Periodic Jacobi operators with complex coefficients

We present certain results on the direct and inverse spectral theory of Jacobi operator with complex periodic coefficients. For instance, we show that any $N$-th degree polynomial whose leading coefficient is $(-1)^N$ Hill discriminant finitely many discrete $N$-periodic Schrödinger operators (Theorem 1). Also, in case where spectrum a closed interval prove result 2) which analog Borg's Theorem for non-self-adjoint case.