# Journal of Operator Theory

Volume 51, Issue 1, Winter 2004 pp. 89-104.

Orthogonality in $\frak{S}_2$ and $\frak{S}_\infty$ spaces and normal derivations**Authors**: Dragoljub J. Keckic

**Author institution:**Faculty of Mathematics, University of Belgrade, Studentski trg 16--18, 11000 Beograd, Yugoslavia

**Summary:**We introduce $\varphi$-Gateaux derivative, and use it to give the necessary and sufficient conditions for the operator $Y$ to be orthogonal (in the sense of James) to the operator $X$, in both spaces ${\frak S}_1$ and ${\frak S}_\infty$ (nuclear and compact operators on a Hilbert space). Further, we apply these results to prove that there exists a normal derivation $\Delta_A$ such that $\overline{\ran\Delta_A}\oplus\ker\Delta_A\neq{\frak S}_1$, and a related result concerning ${\frak S}_\infty$.

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