The Scattered Range Problem of Elementary Operators on \({\cal B}({\cal H})\)

A scattered operator is a bounded linear operator with at most countable spectrum. In this paper, we prove that for any elementary operator on \({\cal B}({\cal H})\), not only for finite length but also for infinite length, if the range of the elementary operator is contained in scattered operators, then the corresponding sum of multipliers is a compact operator. We also prove that for some special classes of elementary operators, such as the elementary operators of length 2, higher order inner derivations and generalized inner derivation, if the range of the elementary operator is contained in the set of scattered operators, then the range is contained in the set of power compact operators. At the same time, the multipliers of the corresponding elementary operators are characterized.