A Review of Some Works in the Theory of Diskcyclic Operators

作者：Bamerni Nareen^*; Kilicman Adem; Noorani Mohd Salmi Md

来源：Bulletin of the Malaysian Mathematical Sciences Society, 2016, 39(2): 723-739.

DOI：10.1007/s40840-015-0137-x

• Hypercyclic operators
• Supercyclic operators
• Diskcyclic operators
• Weighted shift operators

In this paper, we give a brief review concerning diskcyclic operators and then we provide some further characterizations of diskcyclic operators on separable Hilbert spaces. In particular, we show that if x is an element of H has a disk orbit under T that is somewhere dense in H, then the disk orbit of x under T need not be everywhere dense in H. We also show that the inverse and the adjoint of a diskcyclic operator need not be diskcyclic. Moreover, we establish another diskcyclicity criterion and use it to find a necessary and sufficient condition for unilateral backward shifts that are diskcyclic operators. We show that a diskcyclic operator exists on a Hilbert space H over the field of complex numbers if and only if dim(H) = 1 or dim(H) = infinity. Finally, we give a sufficient condition for the somewhere density disk orbit to be everywhere dense.

• 出版日期2016-4