Weak minimizing property and reflexivity

Source abstract

For an operator T : X → Y T:X\to Y denote m ( T ) = inf { ‖ T x ‖ : x ∈ S X } m(T)=\inf \{\|Tx\|:x\in S_{X}\} . A sequence ( x n ) n (x_{n})_{n} in S X S_{X} is said to be minimizing for T T if ‖ T x n ‖ \|Tx_{n}\| tends to m ( T ) m(T) . In 2020 U.S. Chakraborty introduced and studied the following weak minimizing property (WmP): a pair ( X , Y ) (X,Y) of Banach spaces is said to have the WmP if, for every bounded linear operator T : X → Y T: X \to Y that admits a non-weakly null minimizing sequence, the function x ↦ ‖ T x ‖ x \mapsto \|Tx\| attains its minimum on the unit sphere. We present the following new results about the WmP for pairs of infinite-dimensional separable Banach spaces: If ( X , Y ) (X,Y) has the WmP, then X X is reflexive. If X X is reflexive and Y Y does not contain isomorphic copies of X X , then ( X , Y ) (X,Y) has the WmP. If X X is reflexive and Y Y contains an isomorphic copy of X X , then there is an equivalent norm on Y Y such that, for this equivalent norm, ( X , Y ) (X,Y) does not have the WmP. The first result extends to nonseparable X X if and only if X X possesses a countable total set of functionals.