Interpolation theorem for unbounded operators

We prove the following unbounded generalization of the strong interpolation theorem [2, Corollary 3.16] under some extra hypotheses: 1. If h and Ic are self-adjoint operators on a Hilbert space H, Ic is bounded, h greater than or equal to k and h(-), k(+) are compact, then there is a compact operator a such that k less than or equal to a less than or equal to h. 2. If h and k are self-adjoint operators on H, h greater than or equal to k and h(-), k(+) are compact, then for all epsilon > 0 there is a compact operator a such that k - epsilon 1 less than or equal to a less than or equal to h.