In
mathematics, especially
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 (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, a hypercyclic operator on a
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
''X'' is a
bounded linear operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vect ...
''T'': ''X'' → ''X'' such that there is a vector ''x'' ∈ ''X'' such that the sequence is
dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematically ...
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
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
spaces, but 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
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''.
The concept appear ...
), and ''universality''. ''Universality'' in general involves a set of mappings from one
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
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
Józef Marcinkiewicz
Józef Marcinkiewicz (; 30 March 1910 in Cimoszka, near Białystok, Poland – 1940 in Katyn, USSR) was a Polish mathematician.
He was a student of Antoni Zygmund; and later worked with Juliusz Schauder, Stefan Kaczmarz and Raphaël Salem. H ...
, or MacLane in 1952. However, it was not until the 1980s when hypercyclic operators started to be more intensively studied.
Examples
An example of a hypercyclic operator is two times the backward
shift operator
In mathematics, and in particular functional analysis, the shift operator also known as translation operator is an operator that takes a function
to its translation . In time series analysis, the shift operator is called the lag operator.
Shift ...
on the
ℓ2 sequence space, that is the operator, which takes a sequence
:(''a''
1, ''a''
2, ''a''
3, …) ∈ ℓ
2
to a sequence
:(2''a''
2, 2''a''
3, 2''a''
4, …) ∈ ℓ
2.
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 ''T''
''n''''x'' is hypercyclic as well, so there is always a dense set of hypercyclic vectors.
* Moreover, the set of hypercyclic vectors is a
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
''G''δ set, and always contains a dense
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ...
, up to .
* constructed an operator on
ℓ1, such that all the non-zero vectors are hypercyclic, providing a counterexample to the
invariant subspace problem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many va ...
(and even ''invariant subset problem'') in the class of Banach spaces. The problem, whether such an operator (sometimes called ''hypertransitive'', or ''orbit transitive'') exists on a separable Hilbert space, is still open (as of 2014).
References
*
*
*
*{{Citation , last1=Grosse-Erdmann , first1=Karl-Goswin , title=Universal families and hypercyclic operators , doi=10.1090/S0273-0979-99-00788-0 , mr=1685272 , year=1999 , journal=Bulletin of the American Mathematical Society , series=New Series , issn=1088-9485 , volume=36 , issue=3 , pages=345–381, doi-access=free
See also
*
Topological mixing
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''.
The concept appear ...
Functional analysis
Operator theory
Invariant subspaces