## Fractional powers of monotone operators in Hilbert spaces

### Daniel Hauer, Yuan He, Dehui Liu

#### Abstract

In this article, we show that if $$A$$ is a maximal monotone operator on $$H$$ with $$0$$ in the range $$\textrm{Rg}(A)$$ of $$A$$, then for every $$0 < s < 1$$, the Dirichlet problem associated with the Bessel-type equation $A_{1-2s}u:= -\frac{1-2s}{t}u_{t}-u_{tt}+Au\ni 0$ is well-posed for boundary values $$\varphi\in \overline{D(A)}^{\mbox{}_{H}}$$. This allows us to define the Dirichlet-to-Neumann (DtN) map $$\Lambda_{s}$$ associated with $$A_{1-2s}$$ as $\varphi\mapsto \Lambda_{s}\varphi:=-\lim_{t\to 0+}t^{1-2s}u_{t}(t)\qquad\text{in H.}$ The existence of the DtN map $$\Lambda_{s}$$ associated with $$A_{1-2s}$$ is the first step in defining fractional powers $$A^{\alpha}$$ of monotone (possibly, nonlinear and multivalued) operators $$A$$ on $$H$$. We prove that $$\Lambda_{s}$$ is monotone on $$H$$, and if $$\overline{\Lambda}_{s}$$ is the closure of $$\Lambda_{s}$$ in $$H\times H_{w}$$ then we provide conditions implying that $$-\overline{\Lambda}_{s}$$ generates a strongly continuous semigroup on $$\overline{D(A)}^{\mbox{}_{H}}$$. In addition, we show that if $$A$$ is completely accretive, on $$L^{2}(\Sigma,\mu)$$ for a $$\sigma$$-finite measure space $$(\Sigma,\mu)$$, then $$\Lambda_{s}$$ inherits this property from $$A$$.

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

This paper is available as a pdf (488kB) file. It is also on the arXiv: arxiv.org/abs/1805.00134.

 Monday, February 19, 2018 10