  The classical inverse problem of recovering a simply connected smooth planar
domain from the Steklov spectrum \cite{E} is equivalent to the problem of
recovering, up to a conformal equivalence, a positive function $a\in
C^\infty({\mathbb S})$ on the unit circle ${\mathbb S}=\{e^{i\theta}\}$ from
the eigenvalue spectrum of the operator $a\Lambda_e$, where
$\Lambda_e=(-d^2/d\theta^2)^{1/2}$. We introduce $2k$-forms $Z_k(a)\
(k=1,2,\dots)$ in Fourier coefficients of the function $a$ which are called
zeta-invariants. They are uniquely determined by the eigenvalue spectrum of
$a\Lambda_e$. We study some properties of $Z_k(a)$, in particular, their
invariance under the conformal group. Some open questions on zeta-invariants
are posed at the end of the paper.
