On the extremal structure of least upper bound norms and their dual

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• 作者：E. Marques de Sá ; ; Virgı ; ´ ; nia Santos
• 关键词：Linear mappings ; Norms ; Convex sets ; Faces ; Subdifferentials
• 刊名：Linear Algebra and its Applications
• 出版年：2008
• 期刊代码：38_00243795
• 类别：mt
• 出版时间：15 April 2008
• 卷：428
• 期：8-9
• 页码：1928-1938
• 文件大小：166 K

摘要

Given finite dimensional real or complex Banach spaces, E and F, with norms f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml1&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=754edc50bf5911811d8f05f4500ef0d2">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-2/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=12 width=68> and f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml2&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=58e1818d1b1aa322ed217a41c1c8eaef">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-F/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=15 width=71>, we denote by coration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml3&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=dffdf4ce71636cc76dcee04aea9e20c1" title="Click to view the MathML source">N_;c;ν the least upper bound norm induced on f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml4&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=b3e4133489eccfb07f3e86d893074da4">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-J/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=17 width=56>. Some results are given on the extremal structures of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml5&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=1b7d7310d274850b8320b67df03e9782">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-K/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=13 width=14>, the unit ball of coration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml6&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=ef38a511fff526fc81af249cc76009b1" title="Click to view the MathML source">N_;c;ν, of its polar f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml7&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=c9a8c9957453d68ab1cccc30735f9909">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-N/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=19>, and of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml8&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=069cf5599340fc9b4d3949746c597698">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-P/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=16 width=17>, which is the polar of the unit ball of the least upper bound norm coration:none; color:black" href="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml9&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=6aedfe41ced050986d130c6d4ac34045" title="Click to view the MathML source">N_;c;°ν°.

The exposed faces, the extreme points, and a large family of other faces of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml10&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=92f9deaf595fdf0093f47d59fb7832f3">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-3/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=14 width=19> and f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml11&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=869086310d401a959780ad63e3fdeaf1">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-4/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=16 width=17> are presented. It turns out that f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml12&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=7a3f1b09a2035309a50ed57daec02e84">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-5/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=16 width=17> is a subset of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml13&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=8c6988ab957717404b4a5cbb33ec7530">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-6/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=13 width=14>; the set of tangency points of the surfaces of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml14&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=3e7d0517174f9cd45138378184fd5e08">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-7/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=13 width=14> and f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml15&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=05a30574d37eab9187c4e83717b83bca">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-8/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=16 width=17> is completely determined and represented as the union of the exposed faces of f="/science?_ob=MathURL&_method=retrieve&_udi=B6V0R-4RDBFDR-1&_mathId=mml16&_user=10&_cdi=5653&_rdoc=15&_acct=C000050221&_version=1&_userid=10&md5=8f8d647a7b06d1345b8f887f05e95f40">c="http://www.sciencedirect.com/cache/MiamiImageURL/B6V0R-4RDBFDR-1-9/0?wchp=dGLbVzz-zSkzk" alt="Click to view the MathML source" align="absbottom" border="0" height=16 width=17> which are normal to rank-one mappings. We determine sharp bounds on the ranks of mappings in these exposed faces.