On holomorphic functions attaining their norms

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• 作者：Acosta ; Marí ; a D. ; Alaminos ; Jeró ; nimo ; Garcí ; a ; Domingo ; Maestre ; Manuel
• 关键词：Holomorphic function ; Polynomial ; Norm attaining ; Lorentz sequence space
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2004
• 出版时间：September 15, 2004
• 年：2004
• 卷：297
• 期：2
• 页码：625-644
• 全文大小：274 K

文摘

We show that on a complex Banach space X, the functions uniformly continuous on the closed unit ball and holomorphic on the open unit ball that attain their norms are dense provided that X has the Radon–Nikodym property. We also show that the same result holds for Banach spaces having a strengthened version of the approximation property but considering just functions which are also weakly uniformly continuous on the unit ball. We prove that there exists a polynomial such that for any fixed positive integer k, it cannot be approximated by norm attaining polynomials with degree less than k. For X=d_*(ω,1), a predual of a Lorentz sequence space, we prove that the product of two polynomials with degree less than or equal two attains its norm if, and only if, each polynomial attains its norm.