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Computing subdifferential limits of operators on Banach spaces

Rao T. S. S. R. K.

Journal of Applied Analysis

2023

Vol. 29, nr 2

297--302

Let X, Y be real, infinite-dimensional Banach spaces. Let L(X, Y) be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ L(X, Y) is to estimate the limit (which always exists) limt→0+ ‖A + tB‖ − ‖A‖ / t for B ∈ L(X, Y), using the values of B on the state space SA = {τ ∈ L(X, Y)∗ : τ(A) = ‖A‖, ‖τ‖ = 1}. In this paper, we give several examples of Banach spaces, including the ℓp spaces (for 1 < p < ∞) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A∗∗ attains its norm.