  The `operator entanglement' of a quantum operator $O$ is a useful indicator
of its complexity, and, in one-dimension, of its approximability by matrix
product operators. Here we focus on spin chains with a global $U(1)$
conservation law, and on operators $O$ with a well-defined $U(1)$ charge, for
which it is possible to resolve the operator entanglement of $O$ according to
the $U(1)$ symmetry. We employ the notion of symmetry resolved operator
entanglement (SROE) introduced in [PRX Quantum 4, 010318 (2023)] and extend the
results of the latter paper in several directions. Using a combination of
conformal field theory and of exact analytical and numerical calculations in
critical free fermionic chains, we study the SROE of the thermal density matrix
$\rho_\beta = e^{- \beta H}$ and of charged local operators evolving in
Heisenberg picture $O = e^{i t H} O e^{-i t H}$. Our main results are: i) the
SROE of $\rho_\beta$ obeys the operator area law; ii) for free fermions, local
operators in Heisenberg picture can have a SROE that grows logarithmically in
time or saturates to a constant value; iii) there is equipartition of the
entanglement among all the charge sectors except for a pair of fermionic
creation and annihilation operators.
