Fractional powers of a nonlinear analytic differential operator

The nonlinear operator $F(u)=MB(Lu)$ is considered, in which $L$ is an invertible closed linear operator with an everywhere dense domain of definition in a Banach space $E$, $B$ is an analytic operator satisfying strong continuity requirements with respect to the action of $L$ as well as the conditions $B(0)=0$ and $B'(0)=I$, and $M>1$ is an auxiliary number greater than one. Local and global theorems are obtained on the representation of $F$ in the form $F=\mathscr E\circ ML\circ\mathscr E^{-1}$, where $\mathscr E$ and $\mathscr E^{-1}$ are analytic operators, and the real and complex powers $F^\alpha=\mathscr E\circ(ML)^\alpha\circ\mathscr E^{-1}$ are defined. The existence of complex powers is used to obtain an expression for $g(F^{-1}(h))$ in terms of the $g(F^j(h))$ ($j=0,1,\dots$), where $g$ is a functional. It is proved that the results are applicable to nonlinear elliptic differential operators on spaces of periodic functions.
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