Orthogonality of the range and the kernel of some elementary operators

Abstract

We prove the orthogonality of the range and the kernel of an important class of elementary operators with respect to the unitarily invariant norms associated with norm ideals of operators. This class consists of those mappings E : B(H) → B(H), E(X) = AXB + CXD, where B(H) is the algebra of all bounded Hubert space operators, and A, B, C, D are normal operators, such that AC = CA, BD = DB and ker A∩ker C = ker B∩ker D = {0}. Also we establish that this class is, in a certain sense, the widest class for which such an orthogonality result is valid. Some other related results are also given. ©2000 American Mathematical Society.