Back to Search
Start Over
Derivatives of tensor powers and their norms
- 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
- Subjects :
- Mathematics - Functional Analysis
15A15, 15A18, 15A60, 15A69, 47A30, 47A80
Subjects
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