1999/09/17 by Vladimir Kadets, Roman Shvidkoy, Gleb Sirotkin +1 · 1 voice · 1 citation
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Holomorphic and Operator Theory
paper · pdf · doi:10.1090/s0002-9947-99-02377-6
openalex publication_date 1999/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/02
A Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is said to have the Daugavet property if every operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper X right-arrow upper X"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T: X→ X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1"> <mml:semantics> <mml:mn>1</mml:mn> <mml:annotation encoding="application/x-tex">1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfies <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper I d plus upper T double-vertical-bar equals 1 plus double-vertical-bar upper T double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>Id</mml:mi> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">‖\operatorname Id+T‖ = 1+‖T‖</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that then every weakly compact operator satisfies this equation as well and that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains a copy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script l 1"> <mml:semantics> <mml:msub> <mml:mi>ℓ</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">ℓ 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> need not contain a copy of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 1"> <mml:semantics> <mml:msub> <mml:mi>L</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:annotation encoding="application/x-tex">L1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We also study pairs of spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X subset-of upper Y"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X⊂ Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and operators <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T colon upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">T: X→ Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> satisfying <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper J plus upper T double-vertical-bar equals 1 plus double-vertical-bar upper T double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>J</mml:mi> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">‖J+T‖=1+‖T‖</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 J colon upper X right-arrow upper Y"> <mml:semantics> <mml:mrow> <mml:mi>J</mml:mi> <mml:mo>:</mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">J: X→ Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we investigate spaces where the set of operators with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper I d plus upper T double-vertical-bar equals 1 plus double-vertical-bar upper T double-vertical-bar"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>Id</mml:mi> <mml:mo>+</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mo fence="false" stretchy="false">‖</mml:mo> <mml:mi>T</mml:mi> <mml:mo fence="false" stretchy="false">‖</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">‖\operatorname Id+T‖=1+‖T‖</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is as small as possible and give characterisations in terms of a smoothness condition.