The Bishop-Phelps-Bollobás point property

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• 作者：Sheldon Dantas^a ; ^{sheldon.dantas@uv.es" class="auth_mail" title="E-mail the corresponding author} ; Sun Kwang Kim^b ; ^{sunkwang@kgu.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Han Ju Lee^c ; ^{hanjulee@dongguk.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Bishop&ndash ; Phelps theorem ; Bishop&ndash ; Phelps&ndash ; Bollobá ; s property ; Norm attaining ; Bilinear forms
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：444
• 期：2
• 页码：1739-1751
• 全文大小：372 K

文摘

In this article, we study a version of the Bishop–Phelps–Bollobás property. We investigate a pair of Banach spaces (X,Y)$(X,Y)$ such that every operator from X into Y   is approximated by operators which attain their norm at the same point where the original operator almost attains its norm. In this case, we say that such a pair has the Bishop–Phelps–Bollobás point property (BPBpp). We characterize uniform smoothness in terms of BPBpp and we give some examples of pairs (X,Y)$(X,Y)$ which have and fail this property. Some stability results are obtained about ℓ₁${\ell }_1$ and ℓ_∞${\ell }_{\infty }$ sums of Banach spaces and we also study this property for bilinear mappings.