公共不变子空间与紧型条件

本书一书主要总结了算子集合的不变子空间性质，以及类紧算元的相关结果。在算子理论中，我们把紧的拟幂零算子称为Volterra算子。由Volterra算子组成的集合亦称为Volterra集合，如Volterra半群，Volterra代数等。在《公共不变子空间与紧型条件》的第一部分，我们主要讨论Volterra半群，Volterra李代数，Volterra约当代数的不变子空间问题，这些问题都曾经是算子理论、算子李代数中的经典公开问题，在1999-2005年左右得以解决，收录于《公共不变子空间与紧型条件》第一部分。在《公共不变子空间与紧型条件》的第二部分，我们讨论了幂零李代数生成Banach代数是否为Engel代数的这一公开问题，这也是算子李代数的经典问题，至今尚未接近解决，相关部分结果收录于第五章，随后我们把紧算子的相关性质向Banach代数中类紧元集合推广，给出了离散根的定义和性质，最后，我们给出了离散根的扰动理论，这从经典的算子理论中的扰动理论刻画了离散根的本质。除本人研究成果外，《公共不变子空间与紧型条件》亦收录了有名算子理论学者Shulman，Turovskii，Kennedy等专家的从1999到2019年的相关成果。

Preface
Notation
Part I Preliminaries
Chapter 1 Banach Algebras 3
1.1 Jacobson radical and derivation 3
1.2 Analytic properties of the spectrum 5
1.3 Representation theory 6
Chapter 2 Operator Theory 8
2.1 Compact operators 8
2.2 Riesz and scattered operators 10
2.3 Decomposable operator 11
Chapter 3 Lie Algebras 15
3.1 Nilpotent and solvable Lie algebras 15
3.2 Engel algebras 17
3.3 Semisimple Lie algebras 20
Part II Beger-Wang Formulas and Applications
Chapter 4 Joint Spectral Radius 23
4.1 Preliminary properties 23
4.2 Joint quasinilpotence 26
4.3 Analytic properties 29
4.4 Hausdorff measure 31
4.5 Hausdorff and essential spectral radii 32
Chapter 5 Topological Radicals 36
5.1 Preliminary properties 36
5.2 Compactly quasinilpotent radical 37
5.3 Hypocompact radical 45
5.4 The radical rad ^ Rhc 50
Chapter 6 Beger-Wang Formula and Applications 52
6.1 Compactly quasinilpotence 52
6.2 Joint spectral radius on complete chain case 58
6.3 Beger-Wang formula 60
6.4 Coincidence of Hausdorff and essential radii 70
Chapter 7 Generalized Beger-Wang Formulas and Applications 75
7.1 Mixed GBWF 75
7.2 Operator GBWF 80
7.3 Banach algebraic GBWF 81
7.4 Volterra Lie algebra problem 83
Notes 90
Part III Volterra Ideal Theorem and Applications
Chapter 8 Elementary Spectral Manifolds 95
8.1 Preliminary properties 95
8.2 Algebraic and spatial formulas 99
8.3 Applications to scattered operators 103
Chapter 9 Volterra Ideal Problem 112
9.1 A reducibility criterion 112
9.2 Quasi-commutant and quasi-center 114
9.3 Solution of Volterra ideal problem 118
Chapter 10 Lie Algebras of Compact Operators 124
10.1 Engel and E-solvable ideals 124
10.2 ad-compact element 128
10.3 Largest E-solvable ideal 130
Chapter 11 Ad-Compact Lie Algebras 136
11.1 The largest Engel ideal 136
11.2 Irreducible representations by compact operators 138
11.3 E-solvable algebras and E-radical 141
Notes 149
Part IV Lie Algebras Generated by Special Operators
Chapter 12 Essentially Nilpotent Lie Algebras 153
12.1 Two Problems on operator Lie algebras 153
12.2 Nilpotent Lie algebras generated by decomposable operators 154
12.3 Lie algebras generated by quasinilpotent operators 156
12.4 Compact quasinilpotence 159
Chapter 13 Lie Algebras Generated by Operators on Hilbert Spaces 162
13.1 Finite dimensional selfadjoint Lie algebras 162
13.2 Finite dimensional semisimple Lie algebras 166
13.3 Selfadjoint ad-compact E-solvable Lie algebras 170
Chapter 14 Lie Algebras Generated by Jordan Operators 172
14.1 Lie algebras generated by normal operators 172
14.2 Lie algebras generated by Jordan operators 175
Chapter 15 Lie Algebras Generated by Riesz Operators 180
15.1 Engel Lie algebras 180
15.2 E-solvable Lie algebras 183
15.3 Applications to polynomially compact operators 189
Notes 191
Bibliography 192
Index 195