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• W1930584469 abstract "In this paper, we prove the following automatic adjoint theorem: For any sequence spaces <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(X)</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 F left-parenthesis upper Y right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F(Y)</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=upper E left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has the signed-weak gliding hump property and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an infinite matrix which transforms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>E(X)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F left-parenthesis upper Y right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>F(Y)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then the transpose matrix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A prime> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>A’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> transforms <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F left-parenthesis upper Y right-parenthesis Superscript beta> <mml:semantics> <mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>β<!-- β --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>F(Y)^beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula> into <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E left-parenthesis upper X right-parenthesis Superscript beta> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>β<!-- β --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>E(X)^beta</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and for any <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of upper E left-parenthesis upper X right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>xin E(X)</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 T element-of upper F left-parenthesis upper Y right-parenthesis Superscript beta> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi>F</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>Y</mml:mi> <mml:msup> <mml:mo stretchy=false>)</mml:mo> <mml:mi>β<!-- β --></mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Tin F(Y)^beta</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=left-bracket upper A x comma upper T right-bracket equals left-bracket x comma upper A prime upper T right-bracket> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>[</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>]</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy=false>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mi>T</mml:mi> <mml:mo stretchy=false>]</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>[Ax,T]=[x,A’T]</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. That is, the adjoint operator of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> automatically exists and is just the transpose matrix <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A prime> <mml:semantics> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:annotation encoding=application/x-tex>A’</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper A> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=application/x-tex>A</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. From the theorem we obtain a class of infinite matrix topological algebras <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis lamda comma mu right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ<!-- μ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(lambda ,mu )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and prove also a <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=lamda> <mml:semantics> <mml:mi>λ<!-- λ --></mml:mi> <mml:annotation encoding=application/x-tex>lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-multiplier convergence theorem of Orlicz-Pettis type. The theorem improves substantially the famous Stiles’ Orlicz-Pettis theorem." @default.
• W1930584469 created "2016-06-24" @default.
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• W1930584469 date "2001-12-20" @default.
• W1930584469 modified "2023-09-26" @default.
• W1930584469 title "An automatic adjoint theorem and its applications" @default.
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