Stability of unique Hahn–Banach extensions and associated linear projections

Daptari, Soumitra and Paul, Tanmoy and Rao, T. S. S. R. K. (2021) Stability of unique Hahn–Banach extensions and associated linear projections. Linear and Multilinear Algebra. pp. 1-17. ISSN 0308-1087

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Abstract

In this paper, we study two properties viz., property-U and property-SU of a subspace Y of a Banach space X, which correspond to the uniqueness of the Hahn–Banach extension of each linear functional in (Formula presented.) and when this association forms a linear operator of norm-1 from (Formula presented.) to (Formula presented.). It is proved that, under certain geometric assumptions on (Formula presented.) these properties are stable with respect to the injective tensor product; Y has property-U (SU) in Z if and only if (Formula presented.) has property-U (SU) in (Formula presented.). We prove that when (Formula presented.) has the Radon–Nikod (Formula presented.) m Property for (Formula presented.), (Formula presented.) has property-U (property-SU) in (Formula presented.) if and only if Y is so in X. We show that if (Formula presented.) and Y has property-U (SU) in X then Y/Z has property-U (SU) in X/Z. On the other hand, Y has property-SU in X if Y/Z has property-SU in X/Z and (Formula presented.) is an M-ideal in X. This partly solves the 3-space problem for property-SU. We characterize all hyperplanes in (Formula presented.) which have property-SU. We derive necessary and sufficient conditions for all finite codimensional proximinal subspaces of (Formula presented.) which have property-U (SU).

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