Section 12.4

Main types of operators illustrated by examples

[2/100%]

Show

LaTeX

Show LaTeX

Problem 12.110

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)

For the operator $A\left(x_{1}, x_{2}, \ldots \right) = \left(\lambda_{1} x_{1}, \lambda_{2} x_{2}, \ldots \right)$ in the space $l_{2}$ , where $\lambda : = \left\{ \lambda_{n}\right\}_{1}^{\infty } \in l_{\infty }$ , find criteria for the following statements in terms of the sequence $\lambda_{n}$ :

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(a)

$A \in \mathcal{K}\left(l_{2}\right)$ ;

(b)

$A \in \mathcal{S}_{p}\left(l_{2}\right), p \in [1, \infty )$ ;

(c)

$A \in \mathcal{I B}\left(l_{2}\right)$ ;

(d)

$A \in \mathcal{F}\left(l_{2}\right)$ ;

(e)

$A$ is a self-adjoint operator;

(f)

$A$ is a unitary operator;

(g)

$A$ is a projector;

(h)

$A$ is an orthogonal projector;

(i)

$A$ is a partial isometry;

(j)

$A$ is a positive operator;

(k)

$A$ is a normal operator;

(l)

$A$ is an orthogonal reflection.

Problem 12.111

(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)

For the operator $A f(x) = a(x) f(x)$ in the space $L_{2}[0,1]$ , where $a \in L_{\infty }[0,1]$ , find criteria for the following statements in terms of the function $a$ :

Origin: AI-translated from another node content of the same node

Generated from: Stated according to the original source

(a)

$A \in \mathcal{K}\left(L_{2}[0,1]\right)$ ;

(b)

$A \in \mathcal{S}_{1}\left(L_{2}[0,1]\right)$ ;

(c)

$A \in \mathcal{I B}\left(L_{2}[0,1]\right)$ ;

(d)

$A \in \mathcal{F}\left(L_{2}[0,1]\right)$ ;

(e)