An inequality for spherical Cauchy dual tuples

Tom 131 / 2013

Sameer Chavan Colloquium Mathematicum 131 (2013), 265-271 MSC: Primary 47A13, 47B20; Secondary 47A65, 47B47. DOI: 10.4064/cm131-2-8


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.


  • Sameer ChavanDepartment of Mathematics
    Indian Institute of Technology Kanpur
    Kanpur 208016, India

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