Bessel's inequality in tensor space

Abstract

Let A be an n-square complex matrix and define to be the n ^!-square matrix whose entries are , where σ and τ run lexicographically over S_n . If A is positive definite hermitian and χ is a unit n!-tuple then

where c(A) is the largest of the numbers and the summation is over σ∊A_n . For n = 3, if A is not permutation similar to a direct sum and χ is a unit n!-tuple then (χχ) = det(A) iff χ is a multiple of the alternating character. The relationships among recent results of Bapat and Sunder, Chollet, and Gregorac and Hentzel are also discussed.

¹The work of this author was supported by the Air Force Office of Scientific Research under grant AFOSR-83-0150.

¹The work of this author was supported by the Air Force Office of Scientific Research under grant AFOSR-83-0150.

Notes

¹The work of this author was supported by the Air Force Office of Scientific Research under grant AFOSR-83-0150.