\(M\)-ideals and propery \(SU\) (Q2770330)

A closed subspace \(J\) of a Banach space \(X\) is called an M-ideal of \(X\) if there exists a closed subspace \(J_\ast\) of \(X^\ast\) such that \(X^\ast=J^\perp \oplus J_\ast\), and \((1)\) \(\|g+h\|= \|g\|+\|h\|\) for all \(g\in J^\perp\) and \(h\in J_\ast\) [see \textit{P. Harmand, D. Werner} and \textit{W. Werner}, ``M-ideals in Banach spaces and Banach algebras'', Lect. Notes Math. 1547 (1993; Zbl 0789.46011)]. \textit{Eh. Oya} [Math. Notes 43, No. 2, 134-139 (1988; Zbl 0665.46004); ``Extension of functionals and the structure of the space of continuous linear operators'' (1991; Zbl 0783.46016)], and \textit{Eh. Oya} and \textit{M. Põldvere} [Stud. Math. 117, No. 3, 289-306 (1996; Zbl 0854.46014)] introduced the weaker notion of property SU: \(J\) has property SU in \(X\) if, instead of \((1)\), one has \((2)\) \(\|g+h\|> \|h\|\) for all \(0\not=g\in J^\perp\) and \(h\in J_\ast\) (which implies that every continuous linear functional on \(J\) has a unique norm preserving extension to \(X\), the property U of \textit{R. Phelps} [Trans. Am. Math. Soc. 95, 238-255 (1960; Zbl 0096.31102)], and showed that these notions are distinct.NEWLINENEWLINENEWLINEIn the present paper, the authors show that, for every closed subspace \(Z\) of \(Y\), under several assumptions (the space of compact operators \({\mathcal K}(X,Y)\) is an M-ideal of \({\mathcal L}(X,Y)\), \({\mathcal L}(X,Z)\) has property SU in \({\mathcal L}(X,Y)\), and \(d(T, {\mathcal K}(X,Z))=d(T, {\mathcal K}(X,Y))\) for every \(T\in {\mathcal L}(X,Z)\)), then \({\mathcal K}(X,Z)\) is an M-ideal in \({\mathcal L}(X,Z)\) as soon as it has property SU in \({\mathcal L}(X,Z)\).