Operators with diskcyclic vectors subspaces

• Main Authors: Bamerni, Nareen, Kilicman, Adem
• Format: Article
• Language: English
• Published: Taibah University 2015
• Online Access: http://psasir.upm.edu.my/id/eprint/56228/1/Operators%20with%20diskcyclic%20vectors%20subspaces.pdf; http://psasir.upm.edu.my/id/eprint/56228/; http://www.sciencedirect.com/science/article/pii/S1658365515000655#
• Institution: Universiti Putra Malaysia

In this paper, we prove that if T is diskcyclic operator then the closed unit disk multiplied by the union of the numerical range of all iterations of T is dense in H. Also, if T is diskcyclic operator and |λ| ≤ 1, then T – λI has dense range. Moreover, we prove that if α > 1, then 1/αT is hypercyclic in a separable Hilbert space H if and only if T ⊕ αIC is diskcyclic in H ⊕ C. We show at least in some cases a diskcyclic operator has an invariant, dense linear subspace or an infinite dimensional closed linear subspace, whose non-zero elements are diskcyclic vectors. However, we give some counterexamples to show that not always a diskcyclic operator has such a subspace.