Tensor Norms and Operator Ideals (eBook)

A. Defant, K. Floret (Autoren)

eBook Download: PDF

1992 | 1. Auflage
565 Seiten
Elsevier Science (Verlag)
978-0-08-087287-2 (ISBN)

Infos

The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resumé, and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exercises.

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The three chapters of this book are entitled Basic Concepts, Tensor Norms, and Special Topics. The first may serve as part of an introductory course in Functional Analysis since it shows the powerful use of the projective and injective tensor norms, as well as the basics of the theory of operator ideals. The second chapter is the main part of the book: it presents the theory of tensor norms as designed by Grothendieck in the Resume and deals with the relation between tensor norms and operator ideals. The last chapter deals with special questions. Each section is accompanied by a series of exercises.

Inhalt

Front Cover 1
Tensor Norms and Operator Ideals 4
Copyright Page 5
Contents 6
Introduction 14
Chapter I: Basic Concepts 20
1. Bilinear Mappings 20
2. The Algebraic Theory of Tensor Products 28
3. The Projective Norm 39
4. The Injective Norm 59
5. The Approximation Property 71
6. Duality of the Projective and Injective Norm 83
7. The Natural Norm on the p–Integrable Functions 90
8. Absolutely and Weakly p–Summable Series and Averaging Techniques 103
9. Operator Ideals 121
10. Integral Operators 131
11. Absolutely p–Summing Operators 140
Chapter II: Tensor Norms. 159
12. Definition and Examples 159
13. The Five Basic Lemmas 172
14. Grothendieck’s Inequality 179
15. Dual Tensor Norms 190
16. The Bounded Approximation Property 203
17. The Representation Theorem for Maximal Operator Ideals 213
18. ( p , q)–Factorable Operators 236
19. ( p , q)–Dominated Operators 254
20. Projective and Injective Tensor Norms 263
21. Accessible Tensor Norms and Operator Ideals 288
22. Minimal Operator Ideals 300
23. Lgp–Spaces 313
24. Stable Measures 327
25. Composition of Accessible Operator Ideals 340
26. More About Lp and Hilbert Spaces 357
27. Grothendieck’s Fourteen Natural Norms 374
Chapter III: Special Topics 378
28. More Tensor Norms 378
29. The Calculus of Traced Tensor Norms 391
30. The Vector Valued Fourier Transform 407
31. Pisier’s Factorization Theorem 420
32. Mixing Operators 428
33. The Radon–Nikodým Property for Tensor Norms and Reflexivity 443
34. Tensorstable Operator Ideals 458
35. Tensor Norm Techniques for Locally Convex Spaces 482
Appendices: 502
A. Some Structural Properties of Banach Spaces 502
B. Integration Theory 506
C. Representable Operators 521
D. The Radon–Nikodým Property 530
Bibliography 540
List of Symbols 558
Index 568

Textauszug

Chapter I

Basic Concepts

After some remarks on bilinear mappings and the introduction of algebraic tensor products, the projective and injective tensor norms and some of their most important properties are studied in sections 3 – 6. Vector valued p–integrable functions, summability of sequences and averaging methods are treated in section 7 and section 8 – always from the perspective of tensor products. The remaining sections are devoted to the basics of the theory of operator ideals and integral and absolutely p-summing operators between Banach spaces, culminating in the so–called little Grothendieck theorem.

The reader should be familiar with some fundamental tools from Banach space theory, such as the Hahn–Banach theorem, the Mackey theorem / uniform boundedness principle, the weak– and weak–*–topologies, continuous linear operators and the open mapping and closed graph theorems. This clearly includes some simple knowledge about the classical Banach spaces C(K) and Lp, as well as the sequence spaces co and ℓp. In Appendix A some additional information about the structure theory of Banach spaces is collected.

1. Bilinear Mappings

This first section treats bilinear mappings and puts special emphasis on the fact that in many respects they behave differently than linear mappings, although they are intimately related with them.

1.1.

Let E, F, G be vector spaces over the same scalar field = or of the real or complex numbers. A mapping

: E × F → G

is called bilinear if the mappings

x Φ : F → G ? ? ? ? ? ? ? ? ? ? ? and ? Φ y : E → G ? ? ? ? ? ? y ⇝ Φ ( x , y ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? x ⇝ Φ ( x , y )

are linear for each x ∈ E and y ∈ F, in symbols : Φ ∈ Bil(E, F; G) if

x Φ ∈ L ( F , G ) ? and ? Φ y ∈ L ( E , G )

for each x ∈ E and y ∈ F. For simplicity: Bil(E, F) Bil(E, F; ). If E, F, G are normed spaces (or more generally: topological vector spaces), the set of continuous bilinear mappings E × F → G will be denoted by il(E, F; G) and il(E, F) if G = . From

( x , y ) − Φ ( x o , y o ) = Φ ( x − x o , y − y o ) + Φ ( x − x o , y o ) + Φ ( x o , y − y o )

the following result is easily deduced:

Proposition

For Φ ∈ Bil(E, F; G) the following assertions are equivalent:

(a)  Φ is continuous.

