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 $T$ on a Banach space $X$, all functions $f$ holomorphic on a neighbourhood $U$ of the spectrum $\sigma ( T )$ of $T$ (cf. also Spectrum of an operator) are turned into a bounded linear operator $f ( T )$ on $X$ by

$$\begin{equation*} f ( T ) = \frac { 1 } { 2 \pi i } \int _ { \partial U } f ( \lambda ) ( \lambda - T ) ^ { - 1 } d \lambda. \end{equation*}$$

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

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

$$\begin{equation*} \alpha f ( T ) + \beta g ( T ) = ( \alpha f + \beta g ) ( T ), \end{equation*}$$

$$\begin{equation*} f ( T ) g ( T ) = ( f g ) ( T ) , f ( \sigma ( T ) ) = \sigma ( f ( T ) ). \end{equation*}$$

Also, $f ( \lambda ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } \lambda ^ { n }$ on $U$ implies $f ( T ) = \sum _ { n = 0 } ^ { \infty } \alpha _ { n } T ^ { n }$ in the operator norm. If $h ( \lambda ) = g ( f ( \lambda ) )$, then $h ( T ) = g ( f ( T ) )$.

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)