$L^{p}$ and operator norm estimates for the complex time heat operator on homogeneous trees

Let $\mathfrak{X}$ be a homogeneous tree of degree greater than or equal to three. In this paper we study the complex time heat operator ${\mathcal{H}}_{\zeta }$ induced by the natural Laplace operator on $\mathfrak{X}$. We prove comparable upper and lower bounds for the $L^{p}$ norms of its convolution kernel $h_{\zeta }$ and derive precise estimates for the $L^{p}\text{--}L^{r}$ operator norms of ${\mathcal{H}}_{\zeta }$ for $\zeta$ belonging to the half plane $\text{Re}\,\zeta \geq 0.$ In particular, when $\zeta$ is purely imaginary, our results yield a description of the mapping properties of the Schrödinger semigroup on $\mathfrak{X}$.