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].

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