Charts for the Banach manifold of smooth almost complex structures $\mathcal{J}^{l}$ Consider the closure in the $C^l$-topology of the space of smooth almost complex structrues of a symplectic manifold $(M,\omega)$. We will denote this space by $\mathcal{J}^l$.
It's a very used fact in symplectic topology that this space has the structure of a banach manifold. Therefore I would like to understand why this is true. In the book $\textbf{J-Holomorphic Curves and Symplectic Topology}$ in page $49$ the authors say a few words about this. Namelty that the idea is to consider the banach space that are the $C^l$-sections of the bundle $\text{End}(TM,\omega,J)$ and consider the map $Y\mapsto J\exp(-JY)$.
There are a lot of things  to verify here, but first I needed to understand this chart. What do the authors mean by $\exp(-JY)$?  My idea would be to take a point $p$ and a vector $v\in T_pM$ and then look at $\exp_p(-J_pY_p(v))$, but then this would give me a point and hence it doesn't follow well with the $J$ that it's appearing. I guess another idea would be to consider the exponential of operators?
Understanding this then I guess would give me an idea of the the volume of the ball where this chart would be defined and then one could actually check that the transition maps are smooth.
Therefore I would like to know if anyone has any input on what's hapenning here, or knows of any reference where things are done with a little more detail?
Any insight is appreciated, thanks in advance.
 A: I think it's instructive to understend first what is the tangent space of almost complex structures in $\mathbb{R}^{2n}$. If $\{J_t\}_{t\in (-\varepsilon,\varepsilon)}$ is a path of almost complex structures in $\mathbb{R}^{2n}$ with $\frac{dJ_t}{dt}\restriction_{t=0} = X \in \text{End }(\mathbb{R}^{2n})$, you can differentiate the relation $J_t^2 = - Id$ to conclude that $X J = - J X$, ie. $X$ is a complex antilinear matrix. If the complex structure is also compatible with the symplectic structure, you also have to differentiate this relation which will give you one further equation.
From the last paragraph, you can define charts around $J_0$ as $J_0 e^Y$, where $Y$ varies on a neighborhood of zero in the vector space of complex antilinear matrixes. Multiplication by $J$ lives you on the same vector space, so you may as well parametrize the structures as $J_0 e^{-J_0 Y}$ (which, if I remember correctly, simplifies further computations).
Notice that this is not the exponential coming from the metric, it is simply the matrix exponential. The same history applies verbatim to the general case, except that now this computation is carried out on each tangent space. For example $\exp (-J Y)$ is in $\text{GL}_{2n} (T_p M)$ for each $p \in M$.
