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
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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:
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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) |
Dunford integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Dunford_integral&oldid=12378