# Dunford integral

An integral playing a key role in the Riesz–Dunford functional calculus for Banach spaces (cf. Functional calculus.) In this calculus, for a fixed bounded linear operator on a Banach space , all functions holomorphic on a neighbourhood of the spectrum of (cf. also Spectrum of an operator) are turned into a bounded linear operator on by

This integral is called the Dunford integral. It is assumed here that the boundary of consists of a finite number of rectifiable Jordan curves (cf. also Jordan curve), oriented in positive sense.

For suitably chosen domains of and , the following rules of operational calculus hold:

Also, on implies in the operator norm. If , then .

The Dunford integral can be considered as a Bochner integral.

#### References

[a1] | N. Dunford, J.T. Schwartz, "Linear operators" , 1 , Interscience (1958) |

[a2] | K. Yosida, "Functional analysis" , Springer (1980) |

**How to Cite This Entry:**

Dunford integral.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Dunford_integral&oldid=50341