examples of bounded and unbounded operators

The aim of this page is to list examples of bounded (http://planetmath.org/BoundedOperator) and unbounded linear operators.

Bounded

•

Identity operator, Zero operator
•

Shift operators on $\ell^{p}$
•

A linear operator is continuous if and only if it is bounded (see this page (http://planetmath.org/ContinuousLinearMapping)).
•

Any isometry is bounded.
•

A multiplication operator $h(t)\mapsto f(t)h(t)$ , where $f(t)$ is continuous and $h\in L^{p}[0,1]$ .
•

An integral operator $h(t)\mapsto\int_{0}^{1}K(t,s)h(s)\,ds$ , where $\int_{0}^{1}\int_{0}^{1}|K(s,t)|^{2}\,ds\,dt<\infty$ and $h\in L^{2}[0,1]$ . In fact this is a Hilbert-Schmidt operator.
•

The Volterra operator $h(t)\mapsto\int_{0}^{t}h(s)\,ds$ , where $h\in L^{p}[0,1]$ .

•

The derivative is an unbounded operator on the vector space of smooth functions equipped with the $\operatorname{sup}$ -norm.

Title	examples of bounded and unbounded operators
Canonical name	ExamplesOfBoundedAndUnboundedOperators
Date of creation	2013-03-22 15:17:37
Last modified on	2013-03-22 15:17:37
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	12
Author	matte (1858)
Entry type	Example
Classification	msc 47L25