Computing subdifferential limits of operators on Banach spaces

Let X , Y {X,Y} be real, infinite-dimensional Banach spaces. Let ℒ ⁢ ( X , Y ) {{\mathcal{L}}(X,Y)} be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ ( X , Y ) {A\in{\mathcal{L}}(X,Y)} is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ ( X , Y ) , \lim_{t\rightarrow 0^{+}}\frac{\lVert A+tB\rVert-\lVert A\rVert}{t}\quad\text{% for }B\in{\mathcal{L}}(X,Y), using the values of B on the state space S A = { τ ∈ ℒ ⁢ ( X , Y ) ∗ : τ ⁢ ( A ) = ∥ A ∥ , ∥ τ ∥ = 1 } . S_{A}=\bigl{\{}\tau\in{\mathcal{L}}(X,Y)^{\ast}:\tau(A)=\lVert A\rVert,\,% \lVert\tau\rVert=1\bigr{\}}. In this paper, we give several examples of Banach spaces, including the ℓ p {\ell^{p}} spaces (for 1 < p < ∞ {1