##
**Spear operators between Banach spaces.**
*(English)*
Zbl 1415.46002

Lecture Notes in Mathematics 2205. Cham: Springer (ISBN 978-3-319-71332-8/pbk; 978-3-319-71333-5/ebook). xv, 161 p. (2018).

This book is devoted to the concept of spear operators (originally introduced by M. A. Ardalani [Stud. Math. 225, No. 2, 165–171 (2014; Zbl 1316.47006)]), which are defined as follows: Let \(X\) and \(Y\) be Banach spaces and \(G:X \rightarrow Y\) a bounded linear operator. \(G\) is said to be a spear operator if, for every bounded linear operator \(T:X \rightarrow Y\), there exists some scalar \(\omega\) with \(|\omega|=1\) and
\[
\|G+\omega T\|=1+\|T\|.
\]
This notion is closely related to various other concepts from Banach space theory, for instance to the numerical index: the numerical radius of an operator \(T:X \rightarrow X\) is defined by
\[
v(T):=\sup\{|x^*(Tx)|:x\in X,\ x^*\in X^*,\ \|x\|=\|x^*\|=1=x^*(x)\}
\]
and the numerical index of the space \(X\) is
\[
n(X):=\sup\{k\geq 0:v(T)\geq k\|T\| \;\mathrm{for\;all\;operators}\;T:X \rightarrow X\}.
\]
Obviously \(n(X)\leq 1\) and it is known that \(X\) has numerical index 1 if and only if the identity \(\mathrm{id}_X\) is a spear operator.

The equation \[ \max_{|\omega|=1}\|\mathrm{id}_X+\omega T\|=1+\|T\| \] is called the alternative Daugavet equation (aDE), while \[ \|\mathrm{id}_X+T\|=1+\|T\| \] is called the Daugavet equation (DE). The space \(X\) is said to have the Daugavet property/alternative Daugavet property (DP/aDP) if every rank-one operator \(T:X \rightarrow X\) satisfies the DE/aDE.

A sufficient geometrical condition for \(n(X)=1\), introduced by K. Boyko et al. [Math. Proc. Camb. Philos. Soc. Journal Profile 142, No. 1, 93–102 (2007; Zbl 1121.47001)] and called lushness, is the following: For all points \(x\) and \(y\) in the unit sphere of \(X\) and every \(\varepsilon>0\), there is some norm-one functional \(x^*\) on \(X\) such that \(y\in S(x^*,\varepsilon)\) and the distance of \(x\) to the absolutely convex hull of \(S(x^*,\varepsilon)\) is less than \(\varepsilon\), where \(S(x^*,\varepsilon)\) is the slice \[ \{z:\|z\|\leq 1,\,\mathrm{Re}\,x^*(z)>1-\varepsilon\}. \]

Ardalani [loc. cit.] also introduced the concept of numerical range with respect to a general operator \(G\) and proved some results analogous to those for the case \(G=\mathrm{id}_X\).

In the first chapter of the present book, the authors review some of the known results concerning Banach spaces with numerical index one, lushness, DP and aDP.

Chapter 2 deals with Ardalani’s general definition of spear vectors [loc. cit.] (\(G\) being a spear operator then just means that it is a spear vector in the space \(L(X,Y)\) of all bounded linear operators from \(X\) to \(Y\)). Here, the authors also introduce the new notion of spear set.

In Chapter 3, they further introduce the new concepts of lush operators and operators with the aDP (then the space \(X\) is lush/has the aDP if and only if \(\mathrm{id}_X\) is lush/has the aDP). Various results concerning spearness, lushness and the aDP for operators are proved.

Chapter 4 deals with some examples in classical Banach spaces. For instance, lushness of the Fourier transform on \(L^1\) is proved (on arbitrary locally compact abelian groups).

The fifth chapter contains some further results. For example, lush operators are characterised in the case that the domain has the Radon-Nikodým property or the codomain is an Asplund space.

Chapter 6 contains some isometric and isomorphic consequences of the properties under study. For instance, if \(X\) and \(Y\) are real Banach spaces and \(G:X \rightarrow Y\) is an operator with the aDP having infinite rank, then the dual of \(X\) contains an isomorphic copy of \(\ell^1\).

