On the range-Kernel orthogonality of elementary operators. (English) Zbl 1349.47052

Bouali, Said; Bouhafsi, Youssef

Math. Bohem. 140, No. 3, 261-269 (2015).

Let \(\mathcal{L}(\mathcal{H})\) denote the algebra of all bounded linear operators on a complex infinite dimensional Hilbert space \(\mathcal{H}\). For \(A, B \in \mathcal{L}(\mathcal{H})\), the generalized derivation \(\delta_{A,B}\) and the elementary operator \(\Delta_{A,B}\) are defined by \(\delta_{A,B}(X) = AX-XB\) and \(\Delta_{A,B}(X) = AXB-X\) for \(X \in \mathcal{L}(\mathcal{H})\), respectively. In the paper under review, the authors deal with the pairs \((A, B)\) of operators such that the range and the kernel of \(\delta_{A,B}\) are orthogonal for the operator norm, that is, \(\|x+y\|\geq \|y\|\) for \(x\) in the range and \(y\) in the kernel. They also investigate the orthogonality of the range and the kernel of \(\Delta_{A,B}\) in some norm ideals of \(\mathcal{L}(\mathcal{H})\). The authors are inspired by papers of J. Anderson [Proc. Am. Math. Soc. 38, 135–140 (1973; Zbl 0255.47036)] and B. P. Duggal [Proc. Am. Math. Soc. 126, No. 7, 2047–2052 (1998; Zbl 0894.47003)].

Reviewer: Mohammad Sal Moslehian (Mashhad)

MSC:

47B47	Commutators, derivations, elementary operators, etc.
47B10	Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)

Keywords:

derivation; elementary operator; orthogonality; unitarily invariant norm; cyclic subnormal operator; Fuglede-Putnam property

Citations:

Zbl 0255.47036; Zbl 0894.47003