Help with strange notation involving fractions of tensors

I'm currently reading the paper Open strings in background gauge fields by Callan et.al. It is frequently used a notation that is not explained anywhere. If $$F_{\mu\nu}$$ is the electromagnetic field strength tensor, the author write two expressions:

$$\tag1 \left(\frac{1 - F}{1+F} \right)_{\mu \nu},$$

which firstly appears in eq. ($$2.10$$), and

$$\left(\frac{1}{1-F^2}\right)_{\lambda \nu} \tag2,$$ which appears for the first time in eq. ($$2.14$$).

Does anyone know the precise meaning of these expression and how to perform correct manipulations with it (such as derivatives)?

$$f(F)^{\mu}{}_{\nu}$$ refers to a component of the $$(1,1)$$ tensor $$f(F)=\sum_{n=0}^{\infty}a_nF^n$$, where components $$F^{\mu}{}_{\nu}$$ of the $$(1,1)$$ field-strength tensor is viewed as a matrix $$F$$ that can be multiplied and added together.
Here $$f(x)=\sum_{n=0}^{\infty}a_nx^n$$ is an analytic function, e.g. $$f(x)=\frac{1-x}{1+x}$$ or $$f(x)=\frac{1}{1-x^2}$$.
Finally, $$f(F)_{\mu\nu}:=g_{\mu\lambda}f(F)^{\lambda}{}_{\nu}$$.