Let L(H) denote the algebra of operators on a complex infinite dimensional Hilbert space H. For A, B ∈ L(H), the generalized derivation δ A,B and the elementary operator δ A,B are defined by δ A,B(X)=AX-XB and δ A,B}(X)=AXB-X for all X\in L(H). In this paper, we exhibit pairs $(A,B)$ of operators such that the range-kernel orthogonality of δ A,B}$ holds for the usual operator norm. We generalize some recent results. We also establish some theorems on the orthogonality of the range and the kernel of Δ A,B with respect to the wider class of unitarily invariant norms on L(H).
Plasmodia of a Henneguya species measuring 70-900 pm and exhibiting season-dependent stages of development were detected throughout a three-year study on gill myxosporosis of Lake Balaton pikeperch (Stizostedion lucioperca (L.)). Sixty-five out of 160 fish (41%) examined in the period of study were infected by the parasite. Infection was the most prevalent (48%) among pikeperch specimens exceeding 40 cm in length. The highest prevalence of infection (58%) was recorded in 1995-1996 while the lowest (30%) in 1996-1997. The youngest plasmodia appeared in April, and started to develop within the capillaries of the secondary lamellae of the gill filaments. The round or ellipsoidal plasmodia which continued their gradual growth in the subsequent months of the year achieved a size of 800-900 pm by the late autumn months, but remained in intralamellar location throughout the developmental cycle. Mature spores developed in the plasmodia by the end of winter. On the basis of their shape and size, the spores were identified as Henneguya creplini (Gurley, 1894). However, because of the uncertain taxonomy of species assigned to the genus Henneguya the taxonomic position of the parasite requires further study. The host reaction consisting of epithelial proliferation and granulation tissue formation starts around the infected secondary lamella only after the maturation of spores and the disruption of plasmodia.