Hypercyclic operator
Encyclopedia
In mathematics
, especially functional analysis
, a hypercyclic operator on a Banach space
X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense
in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector.
There is no hypercyclic operator in finite-dimensional
spaces, but interestingly, the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic.
The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space
to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity. Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Marcinkiewicz, or MacLane in 1952. However, it was not before 1980s when the hypercyclic operators started to be hugely studied.
on the ℓ2 sequence space, that is the operator, which takes a sequence
∈ ℓ2
to a sequence
∈ ℓ2.
This was proved in 1969 by Rolewicz.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, especially functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, a hypercyclic operator on a Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
X is a bounded linear operator T: X → X such that there is a vector x ∈ X such that the sequence {Tn x: n = 0, 1, 2, …} is dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in the whole space X. In other words, the smallest closed invariant subset containing x is the whole space. Such an x is then called hypercyclic vector.
There is no hypercyclic operator in finite-dimensional
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
spaces, but interestingly, the property of hypercyclicity in spaces of infinite dimension is not a rare phenomenon: many operators are hypercyclic.
The hypercyclicity is a special case of broader notions of topological transitivity (see topological mixing), and universality. Universality in general involves a set of mappings from one topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
to another (instead of a sequence of powers of a single operator mapping from X to X), but has a similar meaning to hypercyclicity. Examples of universal objects were discovered already in 1914 by Julius Pál, in 1935 by Marcinkiewicz, or MacLane in 1952. However, it was not before 1980s when the hypercyclic operators started to be hugely studied.
Examples
An example of a hypercyclic operator is two times the backward shift operatorShift operator
In mathematics, and in particular functional analysis, the shift operator or translation operator is an operator that takes a function to its translation . In time series analysis, the shift operator is called the lag operator....
on the ℓ2 sequence space, that is the operator, which takes a sequence
∈ ℓ2
to a sequence
∈ ℓ2.
This was proved in 1969 by Rolewicz.
Known results
- On every infinite-dimensional separable Banach space there is a hypercyclic operator. On the other hand, there is no hypercyclic operator on a finite-dimensional space, nor on a non-separable Banach space.
- If x is a hypercyclic vector, then Tnx is hypercyclic as well, so there is always a dense set of hypercyclic vectors.
- Moreover, the set of hypercyclic vectors is a connectedConnectednessIn mathematics, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected...
Gδ set, and always contains a dense vector spaceVector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
, up to {0}. constructed an operator on ℓ1, such that all the non-zero vectors are hypercyclic, providing a counterexample to the invariant subspace problem (and even invariant subset problem) in the class of Banach spaces. The problem, whether such an operator (sometimes called hypertransitive, or orbit transitive) exists, is still open (as of 2010).