Perturbations of determinants of nuclear operators in a Hilbert space

Let (Figure presented.) be a complex separable Hilbert space with the unit operator I and {d_k } be an orthonormal basis in (Figure presented.). Let A, Ã be linear operators in (Figure presented.), satisfying the conditions (Figure presented.). It is proved that the determinants satisfy the inequalities (Figure presented.) (Figure presented.). These inequalities refine the well-known ones and enable us to establish upper and lower bounds for the determinants of infinite matrices which are “close” to triangular matrices.