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The inverse problem of recovering a smooth simply connected multisheet planardomain 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 circlefrom the spectrum of the operator $a\Lambda$, where $\Lambda$ is theDirichlet-to-Neumann operator of the unit disk. Zeta-invariants are defined by$Z_m(a)={\rm Tr}[(a\Lambda)^{2m}-(aD)^{2m}]$ for every smooth function $a$. Inthe case of a positive $a$, zeta-invariants are determined by the Steklovspectrum. We obtain some estimate from below for $Z_m(a)$ in the case of a realfunction $a$. On using the estimate, we prove the compactness of a Steklovisospectral family of planar domains in the $C^\infty$-topology. We alsodescribe all real functions $a$ satisfying $Z_m(a)=0$.

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