Spear operators between Banach spaces

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by  Vladimir Kadets

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Spear operators
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between Banach spaces
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Vladimir Kadets
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7 Miguel Martín
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Javier Merí
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2 Antonio Pérez
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1 January 2017
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Spear operators between Banach spaces
Vladimir Kadets
Miguel Martín
Javier Merí
Antonio Pérez
(Kadets)
School of Mathematics and Informatics,
V. N. Karazin Kharkiv National University,
pl. Svobody 4, 61022 Kharkiv, Ukraine.
ORCID: 0000-0002-5606-2679
E-mail address: [email protected]
(Martín)
Departamento de Análisis Matemático,
Facultad de Ciencias,
Universidad de Granada,
18071 Granada, Spain.
ORCID: 0000-0003-4502-798X
E-mail address: [email protected]
(Merí)
Departamento de Análisis Matemático,
Facultad de Ciencias,
Universidad de Granada,
18071 Granada, Spain.
ORCID: 0000-0002-0625-5552
E-mail address: [email protected]
(Pérez)
Departamento de Matemáticas,
Universidad de Murcia,
30100 Espinardo (Murcia), Spain.
ORCID: 0000-0001-8600-7083
E-mail address: [email protected]
2010 Mathematics Subject Classification. Primary 46B04. Secondary 46B20,
46B22, 46B25, 46J10, 47A12, 47A30, 47A99.
Key words and phrases. Banach space, bounded linear operator, Daugavet center,
numerical range of operator; alternative Daugavet property; lush spaces; spear
set; Lipschitz operator.
The research of the first author is done in frames of Ukrainian Ministry of Science
and Education Research Program 0115U000481, and it was partially done during
his stay in Murcia under the support of MINECO/FEDER projects
MTM2014-57838-C2-1-P and partially during his visits to the University of
Granada which were supported by the Spanish MINECO/FEDER project
MTM2015-65020-P. The second and the third authors were partially supported by
Spanish MINECO/FEDER project MTM2015-65020-P, and by Junta de
Andalucía and FEDER grant FQM-185. The fourth author was partially
supported by the MINECO/FEDER project MTM2014-57838-C2-1-P and a
Ph.D. fellowship of “La Caixa Foundation”.
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Abstract
The aim of this manuscript is to study spear operators: bounded linear operators
G between Banach spaces X and Y satisfying that for every other bounded linear
operator T :X −→Y there exists a modulus-one scalar ω such that
(cid:107)G+ωT(cid:107)=1+(cid:107)T(cid:107).
To this end, we introduce two related properties, one weaker called the alternative
Daugavet property (if rank-one operators T satisfy the requirements), and one
stronger called lushness, and we develop a complete theory about the relations
between these three properties. To do this, the concepts of spear vector and spear
setplayanimportantrole. Further,weprovidewithmanyexamplesamongclassical
spaces,beingoneofthemthelushnessoftheFouriertransformonL . Wealsostudy
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the relation of these properties with the Radon-Nikodým property, with Asplund
spaces, withtheduality, andweprovidesomestabilityresults. Further, wepresent
some isometric and isomorphic consequences of these properties as, for instance,
that (cid:96) is contained in the dual of the domain of every real operator with infinite
1
rank and the alternative Daugavet property, and that these three concepts behave
badly with smoothness and rotundity. Finally, we study Lipschitz spear operators
(that is, those Lipschitz operators satisfying the Lipschitz version of the equation
above) and prove that (linear) lush operators are Lipschitz spear operators.

Contents
Chapter 1. Introduction 9
1.1. Motivation 9
1.2. Notation, terminology, and preliminaries 10
1.3. A short account on the results for the identity 12
1.4. A brushstroke on numerical ranges and numerical indices 14
1.5. The structure of the manuscript 16
Chapter 2. Spear vectors and spear sets 19
Chapter 3. Three definitions for operators: spearness, the alternative
Daugavet property, and lushness 27
3.1. A first contact with spear operators 27
3.2. Alternative Daugavet Property 28
3.3. Target operators 31
3.4. Lush operators 37
Chapter 4. Some examples in classical Banach spaces 41
4.1. Fourier transform 41
4.2. Operators arriving to sup-normed spaces 41
4.3. Operators acting from spaces of integrable functions 47
Chapter 5. Further results 53
5.1. Radon-Nikodým Property in the domain or Asplund codomain 53
5.2. Finite-dimensional domain or codomain 55
5.3. Adjoint Operators 57
Chapter 6. Isometric and isomorphic consequences 63
Chapter 7. Lipschitz spear operators 69
Chapter 8. Some stability results 79
8.1. Elementary results 79
8.2. Absolute sums 81
8.3. Vector-valued function spaces 84
8.4. Target operators, lushness and ultraproducts 103
Chapter 9. Open problems 107
Bibliography 109
Index 113
7

