Let $T\in\mathcal{B}(\mathcal{H})$ be a self-adjoint operator acting on a Hilbert space $\mathcal{H}$. Suppose $k\in\mathbb{N}$. Define $$\lambda_k(T)=\sup\left\{\lambda_k(V^*TV):V:\mathbb{C}^k\rightarrow\mathcal{H}\right\}$$ where $\lambda_k(H)$ denotes the $k$-th largest eigenvalue of a self-adjoint matrix $H\in M_n$. Let $$\mu_k(T):=\inf\{\lambda_1(W^*TW): W:\mathcal{H}\rightarrow\mathcal{H} \text{ is an isometry with codimension of ran}W<k\}.$$
I could show that $\lambda_k(T)\leq\mu_k(T)$.
Question: Is $\lambda_k(T)=\mu_k(T)$?