I do not know of such an operator.  But there is the following lovely theorem of [Huber][1]: 

**Theorem:** Two compact hyperbolic surfaces have the same spectrum of the Laplacian if and only if they have the same (a) area and (b) length spectrum. 

For here are Burrin's [lecture notes][2] from a class at ETH - see Section 6.2.


  [1]: https://link.springer.com/article/10.1007/BF01369663
  [2]: https://metaphor.ethz.ch/x/2021/fs/401-4118-21L/ex/spectral.pdf