  Let $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function
satisfying the globally $\log$-H\"older continuous condition and $L$ a
non-negative self-adjoint operator on $L^2(\mathbb R^n)$ whose heat kernels
satisfying the Gaussian upper bound estimates. Let $H_L^{p(\cdot)}(\mathbb
R^n)$ be the variable exponent Hardy space defined via the Lusin area function
associated with the heat kernels $\{e^{-t^2L}\}_{t\in (0,\infty)}$. In this
article, the authors first establish the atomic characterization of
$H_L^{p(\cdot)}(\mathbb R^n)$; using this, the authors then obtain its
non-tangential maximal function characterization which, when $p(\cdot)$ is a
constant in $(0,1]$, coincides with a recent result by Song and Yan [Adv. Math.
287 (2016), 463-484] and further induces the radial maximal function
characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ under an additional
assumption that the heat kernels of $L$ have the H\"older regularity.
