Table of Contents Return to Top =
Historical review
Summable families of 0 (a survey of point set topology)
I. Hilbert space
1. 1 semi-bilinear
1.2 Hermite
1.3 quasi-Hilbert space
1.4 internal product space
1.5 norm, distance and topology on inner product space
1.6 Hilbert space
1.7 standard orthogonal family
1.8 Hilbert dimension
Hilbert sum of 1.9 Hilbert space
Completeness of 1. 10 inner product space
Continuous linear operators on Hilbert space
2. General properties of1continuous linear operator
2.2 Some theorems about continuous linear operators
2.3 continuous linear functional
2.4 continuous bi-semilinear type
2.5 *** Yoke
.2.6 bicontinuous linear operator
2.7 eigenvalue
2.8 spectrum, relaxation formula
2.9 Strong Convergence and Weak Convergence of Linear Operators
Class Ⅲ Special Linear Operators
3. 1 common operator
3.2 Hermite operator
Order between Hermite operators
3.4 forecast
3.5 Decomposition of Identity Mapping
3.6 equidistant operator
3.7 Partial Isometric Operator
ⅳ compact operator
4. 1 compact operator
4.2 Hilbert? Schmidt operator
4.3 Spectral decomposition of normal compact operators
4.4 Application of Integral Equation
Spectral decomposition of ⅴ continuous Hermite operator
5. 1 continuous function calculus
5.2 Application: Polar decomposition of continuous linear operators
5.3 Extension of Function Calculus
5.4 Spectral decomposition of Hermite operator
5.5 Spectral decomposition of normal operators
5.6 Spectral decomposition of unitary operators
5.7 Standard Operators and Multiplication Operators
ⅵ Single parameter unitary operator group
6. Integral of1bounded function on unit mapping decomposition
6.2 Single Parameter Unitary Operator Group
6.3 Application: Botshner Theorem
refer to
Packaging label
postscript
Noun index
↓ Expand everything
Introduction Back to the Top The analysis of Hilbert space and the spectral theory of operators are indispensable tools in many branches of modern mathematics, physics and engineering science, especially in the following fields:
-Theory of partial differential equations;
-Quantum mechanics;
-signal processing;
-Ergodic theory.
John von neumann was one of the pioneers who realized the importance of Hilbert space analysis in quantum mechanics around 1930. Since then, the operator theory in Hilbert space has been developing continuously, and the needs of group representation theory, quantum field theory, quantum statistical mechanics and noncommutative geometry initiated and developed by AlainConnes since 1980s have provided a strong impetus for this development.
Jacques Dismier has a great influence in the field of operator algebra. In addition to his own important contributions in this field, he also made many efforts to spread the work of F.J. Murray and von Neumann. His monographs "Algebraic Operator" and "C * Algebraic Representation" have been necessary books for workers in this field all over the world for decades after their publication. The French school of operator algebra, which he founded and led for a long time, still has great influence in the world. He also directly or indirectly instructed a large number of graduate students. Not only that, he also has outstanding work in other fields of mathematics, such as the representation theory of Lie groups and the theory of envelope algebra. ..
Jacques Desmeyer is not only a great mathematician, but also a well-known excellent teacher. His Cours de Mathematiques du Premier Cycle (a college mathematics course, two volumes, the first volume has a Chinese translation by Higher Education Press) has been used by countless French students. In the master's stage, Jacques Dismier taught the operator spectrum theory in Hilbert space for many years at the University of Paris VI (also known as Pierre and Marie Curie University). The handwritten mimeographed handout he sent to the students is the manuscript of this book. Generations of French students have benefited from this.
Only very simple point set topology and basic knowledge of integral theory are needed. This course gives a clear, elegant and complete description of operator spectrum theory. After describing the basic tools of Hilbert space with elementary methods, all the basic results are gradually involved, until the spectral decomposition of self-yoke operators and the study of single-parameter unitary operator groups: these are some knowledge that all students who want to study mathematics or physics deeply must master.
Unfortunately, this manuscript was not published in France. We have reason to believe that the Chinese version translated by Yao Yijun, one of Jacques Desmeyer's disciples, will benefit a large number of readers in China. This book is bound to become a desk book for teachers, students and researchers in this field.
Claire Anata Hamann-Drahos
Professor, University of Orleans, France ...