General regularized trace formulae for loaded equations

We consider regularized traces for differential operators with the coefficients at the powers of a spectral parameter being the values of an unknown function at prescribed points in its domain. Such differential operators are interpreted as polynomial operator pencils whose coefficients are unbounded fininte-dimensional operators. Basing on the theory of M. V. Keldysh, we construct general regularized trace formulae for such operator pencils. The obtained formulae develop a known result by V. A. Sadovnichii and V. A. Lyubishkin for relative finite-dimensional perturbations of self-adjoint operators.