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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