Hypercyclic operator

Let be $X$ {\ displaystyle X} $X$ Is a topological vector space (for example, a Banach space ). Linear Continuous Operator $T:X\rightarrow X$ ${\ displaystyle T: X \ rightarrow X}$ ${\ displaystyle T: X \ rightarrow X}$ called hypercyclic if an element exists $x\in X$ ${\ displaystyle x \ in X}$ $x \ in X$ such that many $\left\{T^{n}x,n=0,1,2,...\right\}$ ${\ displaystyle \ left \ {T ^ {n} x, n = 0,1,2, ... \ right \}}$ ${\ displaystyle \ left \ {T ^ {n} x, n = 0,1,2, ... \ right \}}$ tight in $X$ ${\ displaystyle X}$ $X$ . This item $x$ ${\ displaystyle x}$ $x$ called a hypercyclic vector for the operator $T$ ${\ displaystyle T}$ $T$ .

The concept of hypercyclicity is a special case of the broader concept of topological transitivity .

Examples

The first example of a hypercyclic operator was received by Birkhoff in 1929.

In 1969, Rolevich proved that the backward shift operator in space is hypercyclic $l^{2}$ ${\ displaystyle l ^ {2}}$ $l^2$ multiplied by a constant $\lambda :|\lambda |>1$ ${\ displaystyle \ lambda: | \ lambda |> 1}$ $\lambda :|\lambda |>1$ translating sequence $(a_{1},a_{2},a_{3},\ldots )\in l^{2}$ ${\ displaystyle (a_ {1}, a_ {2}, a_ {3}, \ ldots) \ in l ^ {2}}$ $(a_{1},a_{2},a_{3},\ldots )\in l^{2}$ in sequence $(\lambda a_{2},\lambda a_{3},\lambda a_{4},\ldots )\in l^{2}$ ${\ displaystyle (\ lambda a_ {2}, \ lambda a_ {3}, \ lambda a_ {4}, \ ldots) \ in l ^ {2}}$ $(\lambda a_{2},\lambda a_{3},\lambda a_{4},\ldots )\in l^{2}$ .

In 1988, Charles Reid came up with an example of an operator in a Banach space. $l^{1}$ ${\ displaystyle l ^ {1}}$ $l^{1}$ such that all its nonzero vectors are hypercyclic. This is a counterexample to the well-known problem of the existence of an invariant subspace for Banach spaces. For Hilbert spaces, the problem remains open.

Links

K.-G. Grosse-Erdmann. Universal families and hypercyclic operators.
Read, CJ (1988), " The invariant subspace problem for a class of Banach spaces, 2: hypercyclic operators ", Israel Journal of Mathematics T. 63 (1): 1–40, MR : 0959046 , ISSN 0021-2172 , DOI 10.1007 / BF02765019

Hypercyclic operator

Examples

Links

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