Orthogonality in script G sign <inf>1</inf> and script G sign <inf>∞</inf> spaces and normal derivations

Abstract

We introduce φ-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 script G sign 1 and script G sign ∞ (nuclear and compact operators on a Hilbert space). Further, we apply these results to prove that there exists a normal derivation Δ A such that ran Δ A ⊕ ker Δ A ≠ script G sign A1 , and a related result concerning script G sign ∞ .