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)|
Dunford integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dunford_integral&oldid=50341