(b)  Φ is continuous at (0, 0).

(c)  There is a constant c ≥ 0 with ‖ Φ(x, y)‖G ≤ c‖x‖E‖y‖F for all (x, y) ∈ E × F.

It is easy to see that

Φ ‖ : = min { c ≥ 0 | c ? as ? in (c)}=sup{ ‖ Φ ( x , y ) ‖ G | x ∈ B E , y ∈ B F }

defines a norm on il(E, F; G) which is even a complete norm if ‖ ·‖G is. Note that continuous bilinear mappings are not uniformly continuous since, e.g., the restriction of 2 → , (x, y) xy to the diagonal is the function ∈ x x2 ∈ .

1.2.

A bilinear mapping Φ ∈ Bil(E, F; G) is separately continuous if all xΦ : F → G and Φy : E → G are continuous.

Theorem

Let E, F, G be normed spaces and E complete. Every separately continuous bilinear mapping Φ ∈ Bil(E, F; G) is continuous.

Proof

The set

: = { z ′ ∘ Φ y | z ′ ∈ B G ′ , y ∈ B F } ⊂ E ′

is σ(E′, E)–bounded since for each x ∈ E

〈 z ′ ∘ Φ y , x 〉 | = | 〈 z ′ , Φ ( x , y ) 〉 | ≤ ‖ z ′ ‖ ‖ Φ ( x , y ) ‖ ≤ ‖ x Φ ‖ .

Mackey's theorem / the uniform boundedness principle shows that D is uniformly bounded, i.e. there is a c ≥ 0 such that for all z′ ∈ BG′ and y ∈ BF

〈 z ′ , Φ ( x , y ) 〉 | = | 〈 z ′ ∘ , Φ y , x 〉 | ≤ c ‖ x ‖ E for all  x ∈ E .

It follows that ‖Φ‖ ≤ c.

1.3.

Some examples of bilinear mappings:

(1)  For x′ ∈ E′ and y′ ∈ F′

x ′ ⊗ – y ′ ] ( x , y ) : = 〈 x ′ , x 〉 〈 y ′ , y 〉

defines a continuous bilinear form and x ′ ⊗ _ y ′ ‖ = ‖ x ′ ‖ ‖ y ′ ‖ . If xn′ and yn′ are in the unit balls and (λn) ∈ ℓ1, then

( x , y ) : = ∑ n = 1 ∞ λ n [ x ′ n ⊗ _ y ′ n ] ( x , y )

is well-defined and φ ‖ ≤ ∑ ∞ n = 1 | λ n | ; bilinear forms of this kind are called nuclear bilinear forms.

(2)  The evaluation map on the space (E, F) of continuous linear operators

( E , F ) × E → F ? ? ? ? ? ? ? ? ? ? ? ( T , x ) ⇝ T x

has norm 1 (if E and F are not trivial).

(3)  If E and F are finite dimensional, then all bilinear mappings E × F → G are continuous (use bases).

(4)  The convolution mapping

1 ( ℝ ) × L 1 ( ℝ ) → L 1 ( ℝ ) ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ( f , g ) ⇝ f * g

is bilinear.

(5)  Take the continuous functions on a compact space K and E a normed space. Then

( K ) × E → C ( K , E ) ? ? ? ? ? ? ? ? ? ( f , x ) ⇝ f ( ⋅ ) x

is bilinear.

1.4.

The mappings

i l ( E , F ) → L ( E , F * ) L ( E , F * ) → B i l ( E , F ) ? ? ? ? ? ? ? φ ⇕ ? ? ? ? ? ? ? L φ ? ? ? ? ? ? ? T ⇕ ? ? ? ? ? ? β T ? ? 〈 L φ x , y 〉 : = φ ( x , y ) β T ( x , y ) : = 〈 T x , y 〉

give isomorphisms of vector spaces and are inverse to each other. Since

φ ‖ = sup { | φ ( x , y ) | | x ∈ B E , y ∈ B F } sup { ‖ L φ x ‖ | x ∈ B E } = ‖ L φ ‖ ∈ [ 0 , ∞ ] ,

this isomorphism reduces for continuous bilinear forms to an isometry of normed spaces

il( E , F ) = 1 L ( E , F ′ ) ‖ L φ ‖ = ‖ φ ‖ ? and ? ‖ β T ‖ = ‖ T ‖.

This relationship is basic for the understanding of the ideas which will be presented in this book: the continuous bilinear forms on E × F are exactly the continuous operators E →F.′

1.5.

Since there is no Hahn-Banach theorem for operators, there is none for bilinear continuous forms in the following sense: Let G ⊂ E be a subspace and φ ∈ il(G, F); does there exist an extension ˜ ...

Produkt-Details

• Erscheint lt. Verlag: 26.11.1992
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Algebra
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie
• Themenwelt: Technik
• ISBN-10: 0-08-087287-5 / 0080872875
• ISBN-13: 978-0-08-087287-2 / 9780080872872