Theoretical Statistics and Mathematics Unit, ISI Delhi

Derivatives of tensor powers and their norms

by Rajendra Bhatia, Priyanka Grover and Tanvi Jain

The norm of the $m$th derivative of the map that takes an operator to its $k$th antisymmetric tensor power is evaluated. The case $m=1$ has been studied earlier by Bhatia and Friedland [R.~Bhatia and S.~Friedland.
\newblock Variation of Grassman powers and spectra.
\newblock {\em Linear Algebra and its Applications}, 40:1--18, 1981]. For this purpose a multilinear version of a theorem of Russo and Dye is proved: it is shown that a positive $m$-linear map between $C^{\ast}$-algebras attains its norm at the $m$-tuple $(I, \, I, \ldots, I).$ Expressions for derivatives of the maps that take an operator to its $k$th tensor power and $k$th symmetric tensor power are also obtained. The norms of these derivatives are computed. Derivatives of the map taking a matrix to its permanent are also evaluated.

isid/ms/2013/10 [fulltext]

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