Cost of solving a matrix equation using the FFT

Question

I am trying to calculate

$$V = (H^TH+I)^{-1} U$$

where $H\in\mathbb{R}^{m\times m}$ is a circulant convolution matrix corresponding to a convolution kernel $h$, and $U\in\mathbb{R}^{m\times n}$. The computation of $V$ can be done using the FFT algorithm.

I am confused about the computational complexity of using the FFT for this kind of problem. I will be most grateful if you could provide some suggestions or ideas for solving this problem. Thank you.