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GEOMETRIC AND FIXED POINT PROPERTIES IN PRODUCTS OF NORMED SPACES

  • M. VEENA SANGEETHA (a1)
Abstract

Given two (real) normed (linear) spaces $X$ and $Y$ , let $X\otimes _{1}Y=(X\otimes Y,\Vert \cdot \Vert )$ , where $\Vert (x,y)\Vert =\Vert x\Vert +\Vert y\Vert$ . It is known that $X\otimes _{1}Y$ is $2$ -UR if and only if both $X$ and $Y$ are UR (where we use UR as an abbreviation for uniformly rotund). We prove that if $X$ is $m$ -dimensional and $Y$ is $k$ -UR, then $X\otimes _{1}Y$ is $(m+k)$ -UR. In the other direction, we observe that if $X\otimes _{1}Y$ is $k$ -UR, then both $X$ and $Y$ are $(k-1)$ -UR. Given a monotone norm $\Vert \cdot \Vert _{E}$ on $\mathbb{R}^{2}$ , we let $X\otimes _{E}Y=(X\otimes Y,\Vert \cdot \Vert )$ where $\Vert (x,y)\Vert =\Vert (\Vert x\Vert _{X},\Vert y\Vert _{Y})\Vert _{E}$ . It is known that if $X$ is uniformly rotund in every direction, $Y$ has the weak fixed point property for nonexpansive maps (WFPP) and $\Vert \cdot \Vert _{E}$ is strictly monotone, then $X\otimes _{E}Y$ has WFPP. Using the notion of $k$ -uniform rotundity relative to every $k$ -dimensional subspace we show that this result holds with a weaker condition on $X$ .

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Some of the results in this article are part of the author’s PhD thesis written at the Indian Institute of Technology Madras, Chennai, India with the financial support of the Council of Scientific and Industrial Research, New Delhi, India.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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