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Universal families and hypercyclic operators. (English) Zbl 0933.47003

From the author’s introduction: Analysis is the study of limiting processes. Not every limiting process, of course, converges, but examples have been found where processes diverge in a maximal way. Such an extreme behaviour is often linked with the phenomenon of universality, which constitutes the topic of this survey.
From an abstract point of view, the phenomenon of universality may be described as follows. We have a topological space \(X\) of objects, a topological space \(Y\) of elements to be approximated and a family (usually a sequence) \(T_\iota: X\to Y\) \((\iota\in I)\) of mappins. Then an object \(x\) in \(X\) is called universal if every element \(y\) in \(Y\) can be approximated by certain \(T_\iota x\), that is, if the set \(\{T_\iota x:\iota\in I\}\) is dense in \(Y\).
Author’s table of contents:
Introduction.
Part I. General Theory.
1. Universal families. 1a. The Universality Criterion; 1b. The set of universal elements; 1c. Universality in linear spaces; 1d. Universal series.
2. Hypercyclic operators. 2a. The Hypercyclicity criterion; 2b. The set of hypercyclic vectors; 2c. Derived hypercyclicity; 2d. Existence of hypercyclic operators.
Part II. Specific Universal Families and Hypercyclic Operators.
3. The real analysis setting. 3a. Universal power and Taylor series; 3b. Universal primitives; 3c. Universal orthogonal series; 3d. Universal series for convergence a.e.; 3e. Further real universalities and hypercyclicities.
4. The complex analysis setting. 4a. Universal and hypercyclic composition operators; 4. Holomorphic monsters; 4c. Hypercyclic differential operators; 4d. Universal power and Taylor series; 4e. Universal matrices.
5. Hypercyclic operators in classical Banach spaces.
6. A concrete universal object: the Riemann zeta-function.
References; Bibliography.

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47A65 Structure theory of linear operators
54H99 Connections of general topology with other structures, applications
47A35 Ergodic theory of linear operators
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
54-02 Research exposition (monographs, survey articles) pertaining to general topology
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References:

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