Fractional powers of monotone operators in Hilbert spaces

Daniel Hauer, Yuan He, Dehui Liu


In this article, we show that for maximal monotone operators \(A\) on \(H\) with \(0\) in the range \(\textrm{Rg}(A)\) of \(A\) and every \(0 < \sigma\le 1/2\), the Dirichlet problem associated with the Bessel-type equation \[ A_{1-2\sigma u}:=-\frac{1-2\sigma} tu_t-u_{tt}+Au\ni 0 \] is well-posed and the associated Dirichlet-to-Neumann map \(\Lambda_{\sigma}\) is a maximal monotone operator on \(H\). This allows to defined fractional powers \(A^{\alpha}\) of nonlinear operators.

Keywords: Monotone operators, Hilbert space, evolution equations, fractional operators.

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