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

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.


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. In the context of operator theory the notion was rst used by Rolewicz [14]. Linear dynamics is mainly concerned with the behaviour of iterates of linear transformations. However, 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), then 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 present work is an extension of the work done by Feldman [11] on Hypercyclicity and somewhere dense orbits.
The concepts of dense orbits is de ned as follows:

Main Results
The extension of Theorem 1.2 from real number to complex number is stated below. The proof of the extended Theorem follows closely the ideas obtained from the real case.
Proof. Our proof relies on reducing a fraction ( In(z ) In(z ) ) to a fraction ( In(a) In(b) ), and then apply Theorem 1.2. We consider several cases as follows: Case 2.1. Suppose that a, b, c, d ∈ R\{ } such that a, b > , a = c and b = d, and In(a+ic) In(b+id) is irrational, then Indeed the one dimensional version of Kronecker's theorem (Theorem 1.1) can be rephrased as follows: If θ is a positive irrational number, then {nθ − k : n, k ∈ N} is dense in R. Now, applying Kronecker's theorem with θ = ln(a+ic) ln(b+id) gives that Multiplying by ln(b + id) on both sides, we obtain, is dense in R. By simplifying we get, Thus, by taking the exponential of the above set, we have that Just as in the case (i) above we have that {nθ − k : n = k ∈ N} is dense in R. By applying Kronecker's theorem with θ = ln(a+ic) ln(b+id) one has n ln(a+ic) ln(b+id) − k : n = k ∈ N is dense in R.
Since a, b = , then we have, Multiplying by ln(id) on both sides, results into, is dense in R. By simplifying, one has, Thus, by taking the exponential of the above set, we see that Hence we have Applying Kronecker's theorem with θ = ln(a+ic) ln(b+id) gives that, Multiplying by ln(b + id) on both sides, we have that, is dense in R. By simplifying, one has, Thus, by taking the exponential of the above set, we see that a n b k : n, k ∈ N is dense in R + .
Case 2.4. If a, b ∈ C\{ } and ( ln(a) n ln(b) k ) is irrational then a n b k : n, k ∈ N is dense in R + . Let a = e qπi and b = e pπi where p, q ∈ R and p ≠ q. Applying Kronecker's theorem with θ = ln(a) ln(b) gives that, that is n ln(e qπi ) ln(e pπi ) − k : n, k ∈ N . By simplifying, one has, Multiplying by ln(p) on both sides, we have that Note: Also if p or q is positive or negative the answer can either be dense in R + or dense in R − but will still be somewhere dense.

Remark 2.1. The above cases are not exhaustive in the sense that the two cases below remain open.
The rst open case is due to the fact that our requirement to reduce ln(z ) ln(z ) to ln(a) ln(b) fails and hence Theorem 1.2 can not be used. Let a, b, c, d ∈ R such that a, b > , c, d ≠ , a ≠ c,  The second open case obeys the reduction of ln(z ) ln(z ) to ln(a)

Case 2.5.
Case 2.6. If a, b = and c, d ∈ {R + \ } such that c, d > , and ln(a+ic) ln(b+id) is irrational, but if n ≠ k, (z ) n (z ) k ∈ C, therefore it can't be dense in R + .
The following result due to Feldman [11] is useful in constructing both examples: Corollary 2.1. ( [11]) If a,b>1 are relatively prime integers, then a n b k : n, k ∈ N is dense in R + .
Hence, {Mn λz } ≠ C and {Mn λz } ≠ ∅. Therefore, {Mn λz } n≥ is contained in the closed unit disk for any positive integer n and {Mn λz } n≥ is somewhere dense in complex space but not dense in C.
The following Corollary due to Ansari [3] can be usedful on our results.

Corollary 2.2. (Ansari's Theorem)([3]) If T is hypercyclic, then for every positive integer n, the operator T n is also hypercyclic; moreover T and T n share the same collection of hypercyclic vectors.
We give the di erent proof of Corollary 2.2 to check whether the set of complex numbers has somewhere dense orbits.

Corollary 2.3. If T is hypercyclic, then for every positive integer n, the operator T n is also hypercyclic; moreover T and T n share the same collection of hypercyclic vectors.
Proof. We shall prove the n = on complex case. Suppose that if T : X −→ X is hypercyclic and n is positive integer that means Ort(T, x) is dense.
By taking closure, we have Also by appling the intersection of interior of closure, we get If x is hypercyclic vector for T then either Orb(T x) or Orb(T , Tx) is somewhere dense that are not dense if and only if Ort(T, x) is dense in X by above condition. Hence, we have Thus T is hypercyclic, if x is hypercyclic vector for T . But also a weaker property than hypercyclicity is the property of admitting a somewhere dense orbit by using Theorem 3.6 (Feldman 2008) and it is obviously, if x ∈ HC(T), then Orb(x, T) is somewhere dense.
We end this paper by giving two examples showing that somewhere dense orbits are not necessarity dense. The rst example was given by Feldman [11] in real Hilbert space, and the second examples is ours which is an extension of the rst example to the complex Hilbert space.  [11], cl(Orb(T, i)) = [ , ∞)i and cl(Orb(T, −i)) = (−∞, ]i. Hence T has somewhere dense orbits that are not dense. Furthermore, Orb(T, i) ∪ Orb(T, −i) is dense in C, but T is not hypercyclic. Thus T is multi-hypercyclic, but not hypercyclic.