Abstract
In this paper, we investigate analytic automorphisms of complex topological vector spaces and their applications to linear and nonlinear transitive operators. We constructed some examples of polynomial automorphisms that show that a natural analogue of the Jacobian Conjecture for infinite dimensional spaces is not true. Also, we prove that any separable Fréchet space supports a transitive analytic operator that is not a polynomial. We found some connections of analytic automorphisms and algebraic bases of symmetric polynomials and applications to hypercyclicity of composition operators.
Highlights
Let X be a complex topological vector space
We found some interesting relations between analytic automorphisms and algebraic bases of symmetric polynomials on Banach spaces and used it for hypercyclic composition operators in spaces of analytic functions
Using the resources in [16] results in the Fréchet algebra of all entire analytic functions H (E ) on E being isometric to the algebra of bounded type symmetric analytic functions Hbs ( L∞ [0, 1]), we show that the composition operator f ( x ) 7→ f ( x + a), f ∈ H (E ) is hypercyclic in H (E )
Summary
Let X be a complex topological vector space. A mapping F : X → X is called an analytic automorphism if F is analytic, bijective and F −1 is analytic. We consider polynomial and analytic automorphisms of topological vector spaces and their applications to linear and nonlinear dynamics. A continuous mapping T : X → X is called hypercyclic on a topological vector space X if there is an element x0 ∈ X for which the orbit under T, Orb ( T, x0 ) = { x0 , Tx0 , T 2 x0 , . ∂Pi necessary an analytic automorphism even if its Jacobi operator is a continuous bijection This gives us a contrast with the finite dimensional case, where the Jacobian Conjecture remains to be unsolved. For details on analytic mappings on topological vector spaces we refer the reader to [23]
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