examples of bounded and unbounded operators

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

Bounded

• Identity operator, Zero operator

• Shift operators on ${\mathrm{\ell }}^p$

• A linear operator is continuous if and only ifit 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 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

• The derivative is an unbounded operator on thevector space of smooth functions equipped with the$\sup$-norm.