Derivatives of tensor powers and their norms

The norm of the mth derivative of the map that takes an operator to its kth 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-algebras attains its norm at the m-tuple (I, I, . . . , I). Expressions for derivatives of the maps that take an operator to its kth tensor power and kth 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.