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We introduce a class of unitarily invariant, locally$\|\cdot\|_1$-dominating, mutually continuous norms with repect to $\tau$ on avon Neumann algebra $\mathcal{M}$ with a faithful, normal, semifinite tracialweight $\tau$. We prove a Beurling-Chen-Hadwin-Shen theorem for$H^\infty$-invariant spaces of $L^\alpha(\mathcal{M},\tau)$, where $\alpha$ isa unitarily invariant, locally $\|\cdot\|_1$-dominating, mutually continuousnorm with respect to $\tau$, and $H^\infty$ is an extension of Arveson'snoncommutative Hardy space. We use our main result to characterize the$H^\infty$-invariant subspaces of a noncommutative Banach function space$\mathcal I(\tau)$ with the norm $\|\cdot\|_{E }$ on $\mathcal{M}$, the crossedproduct of a semifinite von Neumann algebra by an action $\beta$, and$B(\mathcal{H})$ for a separable Hilbert space $\mathcal{H}$.

A Beurling-Chen-Hadwin-Shen Theorem for Noncommutative Hardy Spaces Associated with Semifinite von Neumann Algebras with Unitarily Invariant Norms