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The Daugavet equation for polynomials

Tom 178 / 2007

Yun Sung Choi, Domingo García, Manuel Maestre, Miguel Martín Studia Mathematica 178 (2007), 63-84 MSC: Primary 46G25; Secondary 46B20, 47A12. DOI: 10.4064/sm178-1-4

Streszczenie

We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space $X$, i.e. when the equality $$ \|\mathop{\rm Id}+P\|=1+\|P\| $$ is satisfied for all weakly compact polynomials $P:X\to X$. We show that this is the case when $X=C(K)$, the real or complex space of continuous functions on a compact space $K$ without isolated points. We also study the alternative Daugavet equation $$ \max_{|\omega|=1} \|\mathop{\rm Id} +\omega P\| = 1 + \|P\| $$ for polynomials $P:X\rightarrow X$. We show that this equation holds for every polynomial on the complex space $X=C(K)$ ($K$ arbitrary) with values in $X$. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for $k$-homogeneous polynomials.

Autorzy

  • Yun Sung ChoiDepartment of Mathematics
    POSTECH
    Pohang 790-784, Korea
    e-mail
  • Domingo GarcíaDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    46100 Burjasot (Valencia), Spain
    e-mail
  • Manuel MaestreDepartamento de Análisis Matemático
    Universidad de Valencia
    Doctor Moliner 50
    $46100$ Burjasot (Valencia), Spain
    e-mail
  • Miguel MartínDepartamento de Análisis Matemático
    Facultad de Ciencias
    Universidad de Granada
    18071 Granada, Spain
    e-mail

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