CHAPTER 1
Introduction
1.1. Motivation
Let X and Y be Banach spaces. The main goal of this manuscript is to study
bounded linear operators G : X −→ Y such that for every other bounded linear
operator T :X −→Y there is a modulus-one scalar ω such that the norm equality
(1.1) (cid:107)G+ωT(cid:107)=1+(cid:107)T(cid:107)
holds. In this case, we say that G is a spear operator. When X =Y and G=Id is
the identity operator, the study of this equation goes back to the 1970 paper [27].
It was proved there that for T : X −→ X bounded and linear, the existence of a
modulus-one scalar ω such that the norm equality
(1.2) (cid:107)Id+ωT(cid:107)=1+(cid:107)T(cid:107)
holds,isequivalenttotheequalitybetweenthenumericalradiusofT anditsnorm.
The list of spaces for which the identity is a spear operator (they are called spaces
with numerical index one) contains all C(K) spaces and L (µ) spaces, as well as
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somespacesofanalyticfunctionsandvector-valuedfunctions,whichmotivatedthe
intensive study of this class of spaces in the past decades. Let us say that, as we
will see here, the extension to general operators produces other important (and
very different from the identity) examples ofoperators which are spear. Oneof the
most striking one is the Fourier transform on the L space on a locally compact
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Abelian group (see subsection 4.1).
Theconceptsofnumericalrangeandnumericalradiusofoperators,andtheone
of numerical index of Banach spaces, played an important role in operator theory,
particularlyinthestudyandclassificationofoperatoralgebras(seethefundamental
Bonsall and Duncan books [7, 8], the survey paper [42], and the sections §2.1 and
§2.9 of the recent book [14]). For general operators G, a concept of numerical
range of operators with respect to G has been recently introduced [3], and there is
a relation between (1.1) and numerical ranges, analogous to the case of (1.2). For
instance, a spear operator is geometrically unitary in the strongest possible form.
These concepts provide also a natural motivation for the study of spear operators;
we will give a short account on this in section 1.4.
ThepropertyofGbeingaspearoperatorisformulatedintermsofallbounded
linear operators between two Banach spaces, which leads to several difficulties for
its study in abstract spaces, and also in some concrete spaces. It would be much
moreconvenienttohaveageometricdefinitionofthisproperty(intermsofG),but
unfortunately until now a description of this property in pure geometrical terms
has not been discovered, even for the case when G = Id. In order to manage this
difficulty for this case, two other Banach space properties were introduced: the
alternative Daugavet Property (aDP for short) and lushness (see [56] and [13],
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10 1. INTRODUCTION
respectively). These two properties are of geometric nature, the aDP is weaker
and lushness is stronger than the fact that the identity is a spear operator. On
the other hand, in some classes of Banach spaces these properties are equivalent
(say, in Asplund spaces, in spaces with the Radon-Nikodým property and, more
generally, in SCD spaces introduced in [4]). The study of these two properties has
been crucial in the development of the theory in the case when G = Id. We will
giveashortaccountonthisinsection1.3,wherewealsopresentthemainexamples
and results about SCD spaces.
So,nownaturallyappearsthetaskofre-constructingthetheoryoftheaDPand
lushness in such a way that it could be applied to spear operators. We will do so
here. Onthisway,wenotonlytransfertheknownresultstothenewsetting,butin
fact make much more. Namely, we introduce a unified approach to a huge number
of previously known results, substantially simplify the system of notations and, in
many cases, present the general results for operators G in a more clear way than
it was done earlier for the identity operator. To do so it has been crucial to study
thenewconceptoftargetoperator(seesection3.3),whichplaysthesamerolehere
that the concept of strong Daugavet operator plays for the study of the Daugavet
property [46] and of Daugavet centers [9]. Let us also mention that the aDP, to
beatargetoperatorandlushnessareseparablydeterminedproperties,whilewedo
not know whether the concept of spear operator is. This separable determination
allows us to use the full power of the theory of SCD sets and operators, a task
which will be crucial in the development of the subject.
Finally, anothermotivationtodothisjobwasthepotentialapplicabilityofthe
extended theory to the study of non-linear Lipschitz maps, which would allow to
extend some recent results from [67] and [41]. In particular, this will allow to re-
provetheresultsofthelatterpapersinamorereasonable(i.e.linear)way. Namely,
the standard technique of Lipschitz-free spaces (see [32] or [68]) reduces equation
(1.2) for a non-linear Lipschitz map T :X −→X to an analogous equation for the
linearizationofT, butthislinearizationactsfromtheLipschitz-freespaceF(X)to
X. Hence, in order to use this technique, we are in need of studying equation (1.1)
insteadof (1.2), thatis, tostudyspearoperatorsbetweentwodifferentspaces. See
section 7 for more details.
1.2. Notation, terminology, and preliminaries
By K we denote the scalar field (R or C), and we use the standard notation
T := {λ ∈ K: |λ| = 1} for its unit sphere. We use the letters X, Y, Z for Banach
spaces over K and by subspace we always mean closed subspace. In some cases, we
havetodistinguishbetweentherealandthecomplexcase,butformostresultsthis
difference is insignificant. The closed unit ball and unit sphere of X are denoted
respectively by B and S . We denote the Banach space of all bounded linear
X X
operators from X to Y by L(X,Y), and write L(X) for L(X,X). The identity
operator is denoted by Id, or Id if it is necessary to precise the space. The dual
X
space of X is denoted by X∗, and J : X −→ X∗∗ denotes the natural isometric
X
inclusion of X into its bidual X∗∗. For x ∈ X∗ and y ∈ Y, we write x∗⊗y to
0 0 0 0
denote the rank-one operator given by [x∗⊗y ](x)=x∗(x)y for every x∈X. We
0 0 0 0
write X = Y ⊕ Z and X = Y ⊕ Z to mean that X is, respectively, the (cid:96) -sum
1 ∞ 1
and the (cid:96) -sum of Y and Z. In the first case, we say that Y is an L-summand of
∞
X; in the second case, we say that Y is an M-summand of X.