On a theorem of Roe and Strichartz on the unit complex ball

For \(\nu \in \mathbb {R}\) $\nu \in \mathbb{R}$ , we consider the invariant Laplacians \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ in the unit complex ball \(\mathcal {B}^{n}\) ${\mathcal{B}}^n$ 0.1 \(\begin{aligned} \Delta _{\nu } =4(1-|z|^{2})\left\{ \sum _{i,j=1}^{n}(\delta _{ij}-z_{i}\bar{z_{j}})\dfrac{\partial ^{2}}{\partial z_{i}\partial \bar{z_{j}}} -\nu \sum _{j=1}^{n}\bar{z_{j}}\dfrac{\partial }{\partial \bar{z_{j}}} \right\} . \end{aligned}\) $$\begin{array}{c}\hfill {\mathrm{\Delta }}_{\nu }={4(1-|z|}^2)\left\{\sum _{i,j=1}^n,({\delta }_{\mathit{ij}}-z_i\overline{z_j}),{\displaystyle \frac{{\partial }^2}{\partial z_i\partial \overline{z_j}}},-,\nu ,\sum _{j=1}^n,\overline{z_j},{\displaystyle \frac{\partial }{\partial \overline{z_j}}}\right\}.\end{array}$$ In (Strichartz in Trans Amer Math Soc 338:971–979, 1993) Strichartz has extended the result of Roe (Math Proc Combridge Philos Soc 87:69–73, 1980) to \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ , by showing that if a doubly-infinity sequence \((f_{k})_{k \in \mathbb {Z}}\) ${(f_k)}_{k\in \mathbb{Z}}$ of functions on \(\mathbb {R}^{n}\) ${\mathbb{R}}^n$ satisfies \(f_{k+1}=\Delta f_k\) $f_{k+1}=\mathrm{\Delta }{f}_k$ and uniformly bounded then \(\Delta f_0=-f_0\) $\mathrm{\Delta }{f}_0=-f_0$ . He also proved that this result is not valid in the hyperbolic 3-space. The goal of this paper is to prove that this result holds true in the unit complex ball associated to \(\Delta _{\nu }\) ${\mathrm{\Delta }}_{\nu }$ when a uniform boundedness is modified appropriately.