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Tensor Products and Operators in Spaces of Analytic Functions
Author(s) -
Freniche Francisco J.,
García-Vázquez Juan Carlos,
Rodríguez-Piazza Luis
Publication year - 2001
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s002461070100196x
Subject(s) - tensor product of hilbert spaces , pure mathematics , tensor (intrinsic definition) , algebra over a field , mathematics , tensor product , mathematical analysis , tensor contraction
Let X be an infinite dimensional Banach space. The paper proves the non‐coincidence of the vector‐valued Hardy space H p (T, X ) with neither the projective nor the injective tensor product of H p (T) and X , for 1 < p < ∞. The same result is proved for some other subspaces of L p . A characterization is given of when every approximable operator from X into a Banach space of measurable functions F ( S ) is representable by a function F : S → X * as x ↦ 〈 F (·), x 〉. As a consequence the existence is proved of compact operators from X into H p (T) (1 ⩽ p < ∞) which are not representable. An analytic Pettis integrable function F :T → X is constructed whose Poisson integral does not converge pointwise.