On a Banach space X, a bounded linear operator A is a called subspace convex-cyclic associated with W as a subspace, if the set is dense in W for a vector . In this work, we use Hahn- Banach Theorem to show that the extending linear functional preserve subspace convex-cyclic operator property. Also, the algebraic structures of subspace convex-cyclic vectors can be determined, such as the spectrum.
In our work we have defined an operator called subspace convex-cyclic operator. The property of this newly defined operator relates eigenvalues which have eigenvectors of modulus one with kernels of the operator. We have also illustrated the effect of the subspace convex-cyclic operator when we let it function in linear dynamics and joining it with functional analysis. The work is done on infinite dimensional spaces which may make linear operators have dense orbits. Its property of measure preserving puts together probability space with measurable dynamics and widens the subject to ergodic theory. We have also applied Birkhoff’s Ergodic Theorem to give a modified version of subspace convex-cyclic operator. To work on a separable infinite Hilbert space, it is important to have Gaussian invariant measure from which we use eigenvectors of modulus one to get what we need to have. One of the important results that we have got from this paper is the study of Central Limit Theorem. We have shown that providing Gaussian measure, Central Limit Theorem holds under the certain conditions that are given to the defined operator. In general our work is theoretically new and is combining three basic concepts dynamical system, operator theory and ergodic theory under the measure and statistics theory.
Let H be an infinite dimensional real or complex separable Hilbert space. We introduce a special type of a bounded linear operator T and study its important relation with the invariant subspace problem on H: the operator T is said to be subspace convex-cyclic for a subspace M if there exists a vector whose orbit under T intersects the subspace M in a relatively dense set. We give the sufficient condition for a subspace convex-cyclic transitive operator T to be subspace convex-cyclic. We also give a special type of the Kitai criterion related to invariant subspaces which implies subspace convex-cyclicity. Finally we show a counterexample of a subspace convexcyclic operator which is not subspace convex-cyclic transitive.
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