  We study the trace class perturbations of the whole-line, discrete Laplacian
and obtain a new bound for the perturbation determinant of the corresponding
non-self-adjoint Jacobi operator. Based on this bound, we refine the
Lieb--Thirring inequality due to Hansmann--Katriel. The spectral enclosure for
such operators is also discussed.

Spectral Theory;Classical Analysis and ODEs;Functional Analysis