Similarity orbits in the Calkin algebra

Let B(H) be the algebra of all bounded linear operators on a complex separable infinite dimensional Hilbert space H. Denote by pi the quotient map of B(H) onto the Calkin algebra A(H). In 1984, Apostol et al. raised the following conjecture: If an operator T on H is not similar to a compact perturbation of a Jordan operator, then the similarity orbit of pi(T) in A(H) coincides with the pi-image of the similarity orbit of T. In this paper, we investigate the structure of similarity orbits in the Calkin algebra and give a negative answer to the above conjecture.