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On the Vector‐Valued Hilbert Transform
Author(s) -
Defant Martin
Publication year - 1989
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.19891410123
Subject(s) - mathematics , bounded function , bounded operator , hilbert space , linear operators , operator (biology) , extension (predicate logic) , hilbert transform , linear map , operator space , combinatorics , discrete mathematics , pure mathematics , mathematical analysis , banach space , finite rank operator , computer science , chemistry , biochemistry , spectral density , statistics , repressor , transcription factor , gene , programming language
Let H : L p ( R ) → L p ( R ), 1 < p < ∞ be the real H ILBERT transform. A bounded, linear operator u : E → F ( E, F B ANACH spaces) is a HT‐operator , if the mapping u ⊗ H : E ⊗ L 2 ( R , E) → L 2 ( R , F ) has a bounded, linear extension to L 2 ( R ) → L 2 ( R , F ). For E = F and u = id E B OURGAIN [3] and B URKHOLDER [5] have shown that this holds if and only if E ϵ UMD. We study these HT ‐operators and, in particular, we construct a HT ‐operator which is not UMD ‐factorable. Furthermore, we show that a UMD ‐space E is a H ILBERT space if and only if | id E ⊗ H | = 1.