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34
A Banach space with the Schur and the Daugavet property
 PROC. AMER. MATH. SOC
, 2003
"... We show that a minor refinement of the BourgainRosenthal construction of a Banach space without the RadonNikodym property which contains no bounded δtrees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such s ..."
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Cited by 11 (7 self)
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We show that a minor refinement of the BourgainRosenthal construction of a Banach space without the RadonNikodym property which contains no bounded δtrees yields a space with the Daugavet property and the Schur property. Using this example we answer some open questions on the structure of such spaces; in particular we show that the Daugavet property is not inherited by ultraproducts.
The Daugavet property of C∗algebras, JB∗triples, and of their isometric preduals
, 2004
"... A Banach space X is said to have the Daugavet property if every rankone operator T: X − → X satisfies ‖Id + T ‖ = 1 + ‖T ‖. We give geometric characterizations of this property in the settings of C ∗algebras, JB ∗triples and their isometric preduals. We also show that, in these settings, the Dau ..."
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Cited by 7 (2 self)
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A Banach space X is said to have the Daugavet property if every rankone operator T: X − → X satisfies ‖Id + T ‖ = 1 + ‖T ‖. We give geometric characterizations of this property in the settings of C ∗algebras, JB ∗triples and their isometric preduals. We also show that, in these settings, the Daugavet property passes to ultrapowers, and thus, it is equivalent to an stronger property called the uniform Daugavet property.
The Daugavet property for spaces of Lipschitz functions
, 2005
"... Abstract. For a compact metric space K the space Lip(K) has the Daugavet property if and only if the norm of every f ∈ Lip(K) is attained locally. If K is a subset of an Lpspace, 1 < p < ∞, this is equivalent to the convexity of K. 1. ..."
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Cited by 6 (3 self)
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Abstract. For a compact metric space K the space Lip(K) has the Daugavet property if and only if the norm of every f ∈ Lip(K) is attained locally. If K is a subset of an Lpspace, 1 < p < ∞, this is equivalent to the convexity of K. 1.
Remarks on rich subspaces of Banach spaces
 Studia Math
, 2003
"... Abstract. We investigate rich subspaces of L1 and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products. Dedicated to Professor Aleksander Pe̷lczyński on the occasion of ..."
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Cited by 6 (4 self)
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Abstract. We investigate rich subspaces of L1 and deduce an interpolation property of Sidon sets. We also present examples of rich separable subspaces of nonseparable Banach spaces and we study the Daugavet property of tensor products. Dedicated to Professor Aleksander Pe̷lczyński on the occasion of his 70th birthday 1.
THICKNESS OF THE UNIT SPHERE, ℓ1TYPES, AND THE ALMOST DAUGAVET PROPERTY
, 902
"... Abstract. We study those Banach spaces X for which SX does not admit a finite εnet consisting of elements of SX for any ε < 2. We give characterisations of this class of spaces in terms of ℓ1type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separabl ..."
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Cited by 5 (2 self)
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Abstract. We study those Banach spaces X for which SX does not admit a finite εnet consisting of elements of SX for any ε < 2. We give characterisations of this class of spaces in terms of ℓ1type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separable Banach space X is isomorphic to a space from this class if and only if X contains an isomorphic copy of ℓ1. 1.
Norm equalities for operators on Banach spaces
 Indiana Univ. Math. J
"... Abstract. A Banach space X has the Daugavet property if the Daugavet equation Id+T = 1+ T holds for every rankone operator T : X −→ X. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like g(T ) = f ( T ) for some functions f an ..."
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Cited by 5 (3 self)
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Abstract. A Banach space X has the Daugavet property if the Daugavet equation Id+T = 1+ T holds for every rankone operator T : X −→ X. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like g(T ) = f ( T ) for some functions f and g) lead in fact to the Daugavet property of the space. On the other hand there are equations (for example Id + T = Id − T ) that lead to new, strictly weaker properties of Banach spaces.
The Daugavet equation for polynomials
 Studia Math
"... Abstract. In this paper we study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ‖Id + P ‖ = 1 + ‖P‖ is satisfied for all weakly compact polynomials P: X − → X. We show that this is the case when X = C(K), the real or complex space ..."
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Cited by 4 (2 self)
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Abstract. In this paper we study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ‖Id + P ‖ = 1 + ‖P‖ is satisfied for all weakly compact polynomials P: X − → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation max ω=1 ‖Id + ω P ‖ = 1 + ‖P‖ for polynomials P: X − → X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. The result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for khomogeneous polynomials. In 1963, I. K. Daugavet [13] showed that every compact linear operator T on C[0, 1] satisfies
EXTREMELY NONCOMPLEX C(K) SPACES
 J. MATH. ANAL. APPL.
, 2008
"... We show that there exist infinitedimensional extremely noncomplex Banach spaces, i.e. spaces X such that the norm equality ‖ Id+T 2 ‖ = 1 + ‖T 2 ‖ holds for every bounded linear operator T: X − → X. This answers in the positive Question 4.11 of [Kadets, Martín, Merí, Norm equalities for operator ..."
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Cited by 4 (3 self)
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We show that there exist infinitedimensional extremely noncomplex Banach spaces, i.e. spaces X such that the norm equality ‖ Id+T 2 ‖ = 1 + ‖T 2 ‖ holds for every bounded linear operator T: X − → X. This answers in the positive Question 4.11 of [Kadets, Martín, Merí, Norm equalities for operators, Indiana U. Math. J. 56 (2007), 2385–2411]. More concretely, we show that this is the case of some C(K) spaces with few operators constructed in [Koszmider, Banach spaces of continuous functions with few operators, Math. Ann. 330 (2004), 151–183] and [Plebanek, A construction of a Banach space C(K) with few operators, Topology Appl. 143 (2004), 217–239]. We also construct compact spaces K1 and K2 such that C(K1) and C(K2) are extremely noncomplex, C(K1) contains a complemented copy of C(2 ω) and C(K2) contains a (1complemented) isometric copy of ℓ∞.
THE GROUP OF ISOMETRIES OF A BANACH SPACE AND DUALITY
 J. FUNCT. ANAL.
, 2008
"... We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous oneparameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipativ ..."
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Cited by 4 (2 self)
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We construct an example of a real Banach space whose group of surjective isometries has no uniformly continuous oneparameter semigroups, but the group of surjective isometries of its dual contains infinitely many of them. Other examples concerning numerical index, hermitian operators and dissipative operators are also shown.