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 $\text{Rg}(A)$ 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 $φ\in \overline{D(A)}^{\mbox{}_{H}}$. This allows us to define the Dirichlet-to-Neumann (DtN) operator ${\Lambda }_s$ associated with $A_{1-2s}$ as $$\phi \mapsto {\Lambda }_s\phi :=-\underset{t\to 0+}{\lim }{t}^{1-2s}{u}_t(t)\text{in H.}$$ 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 $A$.

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