Eulerian polynomials are fundamental in combinatorics and algebra. In this paper we study the linear transformation
A
:
R
[
t
]
→
R
[
t
]
\mathcal {A}: \mathbb {R}[t] \to \mathbb {R}[t]
defined by
A
(
t
n
)
=
A
n
(
t
)
\mathcal {A}(t^n) = A_n(t)
, where
A
n
(
t
)
A_n(t)
denotes the
n
n
-th Eulerian polynomial. We give combinatorial, topological and Ehrhart theoretic interpretations of the operator
A
\mathcal {A}
, and investigate questions of unimodality and real-rootedness. In particular, we disprove a conjecture by Brenti (1989) concerning the preservation of real zeros, and generalize and strengthen recent results of Haglund and Zhang (2019) on binomial Eulerian polynomials.