سیستم‌های دینامیکی خطی در فضاهای هیلبرت: ویژگی‌های معمول و مثال‌های صریح (یادداشت‌های انجمن ریاضی آمریکا)

Cover Title page Chapter 1. Introduction 1. General overview 2. Background and notations 2.1. Transitivity, mixing and weak mixing 2.2. Chaos 2.3. Ergodic-theoretic properties 2.4. Frequent hypercyclicity and 𝒰-frequent hypercyclicity 2.5. Properties related to eigenvalues 2.6. Distributional chaos 2.7. The parameter 𝑐(𝑇) 2.8. A last notation 3. Organization of the monograph Chapter 2. Typical properties of hypercyclic operators 1. The strong and strong^{*} topologies 1.1. Why 𝚂𝙾𝚃 and 𝚂𝙾𝚃*? 2. How to prove density results 3. Construction of mixing operators, and density of \gsmxh 4. Topological weak mixing and topological mixing 4.1. Some illustrations 5. Hypercyclic operators without eigenvalues 6. Hypercyclic operators without invariant measures 7. Densely distributionally chaotic operators 8. Summary Chapter 3. Descriptive set-theoretic issues 1. Complexity of the families 𝑇𝑀𝐼𝑋_{𝑀}()̉, \cmh, 𝑈𝐹𝐻𝐶_{𝑀}()̉∩𝐶𝐻_{𝑀}()̉ and 𝑈𝐹𝐻𝐶_{𝑀}()̉ 1.1. Complexity of 𝑇𝑀𝐼𝑋_{𝑀}()̉ 1.2. Complexity of \cmh 1.3. Complexity of 𝑈𝐹𝐻𝐶_{𝑀}()̉ and 𝑈𝐹𝐻𝐶_{𝑀}()̉∩𝐶𝐻_{𝑀}()̉ 2. Some non-Borel sets in \bmh Chapter 4. Ergodicity for upper-triangular operators 1. Definitions and setting 2. Perfect spanning is typical 3. Ergodicity vs ergodicity in the Gaussian sense 3.1. Statement of the main result 3.2. Preliminaries on the eigenvalues of 𝑇_{\la,\om} 3.3. Comeagerness of \pmb{𝒟}_{ℳ} 3.4. An auxiliary result 3.5. Comeagerness of \pmb{ℰ}_{ℳ} 3.6. Some comments on ergodicity and unimodular eigenvalues 4. Additional remarks 4.1. Some natural questions 4.2. More on the operators 𝑇_{\la,\om} Chapter 5. Periodic points at the service of hypercyclicity 1. Precompact orbits and topological mixing 2. Uniform recurrence and topological weak mixing 3. A criterion for 𝒰-frequent hypercyclicity 3.1. Uniform recurrence, almost periodic points and 𝒰-frequent hypercyclicity 3.2. More about 𝒰-frequent hypercyclicity and 𝒸(𝒯) 4. A criterion for frequent hypercyclicity 4.1. Link with the Operator Specification Property Chapter 6. Operators of \cct and of \cpt 1. Operators of \cct: basic facts 2. Operators of \cpt: how to be FHC or UFHC 3. Operators of \cct: how not to be FHC or UFHC 3.1. The main criterion, in abstract form 3.2. How to check the assumptions 4. Operators of \cct: how to be mixing or not mixing Chapter 7. Explicit counterexamples 1. Summary 2. Operators of \cput: FHC does not imply ergodic 2.1. How to be FHC or UFHC 2.2. A word about the OSP 2.3. FHC but not ergodic 3. Operators of \cpdt: UFHC does not imply FHC 3.1. How to be FHC or UFHC 3.2. UFHC but not FHC 4. Operators of \ccut: chaos plus mixing do not imply UFHC 4.1. How to be topologically mixing 4.2. How not to be UFHC 4.3. Chaotic and mixing operators which are not UFHC 5. Chaos plus mixing plus FHC do not imply ergodicity 6. \cct operators with few eigenvalues 7. \cput with many eigenvalues 8. Infinite direct sums of frequently hypercyclic operators Chapter 8. A few questions Short list of abbreviations Bibliography Back Cover Cover Title page Chapter 1. Introduction 1. General overview 2. Background and notations 2.1. Transitivity, mixing and weak mixing 2.2. Chaos 2.3. Ergodic-theoretic properties 2.4. Frequent hypercyclicity and U-frequent hypercyclicity 2.5. Properties related to eigenvalues 2.6. Distributional chaos 2.7. The parameter c(T) 2.8. A last notation 3. Organization of the monograph Chapter 2. Typical properties of hypercyclic operators 1. The strong and strong^{*} topologies 1.1. Why SOT and SOT*? 