Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A2(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|⩽Aexp(−an1/d), where a,A>0 are explicitly given.