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Pointwise absolutely convergent series of operators and related classes of Banach spaces

100%

Kadets L. , Kadets V.

Open Mathematics

2012

tom 10

nr 2

603-608

We study classes of operators represented as a pointwise absolutely convergent series of simpler ones, starting with rank 1 operators. In this short note we address the question, how far the repetition of this procedure can lead.

Sums of SCD sets and their applications to SCD operators and narrow operators

100%

Kadets V. , Shepelska V.

Open Mathematics

2010

tom 8

nr 1

129-134

We answer two open questions concerning the recently introduced notions of slicely countably determined (SCD) sets and SCD operators in Banach spaces. An application to narrow operators in spaces with the Daugavet property is given.

Quotients of Banach Spaces with the Daugavet Property

80%

Kadets V. , Shepelska V. , Werner D.

Bulletin of the Polish Academy of Sciences. Mathematics

2008

tom 56

nr 2

131-147

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak* analogue. We introduce and study analogues of narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L₁[0,1] by an ℓ₁-subspace need not have the Daugavet property. The latter answers in the negative a question posed to us by A. Pełczyński.

Remarks on rich subspaces of Banach spaces

80%

Kadets V. , Kalton N. , Werner D.

tom 159

nr 2

195-206

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.

Properties of lush spaces and applications to Banach spaces with numerical index 1

80%

Boyko K. , Kadets V. , Martín M. , Merí J.

Studia Mathematica

2009

tom 190

nr 2

117-133

The concept of lushness, introduced recently, is a Banach space property, which ensures that the space has numerical index 1. We prove that for Asplund spaces lushness is actually equivalent to having numerical index 1. We prove that every separable Banach space containing an isomorphic copy of c₀ can be renormed equivalently to be lush, and thus to have numerical index 1. The rest of the paper is devoted to the study of lushness just as a property of Banach spaces. We prove that lushness is separably determined, is stable under ultraproducts, and we characterize those spaces of the form X = (ℝⁿ,||·||) with absolute norm such that X-sum preserves lushness of summands, showing that this property is equivalent to lushness of X.