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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.

The embeddability of C0 in spaces of operators

Ghenciu I., Lewis P.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

Vol. 56, no 3-4

239-256

Results of Emmanuele and Drewnowski are used to study the containment of c0 in the space Kw* (X*, Y), as well as the complementation of the space Kw* (X*,Y) of w*-w compact operators in the space Lw*(X*, Y) of w*-w operators from X* to Y.

Complemented spaces of operators

Bator E. M., Lewis P. W.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

413-416

The complementation of various classes of operators in the space L(E, F) of bounded linear operators between Banach spaces E and F is studied.