An approximation problem in the space of bounded operators

2022/03/19 by Mal, Arpita
#46B25 #47L05 #FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B28 #Secondary 41A50

paper · doi:10.48550/arxiv.2203.10266

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 paper \citeRAO21, we obtain sufficient conditions so that for a compact operator T\inL(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(Tx,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^**x₀^**⊥_B Z^⊥⊥⇔ T^**⊥_BL(X^**,Z^⊥⊥), for some x₀^**∈ X^** satisfying ‖T^**x₀^**‖=‖T^**‖‖x₀^**‖, where Z^⊥ is the annihilator of Z. One such situation is when Z is an L¹-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 L₁-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.

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