In joint work with In Sung Hwang and Woo Young Lee, we first consider several questions emerging from the Beurling-Lax-Halmos Theorem, which characterizes the shift-invariant subspaces of vector-valued Hardy spaces. The Beurling-Lax-Halmos Theorem states that a backward shift-invariant subspace is a model space $\mathcal{H}(\Delta) \equiv H_E^2 \ominus \Delta H_{E}^2$, for some operator-valued inner function $\Delta$. Our first question calls for a description of the set $F$ in $H_E^2$ such that $\mathcal{H}(\Delta)=E_F^*$, where $E_F^*$ denotes the smallest backward shift-invariant subspace containing the set $F$. In our pursuit of a general solution to this question, we are naturally led to take into account a canonical decomposition of operator-valued strong $L^2$-functions (in the sense of V.V. Peller).

We then ask: Is every shift-invariant subspace the kernel of a (possibly unbounded) Hankel operator? Further consideration of the above-mentioned question on the structure of shift-invariant subspaces leads us to study and coin a new notion of ``Beurling degree" for an inner function. \ We then establish a connection between the spectral multiplicity of the model operator (the truncated backward shift) and the Beurling degree of the corresponding characteristic function.

Next, we consider the case of multiplicity-free: more precisely, for which characteristic function $\Delta$ of the model operator $T$ does it follow that $T$ is multiplicity-free, i.e., $T$ has multiplicity 1? We prove that if $\Delta$ has a meromorphic pseudo-continuation of bounded type in the complement of the closed unit disk and the adjoint of the flip of $\Delta$ is an outer function, then $T$ is multiplicity-free.

Finally, we focus on rational symbols and study V.P. Potapov's celebrated theorem, that an $n \times n$ matrix function is rational and inner if and only if it can be represented as a finite Blaschke-Potapov product. We extend this result to the operator-valued case. As a corollary, we prove that when $\Delta \in H^\infty(\mathbb{T}, \mathcal{B}(E))$ is a left inner divisor of $z \cdot I_E$, then $\Delta$ is a Blaschke-Potapov factor. \ This requires a new notion of operator-valued rational function in the infinite multiplicity case; that is, $\Phi \in H^\infty(\mathbb{T}, \mathcal{B}(D,E))$ is said to be {\it rational} if $\theta H^2(\mathbb{T},E) \subseteq \ker H_{\Phi^*}$, where $\theta$ is a finite Blaschke product and $H_{\Phi^*}$ denotes the Hankel operator with symbol $\Phi^*$.

Some references:

1. R.E. Curto, I.S. Hwang, D.-O. Kang and W.Y. Lee, Subnormal and quasinormal Toeplitz

operators with matrix-valued rational symbols, Adv. Math. 255(2014), 562–585.

2. R.E. Curto, I.S. Hwang and W.Y. Lee, Matrix functions of bounded type: An interplay

between function theory and operator theory, Mem. Amer. Math. Soc. 260(2019), no.

1253, vi+100.

3. R.E. Curto, I.S. Hwang and W.Y. Lee, The Beurling-Lax-Halmos theorem for infinite multiplicity, J. Funct. Anal. 280(2021), 108884, 101 pp.

4. R.G. Douglas, H. Shapiro, and A. Shields, Cyclic vectors and invariant subspaces for the

backward shift operator, Ann. Inst. Fourier(Grenoble) 20(1970), 37–76.

5. V.V. Peller, Hankel Operators and Their Applications, Springer, New York, 2003.

6. V.P. Potapov, On the multiplicative structure of J-nonexpansive matrix functions, Tr. Mosk.

Mat. Obs. (1955), 125-236 (in Russian); English trasl. in: Amer. Math. Soc. Transl. (2)

15(1966), 131–243.