Fractional powers of dissipative operators
A$ is closed and maximal accretive, $A^{\alpha}$ and $A^{*\alpha}$ are comparable for $0\leqq\alpha<1/2$ ; by this we mean that $A^{t}$ and $A^{*\alpha}$ have the same domain $\mathfrak{D}_{a}$ and that the ratios $\Vert A^{*\alpha}u\Vert/\Vert A^{a}u\Vert$ for $u\in \mathfrak{D}_{a}$ are bounded from above and from below by positive constants.Another result is that $A^{\alpha}$ and $A^{*\alpha}$ have an acute angle for $0\leqq\alpha<1/2$ ; by this is meant that ${\rm Re}(A^{a}u, A^{*\alpha}u)/\Vert A^{a}u\Vert\Vert A^{*\alpha}u\Vert$ is bounded from below by a positive constant (see Sobolevskii [17]).These results are remarkable in view of the fact that nothing is assumed for the relationship between the domains of $A$ and $A^{*}$ themselves or for the angle between $A$ and $A^{*}$ .It follows from these results that $H_{a}=(A^{a\prime}+A^{*\alpha})/2$ is nonnegative self- adjoint and that it is comparable, and has acute angle, with both $A^{a}$ and