An Algorithm for Quartically Hyponormal Weighted Shifts

• Published in: Kyungpook mathematical journal Vol. 51; no. 2; pp. 187 - 194
• Main Authors: Baek, Seung-Hwan, Jung, Il-Bong, Moo, Gyung-Young
• Format: Journal Article
• Language: Korean
• Published: 2011
• Subjects: weighted shifts; cubically hyponormal operators; quartically hyponormal operators; quadratically hyponormal operators
• ISSN: 1225-6951; 0454-8124

Examples of a quartically hyponormal weighted shift which is not 3-hyponormal are discussed in this note. In [7] Exner-Jung-Park proved that if ${\alpha}$(x) : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{53252}{100000}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 4-hyponormal. And, Curto-Lee([5]) improved their result such as that if ${\alpha}(x)$ : $\sqrt{x},\sqrt{\frac{2}{3}},\sqrt{\frac{3}{4}},\sqrt{\frac{4}{5}},{\cdots}$ with 0 < x ${\leq}\;\frac{667}{990}$, then $W_{\alpha(x)}$ is quartically hyponormal but not 3-hyponormal. In this note, we improve slightly Curto-Lee's extremal value by using an algorithm and computer software tool.