Fractional Powers of Monotone Operators in Hilbert Spaces

The aim of this article is to provide a functional analytical framework for defining the fractional powers A(s) for -1 < s < 1 of maximal monotone (possibly multivalued and nonlinear) operators A in Hilbert spaces. We investigate the semigroup {e -(Ast)}t >= 0 generated by -A(s), prove comparison principles and interpolations properties of {e -(Ast)}t >= 0 in Lebesgue and Orlicz spaces. We give sufficient conditions implying that A(s) has a sub-differential structure. These results extend earlier ones obtained in the case s = 1/2 for maximal monotone operators [H. Brezis, equations d'evolution du second ordre associees a des operateurs monotones, Israel J. Math. 12 (1972), 51-60], [V. Barbu, A class of boundary problems for second order abstract differential equations, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 295-319], [V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International, Leiden, 1976], [E. I. Poffald and S. Reich, An incomplete Cauchy problem, J. Math. Anal. Appl. 113 (1986), no. 2, 514-543], and the recent advances for linear operators A obtained in [L. Caffarelli and L Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260], [P. R. Stinga and J. L Torrea, Extension problem and Harnack's inequality for some fractional operators, Comm. Partial Differential Equations 35 (2010), no. 11, 2092-2122].