An Approximation Problem in the Space of Bounded Operators. (25th January 2023) - Mal, Arpita
Abstract: For Banach spaces X, Y, we consider a distance problem in the space of bounded linear operators L ( X, Y ) . Motivated by a recent article, we obtain sufficient conditions so that for a compact operator T ∈ L ( X, Y ) and a closed subspace Z ⊂ Y, the following equation holds, which relates global approximation with local approximation: d ( T, L ( X, Z ) ) = sup d ( T x, Z ) : x ∈ X, ‖ x ‖ = 1 . In some cases, we show that the supremum is attained at an extreme point of the corresponding unit ball. Furthermore, we obtain some situations when the following equivalence holds: T ⊥ B L ( X, Z ) ⇔ T * * x 0 * * ⊥ B Z ⊥ ⊥ ⇔ T * * ⊥ B L ( X * *, Z ⊥ ⊥ ), for some x 0 * * ∈ X * * satisfying ‖ T * * x 0 * * ‖ = ‖ T * * ‖ ‖ x 0 * * ‖, where Z ⊥ is the annihilator of Z . One such situation is when Z is an L 1 − predual space and an M − ideal in Y and T is a multi-smooth operator of finite order. Another such situation is when X is an abstract L 1 − space and T is a multi-smooth operator of finite order. Finally, as a consequence of the results, we obtain a sufficient condition for proximinality of a subspace Z in Y .