sub-space restricted minimum eigenvalue of Hadamard product of two PSD matrices

Question

Let $\mathbf{A},\mathbf{B}\in\mathbb{R}^{n\times n}$ be two positive semidefinite matrices. Also let $\mathbf{A}\circ \mathbf{B}$ denote the Hadamard product of $\mathbf{A}$ and $\mathbf{B}$. A classical result states that \begin{align*} \lambda_{\min}(\mathbf{A}\circ \mathbf{B})\ge \lambda_{\min}(\mathbf{A})\lambda_{\min}(\mathbf{B}) \end{align*} I am wondering whether a "subspace-restricted" version of this result is true. More specifically let $\mathbf{U}\in\mathbb{R}^{n\times k}$ be an orthonormal matrix obeying $\mathbf{U}^T\mathbf{U}=\mathbf{I}_k$ with $k\le n$. I am wondering if the following inequality holds

\begin{align*} \lambda_{\min}(\mathbf{U}^T(\mathbf{A}\circ \mathbf{B})\mathbf{U})\ge \lambda_{\min}(\mathbf{U}^T\mathbf{A}\mathbf{U})\lambda_{\min}(\mathbf{U}^T\mathbf{B}\mathbf{U}) \end{align*}

Answer 1

As stated the assertion is false e.g. let first column of A equal to 0 and second column of B equal to 0. take $\mathbf{u}=\frac{(\mathbf{e}_1+\mathbf{e}_2)}{\sqrt{2}}$