Book contents
- Frontmatter
- Contents
- Introduction
- 0 Banach space background
- 1 Finite rank operators: trace and 1-nuclear norm
- 2 Finite sequences of elements: the quantities μ1, μ2
- 3 The summing norms
- 4 Other nuclear norms: duality with the summing norms
- 5 Pietsch's theorem and its applications
- 6 Averaging; type 2 and cotype 2 constants
- 7 More averaging: Khinchin's inequality and related results
- 8 Integral methods; Gaussian averaging
- 9 2-dominated spaces
- 10 Grothendieck's inequality
- 11 The interpolation method for Grothendieck-type theorems
- 12 Results connected with the basis constant
- 13 Estimation of summing norms using a restricted number of elements
- 14 Piseer's theorem for π2, 1
- 15 Tensor products of operators
- 16 Trace duality revisited: integral norms
- 17 Applications of local reflexivity
- 18 Cone-summing norms
- References
- List of symbols
- Index
18 - Cone-summing norms
Published online by Cambridge University Press: 12 October 2009
- Frontmatter
- Contents
- Introduction
- 0 Banach space background
- 1 Finite rank operators: trace and 1-nuclear norm
- 2 Finite sequences of elements: the quantities μ1, μ2
- 3 The summing norms
- 4 Other nuclear norms: duality with the summing norms
- 5 Pietsch's theorem and its applications
- 6 Averaging; type 2 and cotype 2 constants
- 7 More averaging: Khinchin's inequality and related results
- 8 Integral methods; Gaussian averaging
- 9 2-dominated spaces
- 10 Grothendieck's inequality
- 11 The interpolation method for Grothendieck-type theorems
- 12 Results connected with the basis constant
- 13 Estimation of summing norms using a restricted number of elements
- 14 Piseer's theorem for π2, 1
- 15 Tensor products of operators
- 16 Trace duality revisited: integral norms
- 17 Applications of local reflexivity
- 18 Cone-summing norms
- References
- List of symbols
- Index
Summary
Elementary theory
Ideas connected with positivity have permeated a good deal of the work in this book. For operators defined on a normed lattice, it is natural to consider a “summing” norm that is defined in a way that pays attention to the order structure. The simplest way to do this is to restrict to positive elements in the definition of φ1. The resulting “cone-summing” norm gives rise to a theory that parallels closely (and in places more simply) the most successful parts of the theory of 1-summing and 2-summing norms. It also provides a proper setting for our sporadic earlier remarks on positive operators. The concept and the basic results are due to Schlotterbeck (1971).
To set the scene, we need a few very elementary concepts and results relating to normed lattices. The definition was given in Section 0. The set {x: x ≥ 0) in a linear lattice X is called the positive cone, and will be denoted by X+. The supremum of the two elements x, y is denoted by x ∨ y, the infimum by x ∧ y.
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- Chapter
- Information
- Summing and Nuclear Norms in Banach Space Theory , pp. 160 - 169Publisher: Cambridge University PressPrint publication year: 1987