Gâteaux derivative of the norm in \(\mathcal{K}(X;Y)\) (Q728036)

Let \(X\) be a normed space, and let \(x,y \in X\). Let \(\varphi \in [0, 2\pi)\), or let \(\varphi \in \{0,-\pi\}\) if the space \(X\) is over \(\mathbb{R}\). The \(\varphi\)-Gâteaux derivative of the norm at \(x\) in the \(\varphi,y\)-direction is defined by \[ D_\varphi(x,y) := \lim_{t \to 0^+} \frac{\|x + t e^{i\varphi} y\| - \|x\|}{t}. \] \textit{D. Kečkić} [J. Oper. Theory 51, No. 1, 89--104 (2004; Zbl 1068.46024)] has shown that, if \(H\) is a complex Hilbert space and \(A,B\) are compact operators on \(H\), then \(D_\varphi(A,B) = \max\{ D_\varphi(Ay,By) : y \in M(A)\}\), where \(M(A) = \{y \in S_H : \|Ay\| = \|A\|\}\). In this paper, the above mentioned result is generalized to compact operators from a Banach space \(X\) to a Banach space \(Y\). Let \(K(X,Y)\) denote the space of all such operators. It is shown that, if \(A,B \in K(X,Y)\) with \(A \neq 0\), then \[ D_\varphi(A,B) = \sup\{ D_\varphi(A^*y^*,B^*y^*) : y^* \in M(A^*) \cap \text{Ext}(S_{Y^*}) \}, \] where \(\text{Ext}(S_{Y^*})\) is the set of extreme points of \(S_{Y^*}\). If \(X\) is assumed to be a reflexive Banach space, then, for \(A,B \in K(X,Y)\), \[ D_\varphi(A,B) = \max\{ D_\varphi(Ay,By) : y \in M(A)\}, \] just as in the Hilbert space case. In the final section, these results are used to characterize orthogonality in the sense of Birkhoff in the space \(K(X,Y)\).