In Chapter 7, the authors consider a nonlinear generalisation of spear operators called Lipschitz spear operators.

Chapter 8 deals with some stability results for the properties under study (for instance in absolute sums and vector-valued \(C(K)\)-, \(L^1\)- and \(L^{\infty}\)-spaces).

Finally, some open problems are listed in Chapter 9.

This book will certainly be of interest to all researchers who specialise in Banach space theory.

The equation \[ \max_{|\omega|=1}\|\mathrm{id}_X+\omega T\|=1+\|T\| \] is called the alternative Daugavet equation (aDE), while \[ \|\mathrm{id}_X+T\|=1+\|T\| \] is called the Daugavet equation (DE). The space \(X\) is said to have the Daugavet property/alternative Daugavet property (DP/aDP) if every rank-one operator \(T:X \rightarrow X\) satisfies the DE/aDE.

A sufficient geometrical condition for \(n(X)=1\), introduced by K. Boyko et al. [Math. Proc. Camb. Philos. Soc. Journal Profile 142, No. 1, 93–102 (2007; Zbl 1121.47001)] and called lushness, is the following: For all points \(x\) and \(y\) in the unit sphere of \(X\) and every \(\varepsilon>0\), there is some norm-one functional \(x^*\) on \(X\) such that \(y\in S(x^*,\varepsilon)\) and the distance of \(x\) to the absolutely convex hull of \(S(x^*,\varepsilon)\) is less than \(\varepsilon\), where \(S(x^*,\varepsilon)\) is the slice \[ \{z:\|z\|\leq 1,\,\mathrm{Re}\,x^*(z)>1-\varepsilon\}. \]

Ardalani [loc. cit.] also introduced the concept of numerical range with respect to a general operator \(G\) and proved some results analogous to those for the case \(G=\mathrm{id}_X\).

In the first chapter of the present book, the authors review some of the known results concerning Banach spaces with numerical index one, lushness, DP and aDP.

Chapter 2 deals with Ardalani’s general definition of spear vectors [loc. cit.] (\(G\) being a spear operator then just means that it is a spear vector in the space \(L(X,Y)\) of all bounded linear operators from \(X\) to \(Y\)). Here, the authors also introduce the new notion of spear set.

In Chapter 3, they further introduce the new concepts of lush operators and operators with the aDP (then the space \(X\) is lush/has the aDP if and only if \(\mathrm{id}_X\) is lush/has the aDP). Various results concerning spearness, lushness and the aDP for operators are proved.

Chapter 4 deals with some examples in classical Banach spaces. For instance, lushness of the Fourier transform on \(L^1\) is proved (on arbitrary locally compact abelian groups).

The fifth chapter contains some further results. For example, lush operators are characterised in the case that the domain has the Radon-Nikodým property or the codomain is an Asplund space.

Chapter 6 contains some isometric and isomorphic consequences of the properties under study. For instance, if \(X\) and \(Y\) are real Banach spaces and \(G:X \rightarrow Y\) is an operator with the aDP having infinite rank, then the dual of \(X\) contains an isomorphic copy of \(\ell^1\).

In Chapter 7, the authors consider a nonlinear generalisation of spear operators called Lipschitz spear operators.

Chapter 8 deals with some stability results for the properties under study (for instance in absolute sums and vector-valued \(C(K)\)-, \(L^1\)- and \(L^{\infty}\)-spaces).

Finally, some open problems are listed in Chapter 9.

This book will certainly be of interest to all researchers who specialise in Banach space theory.

Reviewer: Jan-David Hardtke (Leipzig)

### MSC:

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

46B04 | Isometric theory of Banach spaces |

46B20 | Geometry and structure of normed linear spaces |

47A12 | Numerical range, numerical radius |

47A30 | Norms (inequalities, more than one norm, etc.) of linear operators |

47B37 | Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.) |

46B25 | Classical Banach spaces in the general theory |

46B22 | Radon-Nikodým, Kreĭn-Milman and related properties |

46J10 | Banach algebras of continuous functions, function algebras |