Dunford integral

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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.


[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:
This article was adapted from an original article by J. de Graaf (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article