  This paper studies algebraic and analytic structures associated with the
Lerch zeta function. It defines a family of two-variable Hecke operators $\{
T_m: \, m \ge 1\}$ given by $T_m(f)(a, c) = \frac{1}{m} \sum_{k=0}^{m-1}
f(\frac{a+k}{m}, mc)$ acting on certain spaces of real-analytic functions,
including Lerch zeta functions for various parameter values. It determines the
action of various related operators on these function spaces. It characterizes
Lerch zeta functions (for fixed $s$ in the following way. It shows that there
is for each $s \in {\bf C}$ a two-dimensional vector space spanned by linear
combinations of Lerch zeta functions is characterized as a maximal space of
simultaneous eigenfunctions for this family of Hecke operators. This result is
an analogue of a result of Milnor for the Hurwitz zeta function. We also relate
these functions to a linear partial differential operator in the $(a,
c)$-variables having the Lerch zeta function as an eigenfunction.
