Somewhere Dense Orbit that is not Dense on a Complex Hilbert Space

Neema Wilberth,Marco Mpimbo,Santosh Kumar

doi:10.1515/conop-2019-0005

Abstract

Abstract In this paper, we present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit and is not dense. We give the solution to the question stated in [11]: “Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit that is not dense?” We do so by extending the results due to Feldman [11] and Leòn-Saavedra [12] to complex Hilbert space. Further illustrative examples of somewhere dense orbits are given to support the results.

Highlights

An n-tuple of operators means a nite sequence of length n of commuting continuous linear operators
In this paper, we present the existence of n-tuple of operators on complex Hilbert space that has a somewhere dense orbit and is not dense
We give the solution to the question stated in [11]: “Is there n-tuple of operators on a complex Hilbert space that has a somewhere dense orbit that is not dense?” We do so by extending the results due to Feldman [11] and Leòn-Saavedra [12] to complex Hilbert space

Summary

Introduction

An n-tuple of operators means a nite sequence of length n of commuting continuous linear operators. The concept of orbit comes from the theory of dynamical systems. A new phenomenon appears in an in nite-dimensional setting: linear operators may have dense orbits. Suppose that T is continuous linear operator on a topological vector space X over the eld F(=R or C), for an element x ∈ X, the orbit of x under T is Orb(T, x) = {x, T x, T x, ...} where x ∈ X is a xed vector. The concepts of dense orbits is de ned as follows: De nition 1.1. [8](Kronecker’s theorem) If x is a positive irrational number, the sequence {kx−s : k, s ∈ N} is dense in R The following result due to Boyd [8] will be useful in this paper: Theorem 1.1. [8](Kronecker’s theorem) If x is a positive irrational number, the sequence {kx−s : k, s ∈ N} is dense in R

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Main Results

Hence we have cn dk