2. How to prove density results 3. Construction of mixing operators, and density of \gsmxh 4. Topological weak mixing and topological mixing 4.1. Some illustrations 5. Hypercyclic operators without eigenvalues 6. Hypercyclic operators without invariant measures 7. Densely distributionally chaotic operators 8. Summary Chapter 3. Descriptive set-theoretic issues 1. Complexity of the families TMIX_{M}()̉, \cmh, UFHC_{M}()̉∩CH_{M}()̉ and UFHC_{M}()̉ 1.1. Complexity of TMIX_{M}()̉ 1.2. Complexity of \cmh 1.3. Complexity of UFHC_{M}()̉ and UFHC_{M}()̉∩CH_{M}()̉ 2. Some non-Borel sets in \bmh Chapter 4. Ergodicity for upper-triangular operators 1. Definitions and setting 2. Perfect spanning is typical 3. Ergodicity vs ergodicity in the Gaussian sense 3.1. Statement of the main result 3.2. Preliminaries on the eigenvalues of T_{\la,\om} 3.3. Comeagerness of \pmb{D}_{M} 3.4. An auxiliary result 3.5. Comeagerness of \pmb{E}_{M} 3.6. Some comments on ergodicity and unimodular eigenvalues 4. Additional remarks 4.1. Some natural questions 4.2. More on the operators T_{\la,\om} Chapter 5. Periodic points at the service of hypercyclicity 1. Precompact orbits and topological mixing 2. Uniform recurrence and topological weak mixing 3. A criterion for U-frequent hypercyclicity 3.1. Uniform recurrence, almost periodic points and U-frequent hypercyclicity 3.2. More about U-frequent hypercyclicity and c(T) 4. A criterion for frequent hypercyclicity 4.1. Link with the Operator Specification Property Chapter 6. Operators of \cct and of \cpt 1. Operators of \cct: basic facts 2. Operators of \cpt: how to be FHC or UFHC 3. Operators of \cct: how not to be FHC or UFHC 3.1. The main criterion, in abstract form 3.2. How to check the assumptions 4. Operators of \cct: how to be mixing or not mixing Chapter 7. Explicit counterexamples 1. Summary 2. Operators of \cput: FHC does not imply ergodic 2.1. How to be FHC or UFHC 2.2. A word about the OSP 2.3. FHC but not ergodic 3. Operators of \cpdt: UFHC does not imply FHC 3.1. How to be FHC or UFHC 3.2. UFHC but not FHC 4. Operators of \ccut: chaos plus mixing do not imply UFHC 4.1. How to be topologically mixing 4.2. How not to be UFHC 4.3. Chaotic and mixing operators which are not UFHC 5. Chaos plus mixing plus FHC do not imply ergodicity 6. \cct operators with few eigenvalues 7. \cput with many eigenvalues 8. Infinite direct sums of frequently hypercyclic operators Chapter 8. A few questions Short list of abbreviations Bibliography Back Cover "We solve a number of questions pertaining to the dynamics of linear operators on Hilbert spaces, sometimes by using Baire category arguments and sometimes by constructing explicit examples. In particular, we prove the following results. (i) A typical hypercyclic operator is not topologically mixing, has no eigenvalues and admits no non-trivial invariant measure, but is densely distributionally chaotic. (ii) A typical upper-triangular operator with coefficients of modulus 1 on the diagonal is ergodic in the Gaussian sense, whereas a typical operator of the form "diagonal with coefficients of modulus 1 on the diagonal plus backward unilateral weighted shift" is ergodic but has only countably many unimodular eigenvalues; in particular, it is ergodic but not ergodic in the Gaussian sense. (iii) There exist Hilbert space operators which are chaotic and U-frequently hypercyclic but not frequently hypercyclic, Hilbert space operators which are chaotic and frequently hypercyclic but not ergodic, and Hilbert space operators which are chaotic and topologically mixing but not U-frequently hypercyclic. We complement our results by investigating the descriptive complexity of some natural classes of operators defined by dynamical properties"-- Provided by publisher