a(t) x(t)$ a partial isometry in the space $L_{2}[0,1]$?

Find the polar decomposition $A = W S = R W$ for the operator $A \in \mathcal{B}(H)$ in the case

Prove that for any function $a \in L_{\infty }[0,1]$ the operator $(A x)(t) = a(t) x(t)$ is normal in $L_{2}[0,1]$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $u : X \to \mathbb {F}$ be an $\Omega$-measurable function such that $\sup \left\{ \left|u(x)\right| : x \in X\right\} < \infty$. Show that $U : L^2(X, \Omega , \mu ) \to L^2(X, \Omega , \mu )$ defined by $Uf = uf$ is an isometry if and only if $\left|u(x)\right| = 1$ a.e. $[\mu ]$, in which case $U$ is surjective.

Show that a multiplication operator $M_{\phi }$ satisfies $M_{\phi }^{2} = M_{\phi }$ if and only if $\phi$ is a characteristic function.

Theorem 1.5 (context for $M_\phi$): Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and put $\mathcal{H} = L^{2}(X, \Omega , \mu ) \equiv L^{2}(\mu )$. If $\phi \in L^{\infty }(\mu )$, define $M_{\phi }: L^{2}(\mu ) \rightarrow L^{2}(\mu )$ by $M_{\phi } f = \phi f$. Then $M_{\phi } \in \mathcal{B}(L^{2}(\mu ))$ and $\left\| M_{\phi }\right\| = \left\| \phi \right\|_{\infty }$.

Verify the statement in Example 2.8: Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $M_{\phi }$ be the multiplication operator with symbol $\phi$ (1.5). Then $M_{\phi }^{*}$ is $M_{\overline{\phi }}$, the multiplication operator with symbol $\overline{\phi }$.

Find the adjoint of a diagonal operator (Exercise 1.8).

Exercise 1.8 (context): Let $\left\{ e_{n}\right\}$ be the usual basis for $l^{2}$ and let $\left\{ \alpha_{n}\right\}$ be a sequence of scalars. Show that there is a bounded operator $A$ on $l^{2}$ such that $Ae_{n} = \alpha_{n} e_{n}$ for all $n$ if and only if $\left\{ \alpha_{n}\right\}$ is uniformly bounded, in which case $\left\| A\right\| = \sup \left\{ \left|\alpha_{n}\right|: n \geq 1\right\}$. This type of operator is called a diagonal operator or is said to be diagonalizable.

Show that a diagonalizable operator is normal.

Verify the statement made in (1.5): let $(X, \Omega , \mu )$ and $M_\phi : L^p(\mu ) \to L^p(\mu )$ be as in Example III.2.2. If $1 \le p < \infty$ and $1/p + 1/q = 1$, then $M_\phi^*: L^q(\mu ) \to L^q(\mu )$ is given by $M_\phi^* f = \phi f$. That is, $M_\phi^* = M_\phi$.

Find the polar decomposition of each of the following operators:

Let $e_0, e_1, \ldots$ be an orthonormal basis for $\mathcal{H}$ and let $\alpha_0, \alpha_1, \ldots$ be complex numbers. Define

and let $Ah = \sum_{n=0}^{\infty } \alpha_n \left\langle h, e_n \right\rangle e_n$ for $h$ in $\mathcal{D}$. Verify that $A \in \mathcal{C}(\mathcal{H})$ with $\operatorname {dom} A = \mathcal{D}$, that $\operatorname {dom} A^* = \mathcal{D}$, and that $A^*h = \sum_{n=0}^{\infty } \overline{\alpha_n} \left\langle h, e_n \right\rangle e_n$ for all $h$ in $\mathcal{D}$.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $\phi : X \to \mathbb {C}$ be an $\Omega$-measurable function. Let $\mathcal{D} = \left\{ f \in L^2(\mu ) : \phi f \in L^2(\mu )\right\}$ and define $Af = \phi f$ for all $f$ in $\mathcal{D}$. Verify that $A \in \mathcal{C}(L^2(\mu ))$, that $\operatorname {dom} A^* = \mathcal{D}$, and that $A^*f = \overline{\phi } f$ for $f$ in $\mathcal{D}$.

Let $e_0, e_1, \ldots$ be an orthonormal basis for $\mathcal{H}$ and let $\alpha_0, \alpha_1, \ldots$ be complex numbers. Define

and let $Ah = \sum_{n=0}^{\infty } \alpha_n \left\langle h, e_n \right\rangle e_n$ for $h$ in $\mathcal{D}$. Find the Cayley transform of $A$ when each $\alpha_n$ is real.

Let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space and let $\phi : X \to \mathbb {C}$ be an $\Omega$-measurable function. Let $\mathcal{D} = \left\{ f \in L^2(\mu ) : \phi f \in L^2(\mu )\right\}$ and define $Af = \phi f$ for all $f$ in $\mathcal{D}$. Find the Cayley transform of $A$ when $\phi$ is real valued.

Fix $\lambda \in \ell^{\infty }$, and let $M_{\lambda }$ be the multiplication operator defined in Exercise 1.66. Find $M_{\lambda }^{*}$, and determine when $M_{\lambda }$ is self-adjoint, positive, or positive definite.

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

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

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