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• W1590186543 abstract "In this paper, we discuss finite rank operators in a closed maximal triangular algebra <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper S> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {S}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Based on the following result that each finite rank operator of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper S> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {S}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can be written as a finite sum of rank one operators each belonging to <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper S> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {S}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, we proved that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis script upper S intersection script upper F script left-parenthesis script upper H script right-parenthesis right-parenthesis Superscript w Super Superscript asterisk Baseline equals left-brace upper T element-of script upper B script left-parenthesis script upper H script right-parenthesis colon upper T upper N subset-of-or-equal-to upper N Subscript tilde Baseline comma for-all upper N element-of script upper N right-brace> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:mo>∩<!-- ∩ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> <mml:mo class=MJX-tex-caligraphic mathvariant=script stretchy=false>(</mml:mo> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> <mml:mo class=MJX-tex-caligraphic mathvariant=script stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msup> <mml:mi>w</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>T</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>B</mml:mi> <mml:mo class=MJX-tex-caligraphic mathvariant=script stretchy=false>(</mml:mo> <mml:mi class=MJX-tex-caligraphic mathvariant=script>H</mml:mi> <mml:mo class=MJX-tex-caligraphic mathvariant=script stretchy=false>)</mml:mo> </mml:mrow> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi>T</mml:mi> <mml:mi>N</mml:mi> <mml:mo>⊆<!-- ⊆ --></mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∼<!-- ∼ --></mml:mo> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>∀<!-- ∀ --></mml:mi> <mml:mi>N</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>N</mml:mi> </mml:mrow> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>({mathcal {S}}cap {mathcal {F(H)}})^{w^{*}}={Tin {mathcal {B(H)}}: TNsubseteq N_{sim }, forall Nin mathcal {N}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript tilde Baseline equals upper N> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∼<!-- ∼ --></mml:mo> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{sim }=N</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d i m upper N minus upper N Subscript minus Baseline less-than-or-equal-to 1> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>m</mml:mi> <mml:mi>N</mml:mi> <mml:mo>⊖<!-- ⊖ --></mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msub> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>dim Nominus N_{-}leq 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N Subscript tilde Baseline equals upper N Subscript minus> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>∼<!-- ∼ --></mml:mo> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>N_{sim }=N_{-}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d i m upper N minus upper N Subscript minus Baseline equals normal infinity> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>i</mml:mi> <mml:mi>m</mml:mi> <mml:mi>N</mml:mi> <mml:mo>⊖<!-- ⊖ --></mml:mo> <mml:msub> <mml:mi>N</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo>−<!-- − --></mml:mo> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>dim Nominus N_{-}=infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also proved that the Erdos Density Theorem holds in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper S> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {S}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper S> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>S</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{mathcal {S}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strongly reducible." @default.
• W1590186543 created "2016-06-24" @default.
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• W1590186543 date "2002-10-01" @default.
• W1590186543 modified "2023-10-18" @default.
• W1590186543 title "Finite rank operators in closed maximal triangular algebras II" @default.
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