Title: On extension of isometries between unit spheres of π΄πΏ_{π}-spaces (0<π<β)
Abstract:In this paper, we study the extension of isometries between unit spheres of atomic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A u...In this paper, we study the extension of isometries between unit spheres of atomic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper L Subscript p"><mml:semantics><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">AL_p</mml:annotation></mml:semantics></mml:math></inline-formula>-spaces<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 greater-than p greater-than normal infinity comma p not-equals 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>></mml:mo><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi mathvariant="normal">β</mml:mi><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>p</mml:mi><mml:mo>β </mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(0>p>\infty ,\ p\neq 2)</mml:annotation></mml:semantics></mml:math></inline-formula>. We find a condition under which an isometry<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"><mml:semantics><mml:mi>T</mml:mi><mml:annotation encoding="application/x-tex">T</mml:annotation></mml:semantics></mml:math></inline-formula>between unit spheres can be linearly isometrically extended. Moreover, we prove that every onto isometry between unit spheres of atomic<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A upper L Subscript p"><mml:semantics><mml:mrow><mml:mi>A</mml:mi><mml:msub><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msub></mml:mrow><mml:annotation encoding="application/x-tex">AL_p</mml:annotation></mml:semantics></mml:math></inline-formula>-spaces<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 0 greater-than p greater-than normal infinity comma p not-equals 2 right-parenthesis"><mml:semantics><mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mn>0</mml:mn><mml:mo>></mml:mo><mml:mi>p</mml:mi><mml:mo>></mml:mo><mml:mi mathvariant="normal">β</mml:mi><mml:mo>,</mml:mo><mml:mtext> </mml:mtext><mml:mi>p</mml:mi><mml:mo>β </mml:mo><mml:mn>2</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:mrow><mml:annotation encoding="application/x-tex">(0>p>\infty ,\ p\neq 2)</mml:annotation></mml:semantics></mml:math></inline-formula>can be linearly isometrically extended to the whole space.Read More