An inequality for spherical Cauchy dual tuples

Sameer Chavan¹

¹ Department of Mathematics Indian Institute of Technology Kanpur Kanpur 208016, India

Colloquium Mathematicum, Tome 131 (2013) no. 2, pp. 265-271

Résumé

Let $T$ be a spherical $2$-expansive $m$-tuple and let $T^{\mathfrak s}$ denote its spherical Cauchy dual. If $T^{\mathfrak s}$ is commuting then the inequality $$ \sum _{|\beta |=k} (\beta !)^{-1} {(T^{\mathfrak s})}^{\beta }{(T^{\mathfrak s})^*}^{\beta }\leq \left ({k+m-1\atop k}\right) \sum_{|\beta |=k} (\beta !)^{-1} {(T^{\mathfrak s})^*}^{\beta }(T^{\mathfrak s})^{\beta } $$ holds for every positive integer $k.$ In case $m=1,$ this reveals the rather curious fact that all positive integral powers of the Cauchy dual of a $2$-expansive (or concave) operator are hyponormal.

Détail

DOI : 10.4064/cm131-2-8

Keywords: spherical expansive m tuple mathfrak denote its spherical cauchy dual mathfrak commuting inequality sum beta beta mathfrak beta mathfrak * beta leq m atop right sum beta beta mathfrak * beta mathfrak beta holds every positive integer reveals rather curious positive integral powers cauchy dual expansive concave operator hyponormal