Spectral Problems Of Jacobi Operators In Limit-Circle Case

Abstract

This paper investigates the minimal symmetric operator bounded from below and generated by the real infinite Jacobi matrix in the Weyl-Hamburger limitcircle case. It is shown that the inverse operator and resolvents of the selfadjoint, maximal dissipative and maximal accumulative extensions of this operator are nuclear (or trace class) operators. Besides, we prove that the resolvents of the maximal dissipative operators generated by the infinite Jacobi matrix, which has complex entries, are also nuclear (trace class) operators and that the root vectors of these operators form a complete system in the Hilbert space.