# When an Hilbert–Schmidt operator is invertible? How to construct the kernel of the inverse?

This question is motivated by the following example. I have the following integral operator defined on $L^2(\mathbb{T}^n)\to L^2(\mathbb{T}^n)$ $$T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \hat{f}(k) e^{ik\cdot x}$$ where $\hat f(k)$ is the $k$-Fourier coefficent of $f$, and $\sigma(k,\cdot)$ is invertible for any $k\neq O$. For example, if $\sigma$ do not depend on $x$, we get a multiplication operator $$T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k) \ \hat{f}(k) e^{ik\cdot x}$$ thanks to the assumption that $\sigma(k)$ is invertible we can construct an inverse of $T$ (I am not interested in continuity for now). This suggested me to formally invert the general $T$ as

$$A g(x)= \sum_{k\in \mathbb{Z}} \frac{1}{\sigma(k,x)} \hat g (k)e^{i k\cdot x}$$ I do not see, though, a simple manner to prove that this constitutes an inverse (at least where defined).

1)Consider an integral operator of the form $$Tf(x) = \int_X f(y)K(y,x)d\mu(y)$$ from $L^2(X,\mu)$ to itself. I wonder how, and under which hypothesis, we can construct a formal inverse using the kernel $K$.

2) Consider the operator defined above on $L^2(\mathbb{T}^n)$, the one in the form $$T f(x) = \sum_{k \in \mathbb{Z}^n} \sigma(k,x) \ \hat{f}(k) e^{ik\cdot x},$$ how can we construct a formal inverse?

3) If we have an integral transform with kernels $K$ which is invertible with $K^{-1}$ kernel of the inverse, can we construct a formal inverse of $$Tf(x) = \int_X\int_X f(y) K(y,\xi) K^{-1}(x,\xi)\sigma(x,\xi) d\mu(y) d\mu(\xi)$$ ?

This question is motivated by an assertion that I have found that suggests that if a linear differential operator $D:C^{\infty}(\mathbb{T}^n)\to C^{\infty}(\mathbb{T}^n)$ is elliptic, then, we can construct an inverse of $D$ restricted to non-constant functions.

• The operator $A$ that you define in your third equation is not a "formal inverse" of the operator $T$ that you define in the first line, but, if I recall correctly, they are inverses of each other "up to an smoothing operator" meaning that the composition will be the $id$ with a smooth correction term. You should look a text in pseudodifferential operator, like Taylor's Pseudodifferential operators and nonlinear PDE's or Stein's Harmonic analysis, Chapter VI. – Adrián González-Pérez May 23 '18 at 12:07