Abstract

Abstract. This is a survey paper on additive combinations of certain special-type operators on a Hilbert space. We consider (finite) linear combinations, sums, convex combinations and/or averages of operators from the classes of diagonal operators, unitary operators, isometries, projections, symmetries, idempotents, square-zero operators, nilpotent operators, quasinilpotent operators, involutions, commutators, self-commutators, norm-attaining operators, numerical-radius-attaining operators, irreducible operators and cyclic operators. In each case, we are mainly concerned with the characterization of such combinations and the minimal number of the special operators required in them. We will omit the proofs of most of the results here but give some indication or brief sketch of the ideas behind and point out the remaining open problems. 1. Introduction. In the study of operators on Hilbert spaces, the Cartesian decomposition is a fundamental and useful tool. In this case, we express an operator T as an additive combination of two Hermitian operators: T = T 1 + iT 2 , where T 1 and T 2 are the real and imaginary parts of T . In operator theory, there are many other results of a similar nature: certain operators can be expressed as an additive combination of special-type operators. The aim of such decompositions is quite obvious. We try to gain insight into the structure of operators in the former class through our understanding of operators in the latter. Results of this nature are interesting by themselves and they often open a new vista of the operator terrain. The purpose of this paper is to survey such results, indicate their applications and point out the remaining open problems.