An Analog of the Cauchy–Schwarz Inequality for Hadamard Products and Unitarily Invariant Norms

The authors show that for any unitarily invariant norm $\| \cdot \|$ on $M_n $ (the space of n-by-n complex matrices) \[ (1)\qquad \| A^ * B \|^2 \leq \| A^* A \| \| B^* B \| \quad \text{for all}\, A,B \in M_{m,n} \] and \[ \| A \circ B \|^2 \leq \| A^* A \| \| B^* B \|\quad{\text{for all}}\,A,B \in M_n , \] where $ \circ $ denotes the Hadamard (entrywise) product. These results are a consequence of an inequality for absolute norms on $C^n$\[( 2 )\qquad \| x \circ y \|^2 \leq \| x \circ \bar x \| \| y \circ \bar y \| \quad \text{for all}\,x, y \in C^n \]. The authors also characterize the norms on $C^n $ that satisfy (2), characterize the unitary similarity invariant norms on $M_n $ that satisfy (1), and obtain related results on norms on $C^n $ and unitary similarity invariant norms on $M_n $ that are of independent interest.