Uniqueness of norm-preserving extensions of functionals on the space of compact operators
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Julia Martsinkevitš [1] ; Märt Põldvere [1]
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[1] University of Tartu
University of Tartu
Tartu linn, Estonia
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- Localización: Mathematica scandinavica, ISSN 0025-5521, Vol. 125, Nº 1, 2019, págs. 67-83
- Idioma: inglés
- DOI: 10.7146/math.scand.a-112071
- Texto completo no disponible (Saber más ...)
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Resumen
Godefroy, Kalton, and Saphar called a closed subspace Y of a Banach space Z an ideal if its annihilator Y⊥ is the kernel of a norm-one projection P on the dual space Z∗. If Y is an ideal in Z with respect to a projection on Z∗ whose range is norming for Z, then Y is said to be a strict ideal. We study uniqueness of norm-preserving extensions of functionals on the space K(X,Y) of compact operators between Banach spaces X and Y to the larger space K(X,Z) under the assumption that Y is a strict ideal in Z. Our main results are: (1) if y∗ is an extreme point of BY∗ having a unique norm-preserving extension to Z, and x∗∗∈BX∗∗, then the only norm-preserving extension of the functional x∗∗⊗y∗∈K(X,Y)∗ to K(X,Z) is x∗∗⊗z∗ where z∗∈Z∗ is the only norm-preserving extension of y∗ to Z; (2) if K(X,Y) is an ideal in K(X,Z) and Y has Phelps' property U in its bidual Y∗∗ (i.e., every bounded linear functional on Y admits a unique norm-preserving extension to Y∗∗), then K(X,Y) has property U in K(X,Z) whenever X∗∗ has the Radon-Nikodým property.
