Convergence of $c^* Ac$ in second quantization

Let $A$ denote a bounded operator on a complex separable Hilbert space $mathscr{H}$. Let $mathscr{F} = bigoplus mathscr{F}_N$ be the Fock space generated by $mathscr{H}$ where $mathscr{F}_N$ is the $N$-particle Hilbert space. Let $dGamma (A):mathscr{F}_N to mathscr{F}_N$ be defined as
$$dGamma (A) =Aotimes Iotimes cdots otimes I+cdots+Iotimescdotsotimes Iotimes A$$
so that (restricted to $mathscr{F}_N$) is bounded linear (i.e., we don’t have to worry about domain problems). In physics, we usually use creation and annihilation operators to represent $dGamma (A)$, i.e.,
$$
c^* Ac equivsum_nc^*(A varphi_n)c(varphi_n) =sum_{m,n} A_{mn} c_m^* c_n
$$
where $varphi_n =|nrangle$ is an orthonormal basis of $mathscr{H}$.

Question. What kind of convergence does $c^* Ac$ have relative to $dGamma (A)$, i.e., in what topology does $sum_{nle N} c^*(A varphi_n)c(varphi_n) to dGamma (A)$ as $Nto infty$?

EDIT. Of course, if $mathscr{H}$ is finite-dim, then the summation is finite and the 2 are equal. However, in infinite-dim, the best that I can do is that if $g$ is in the span of pure tensors in $mathscr{F}^N$, then
$$
c^*Ac|grangle
$$
is well-defined and equal to $dGamma(A)|grangle$. This seems very close to strong-operator topology convergence, but not quite. Maybe there’s a way to make this stronger?