Subdifferential set of an operator

Abstract

AbstractLet X, Y be complex Banach spaces. Let $${\mathcal L}(X,Y)$$ L ( X , Y ) be the space of bounded operators. An important aspect of understanding differentiability is to study the subdifferential of the norm at a point, say $$x \in X$$ x ∈ X , this is the set, $$\{f \in X^*:\Vert f\Vert =1~,f(x)=\Vert x\Vert \}$$ { f ∈ X ∗ : ‖ f ‖ = 1 , f ( x ) = ‖ x ‖ } . See page 7 in Deville et al. (Pitman Monographs and Surveys in Pure and Applied Mathematics. 64. Harlow: Longman Scientific and Technical. New York: John Wiley and Sons, Inc. 1993). Motivated by recent results of Singla (Singla in Linear Alg. Appl. 629:208–218, 2021) in the context of Hilbert spaces, for $$T \in {{\mathcal {L}}}(X,Y)$$ T ∈ L ( X , Y ) , we determine the subdifferential of the operator norm at T, $$\partial _T = \{\Lambda \in {{\mathcal {L}}}(X,Y)^*: \Lambda (T) = \Vert T\Vert ~,~\Vert \Lambda \Vert =1\}$$ ∂ T = { Λ ∈ L ( X , Y ) ∗ : Λ ( T ) = ‖ T ‖ , ‖ Λ ‖ = 1 } . Our approach is based on the ‘position’ of the space of compact operators and the Calkin norm of T. Our ideas give a unified approach and extend several results from Singla (Linear Alg. Appl. 629:208–218, 2021) to the case of $$\ell ^p$$ ℓ p -spaces for $$1<p<\infty $$ 1 < p < ∞ . We also investigate the converse, using the structure of the subdifferential set to decide when the Calkin norm is a strict contraction. As an application of these ideas, we partially solve the open problem of relating the subdifferential of the operator norm at a compact operator T to that of $$T(x_0)$$ T ( x 0 ) , where $$x_0$$ x 0 is a unit vector where T attains its norm.