Back to Search Start Over

Derivatives of tensor powers and their norms

Authors :
Bhatia, Rajendra
Grover, Priyanka
Jain, Tanvi
Source :
The Electronic Journal of Linear Algebra, Volume 26, pp. 604-619, September 2013
Publication Year :
2011

Abstract

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, Variation of Grassman powers and spectra, 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 $(I, \, I, ..., I).$ 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.<br />Comment: 15 pages

Details

Database :
arXiv
Journal :
The Electronic Journal of Linear Algebra, Volume 26, pp. 604-619, September 2013
Publication Type :
Report
Accession number :
edsarx.1102.2414
Document Type :
Working Paper