Fractional powers of monotone operators in Hilbert spaces

Abstract

In this article, we show that if $A$ is a maximal monotone operator on a Hilbert space $H$ 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) operator $\Lambda_{s}$ 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 operator $\Lambda_{s}$ associated with $A_{1-2s}$ is the first step to define 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 sufficient 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 $A$.