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• W4310415723 abstract "We study geometric and isometric properties of two classes of sequence spaces. First, we consider the Banach sequence space <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Subscript n> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>X_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the completion of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=c 00> <mml:semantics> <mml:msub> <mml:mi>c</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mn>00</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>c_{00}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> relatively to the norm <disp-formula content-type=math/mathml> [ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-vertical-bar x double-vertical-bar Subscript n Baseline equals sup left-brace right-brace comma comma sigma-summation sigma-summation element-of element-of iF vertical-bar vertical-bar xi comma with comma separator separator Fof cardinality at mostn period> <mml:semantics> <mml:mrow> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>x</mml:mi> <mml:msub> <mml:mo fence=false stretchy=false>‖<!-- ‖ --></mml:mo> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo movablelimits=true form=prefix>sup</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:munder> <mml:mo>∑<!-- ∑ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>F</mml:mi> </mml:mrow> </mml:munder> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo stretchy=false>|</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mtext> with </mml:mtext> <mml:mo>,</mml:mo> <mml:mi>F</mml:mi> <mml:mtext> of cardinality at most </mml:mtext> <mml:mi>n</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>|x|_n = sup left {sum _{i in F}|x_i|, text { with }, F text { of cardinality at most } nright }.</mml:annotation> </mml:semantics> </mml:math> ] </disp-formula> For this space, we show that the isometry group of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Subscript n> <mml:semantics> <mml:msub> <mml:mi>X</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>X_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consists of standard isometries. Second, we consider a Tsirelson-type space, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T left-bracket theta comma upper A Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>[</mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>T[theta , A_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (<inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=0 greater-than theta greater-than 1> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>></mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>0>theta >1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) associated with the regular family <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A Subscript n> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding=application/x-tex>A_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, comprised of all subsets of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper N> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with at most <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=n> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding=application/x-tex>n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> elements. For this space, we characterize the extreme points and show that for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=theta less-than-or-equal-to StartFraction 1 Over n EndFraction> <mml:semantics> <mml:mrow> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>n</mml:mi> </mml:mfrac> </mml:mrow> <mml:annotation encoding=application/x-tex>theta leq frac {1}{n}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T left-bracket theta comma upper A Subscript n Baseline right-bracket> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>[</mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>T[theta , A_n]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> supports standard isometries. We also derive the form for the surjective isometries of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T left-bracket theta comma upper A 3 right-bracket> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>[</mml:mo> <mml:mi>θ<!-- θ --></mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>3</mml:mn> </mml:msub> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>T[theta , A_3]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>." @default.
• W4310415723 created "2022-12-10" @default.
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• W4310415723 date "2023-07-21" @default.
• W4310415723 modified "2023-09-23" @default.
• W4310415723 title "Geometric and isometric properties of two classes of sequence spaces" @default.
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• W4310415723 doi "https://doi.org/10.1090/proc/16347" @default.
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