Alpay et al: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis


by Daniel Alpay, Maria Elena Luna-Elizarrarás, Michael Shapiro, Daniele C. Struppa

  • Series: SpringerBriefs in Mathematics
  • Paperback: 95 pages
  • Publisher: Springer; 2014 edition (April 30, 2014)
  • Language: English
  • ISBN-10: 3319051091
  • ISBN-13: 978-3319051093
  • Price: $46.51
  • Same (?) arxiv book: http://arxiv.org/pdf/1304.0781v1.pdf

Description: This book provides the foundations for a rigorous theory of functional analysis with bicomplex scalars. It begins with a detailed study of bicomplex and hyperbolic numbers and then defines the notion of bicomplex modules. After introducing a number of norms and inner products on such modules (some of which appear in this volume for the first time), the authors develop the theory of linear functionals and linear operators on bicomplex modules. All of this may serve for many different developments, just like the usual functional analysis with complex scalars and in this book it serves as the foundational material for the construction and study of a bicomplex version of the well known Schur analysis.

Table of contents (PDF):

1 Bicomplex and Hyperbolic Numbers. . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Bicomplex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Conjugations and Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 The Euclidean Norm on BC . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.4 Idempotent Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.5 A Partial Order on D and a Hyperbolic-Valued Norm . . . . . . . . 11

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Bicomplex Functions and Matrices . . . . . . . . . . . . . . . . . . . . . . . . 19

2.1 Bicomplex Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . 19

2.2 Bicomplex Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 BC-Modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 BC-Modules and Involutions on them . . . . . . . . . . . . . . . . . . . 31

3.2 Constructing a BC-Module from Two Complex

Linear Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

4 Norms and Inner Products on BC-Modules. . . . . . . . . . . . . . . . . . 37

4.1 Real-Valued Norms on Bicomplex Modules . . . . . . . . . . . . . . . 37

4.2 D-Valued Norm on BC-Modules . . . . . . . . . . . . . . . . . . . . . . . 39

4.3 Bicomplex Modules with Inner Product . . . . . . . . . . . . . . . . . . 42

4.4 Inner Products and Cartesian Decompositions . . . . . . . . . . . . . . 47

4.5 Inner Products and Idempotent Decompositions . . . . . . . . . . . . 49

4.6 Complex Inner Products on X Induced

by Idempotent Decompositions . . . . . . . . . . . . . . . . . . . . . . . . 53

4.7 The Bicomplex Module BCn. . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.8 The Ring HðCÞ of Biquaternions as a BC-Module . . . . . . . . . . 58

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5 Linear Functionals and Linear Operators on BC-Modules. . . . . . . 63

5.1 Bicomplex Linear Functionals. . . . . . . . . . . . . . . . . . . . . . . . . 63

5.2 Polarization Identities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.3 Linear Operators on BC-Modules . . . . . . . . . . . . . . . . . . . . . . 73

Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Schur Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.1 A Survey of Classical Schur Analysis . . . . . . . . . . . . . . . . . . . 79

6.2 The Bicomplex Hardy Space. . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.3 Positive Definite Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

6.4 Schur Multipliers and Their Characterizations

in the BC-Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.5 An Example: Bicomplex Blaschke Factors . . . . . . . . . . . . . . . . 91

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Source: http://www.amazon.com/Functional-Analysis-Bicomplex-SpringerBriefs-Mathematics/dp/3319051091, http://arxiv.org/pdf/1304.0781v1.pdf

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