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公共不變子空間與緊型條件(英) 版權(quán)信息
- ISBN：9787030687128
- 條形碼：9787030687128 ; 978-7-03-068712-8
- 裝幀：一般膠版紙
- 冊(cè)數(shù)：暫無(wú)
- 重量：暫無(wú)
- 所屬分類：>
公共不變子空間與緊型條件(英) 內(nèi)容簡(jiǎn)介
主要總結(jié)了算子集合的不變子空間性質(zhì)，以及類緊算元的相關(guān)結(jié)果。在算子理論中，我們把緊的擬冪零算子稱為Volterra算子。由Volterra算子組成的集合亦稱為Volterra集合，如Volterra半群，Volterra代數(shù)等。在本書的部分，我們主要討論Volterra半群，Volterra李代數(shù)，Volterra約當(dāng)代數(shù)的不變子空間問(wèn)題，這些問(wèn)題都曾經(jīng)是算子理論、算子李代數(shù)中的經(jīng)典公開問(wèn)題，在1999-2005年左右得以解決，收錄于本書部分。在本書的第二部分，我們討論了冪零李代數(shù)生成Banach代數(shù)是否為Engel代數(shù)的這一公開問(wèn)題，這也是算子李代數(shù)的經(jīng)典問(wèn)題，至今尚未接近解決，相關(guān)部分結(jié)果收錄于第五章，隨后我們把緊算子的相關(guān)性質(zhì)向Banach代數(shù)中類緊元集合推廣，給出了離散根的定義和性質(zhì)，很后，我們給出了離散根的擾動(dòng)理論，這從經(jīng)典的算子理論中的擾動(dòng)理論刻畫了離散根的本質(zhì)。除本人研究成果外，本文亦收錄了有名算子理論學(xué)者Shulman,Turovskii,Kennedy等專家的從1999到2019年的相關(guān)成果。
公共不變子空間與緊型條件(英) 目錄
Contents
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
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
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公共不變子空間與緊型條件(英) 節(jié)選
Part I Preliminaries Chapter 1 Banach Algebras 1.1 Jacobson radical and derivation In this book， we assume that every vector space is over the complex field C. A complex associative algebra is a vector space A over the complex field C， with a multiplication satisfying the following properties: x(yz) = (xy)z， x(y + z) = xy + xz， (y + z)x = yx + zx， λ(xy) = (λx)y = x(λy)， for all x， y， z 2 A and λ 2 C. If moreover A is a normed space for a norm ||.|| and satisfies the norm inequality for all x， y∈A， we say that A is a normed algebra. Furthermore， If A is a Banach space， we say that A is a Banach algebra. If there is an element in A， denoted by 1， with 1x = x1 = x， for every x ∈A. Then A is called unital， and 1 is called the unit. If a normed algebra A is not unital， it is always possible to imbed it isometrically in the normed algebra with unit as in [1， Chapter III， Section 1]. In the following of this section， let A be a unital Banach algebra. For some x 2 A， if there is y∈ A， such that xy = yx = 1， then we call x is invertible in A. The set of all the invertible elements in A is denoted by G(A). Then we can define the spectrum of x in A， denoted by σA(x) (or σ(x) for brief) as follows. It is well known that σ(x) is nonempty and compact by [1， Theorem 3.2.8]. The spectral radius of x in A， denoted by ρA(x) (or ρ(x) for brief) is defined by . Then by Gelfand’s Theorem [1， Theorem 3.2.8]. If ρ(x) = 0， then x is called quasinilpotent. We also need the holomorphic functional calculus， which is also called Riesz functional calculus. One can find the information in [1， Chapter III， Section 3].
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