# 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

• 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 $h\in L^{p}[0,1]$.

## Unbounded

Title examples of bounded and unbounded operators ExamplesOfBoundedAndUnboundedOperators 2013-03-22 15:17:37 2013-03-22 15:17:37 matte (1858) matte (1858) 12 matte (1858) Example msc 47L25