Sharp Schatten Constants for the Subquadratic Range

Determine the fixed-dimensional best constant c^{lin}_{p,m,n}, or the all-dimensional constant c^{lin}_p(m)=sup_{n\ge1} c^{lin}_{p,m,n}, for 1<p<2, and characterize the corresponding extremal cases.

Background

The paper proves that, for every Schatten exponent p, number of summands m, and matrix dimension n, the sharp constant for inequalities involving arbitrary nonnegative concave functions is exactly the sharp constant for the underlying linear absolute-value inequality. Consequently, unresolved questions about the linear constants directly determine the unresolved cases for the full class of concave functions.

For 1<p<2, the candidate formula proposed by Tang–Zhang is reported to fail. Counterexamples are known for two matrices throughout this range, and Pang is cited as having established failure for every m\ge2. The paper therefore identifies the correct fixed-dimensional and all-dimensional sharp constants, together with the extremal configurations attaining them, as unresolved.