LINEAR INDEPENDENCE OF QUANTUM TRANSLATES

Source abstract

Abstract If A is in the p -Schatten class on R n ${\mathbb {R}}^n$ double struck upper R Superscript n , 1 ≤ p ≤ 4 n / ( 2 n − 1 ) $1\leq p \leq {4n}/{(2n-1)}$ 1 less than or equals p less than or equals 4 n divided by left parenthesis 2 n minus 1 right parenthesis , then the quantum translates of A are linearly independent. Moreover, there exists a nonzero operator in the p -Schatten class on R n ${\mathbb {R}}^n$ double struck upper R Superscript n , p > 4 n / ( 2 n − 1 ) $p>{4n}/{(2n-1)}$ p greater than 4 n divided by left parenthesis 2 n minus 1 right parenthesis , whose quantum translates are linearly dependent.