Almost overcomplete and almost overtotal sequences in Banach spaces II

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• 作者：Vladimir P. Fonf^a ; ¹ ; ^{fonf@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Jacopo Somaglia^b ; ² ; ⁵ ; ^{jacopo.somaglia@unimi.it" class="auth_mail" title="E-mail the corresponding author} ; Stanimir Troyanski^c ; ^d ; ³ ; ^{stroya@um.es" class="auth_mail" title="E-mail the corresponding author} ; Clemente Zanco^b ; ⁴ ; ⁵ ; ^{clemente.zanco@unimi.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Almost overcomplete sequence ; Almost overtotal sequence
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 February 2016
• 年：2016
• 卷：434
• 期：1
• 页码：84-92
• 全文大小：310 K

文摘

A sequence in a separable Banach space X   〈resp. in the dual space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si1.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=e42382bcf24ada16b60968a2cb3f228b" title="Click to view the MathML source">X^⁎class="mathContainer hidden">class="mathCode">$X^{⁎}$〉 is said to be overcomplete (OC in short) 〈resp. overtotal (OT in short) on X〉 whenever the linear span of each subsequence is dense in X 〈resp. each subsequence is total on X〉. A sequence in a separable Banach space X   〈resp. in the dual space class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si1.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=e42382bcf24ada16b60968a2cb3f228b" title="Click to view the MathML source">X^⁎class="mathContainer hidden">class="mathCode">$X^{⁎}$〉 is said to be almost overcomplete (AOC in short) 〈resp. almost overtotal (AOT in short) on X〉 whenever the closed linear span of each subsequence has finite codimension in X 〈resp. the annihilator (in X) of each subsequence has finite dimension〉. We provide information about the structure of such sequences. In particular it can happen that, an AOC 〈resp. AOT〉 given sequence admits countably many not nested subsequences such that the only subspace contained in the closed linear span of every of such subsequences is the trivial one 〈resp. the closure of the linear span of the union of the annihilators in X of such subsequences is the whole X〉. Moreover, any AOC   sequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si2.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=4c7752802be95f9e1891447454e69ae6" title="Click to view the MathML source">{x_n}_n∈Nclass="mathContainer hidden">class="mathCode">${\{x_n\}}_{n\in \mathbb{N}}$ contains some subsequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si121.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=1df31ff242d7f639a90b09c96248f747" title="Click to view the MathML source">{x_nj}_j∈Nclass="mathContainer hidden">class="mathCode">${\{x_{n_j}\}}_{j\in \mathbb{N}}$ that is OC   in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si122.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=adb4b3b6c2dd96af5c4503948d35d5b8" title="Click to view the MathML source">[{x_nj}_j∈N]class="mathContainer hidden">class="mathCode">$[{\{x_{n_j}\}}_{j\in \mathbb{N}}]$; any AOT   sequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si139.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=a80aaf4f2e039e7cad90ed139d1ac263" title="Click to view the MathML source">{f_n}_n∈Nclass="mathContainer hidden">class="mathCode">${\{f_n\}}_{n\in \mathbb{N}}$ contains some subsequence class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si147.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=ab1623e09a640ced0343e0417ffb96c1" title="Click to view the MathML source">{f_nj}_j∈Nclass="mathContainer hidden">class="mathCode">${\{f_{n_j}\}}_{j\in \mathbb{N}}$ that is OT on any subspace of X   complemented to class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X15008185&_mathId=si162.gif&_user=111111111&_pii=S0022247X15008185&_rdoc=1&_issn=0022247X&md5=16e2183cf6d1207a40e0f895d95474ac">class="imgLazyJSB inlineImage" height="23" width="60" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X15008185-si162.gif">class="mathContainer hidden">class="mathCode">${\{f_{n_j}\}}_{j\in \mathbb{N}}^{\top }$.