The Beurling-Lax-Halmos theorem for infinite multiplicity

In this paper, we consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes shift-invariant subspaces of vector-valued Hardy spaces. The Theorem states that a backward subspace is model space $\mathcal{H}(\Delta) \equiv H_E^2 \ominus \Delta H_{E}^2$, for some inner function $\Delta$. Our first question calls description set $F$ in $H_E^2$ such $\mathcal{H}(\Delta)=E_F^*$, where $E_F^*$ denotes smallest containing $F$. our pursuit general solution to question, are naturally led take into account canonical decomposition operator-valued strong $L^2$-functions. Next, ask: Is every kernel (possibly unbounded) Hankel operator? As know, operator shift-invariant, so above equivalent seeking equation $\ker H_{\Phi}^*=\Delta H_{E^{\prime}}^2$, $\Delta$ an satisfying $\Delta^* \Delta=I_{E^{\prime}}$ almost everywhere on unit circle $\mathbb{T}$ and $H_{\Phi}$ with symbol $\Phi$. Consideration structure leads us study coin new notion "Beurling degree" function. We then establish deep connection between spectral multiplicity Beurling degree corresponding characteristic At same time, meromorphic pseudo-continuations bounded type functions, use operators (truncated shifts) separable complex Hilbert particular, multiplicity-free case.