Steklov zeta-invariants and a compactness theorem for isospectral families of planar domains

The inverse problem of recovering a smooth simply connected multisheet planar domain from its Steklov spectrum is equivalent to the problem of determination, up to a gauge transform, of a smooth positive function a on the unit circle from the spectrum of the operator aΛ, where Λ is the Dirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by Zm(a)=Tr[(aΛ)2m−(aD)2m] for every smooth function a. In the case of a positive a, zeta-invariants are determined by the Steklov spectrum. We obtain some estimate from below for Zm(a) in the case of a real function a. On using the estimate, we prove the compactness of a Steklov isospectral family of planar domains in the C∞-topology. We also describe all real functions a satisfying Zm(a)=0.