The famous Wold decomposition gives a complete structure of an isometry on a Hilbert space. Berger, Coburn, and Lebow (BCL) obtained a structure for a tuple of commuting isometries acting on a Hilbert space. In this talk, we shall discuss a structure of a pair of commuting $C_0$-semigroups of isometries and obtain a BCL type result.

The right-shift-semigroup $\mathcal S^\mathcal E=(S^\mathcal E_t)_{t\ge 0}$ on $L^2(\mathbb R_+,\mathcal E)$ for any Hilbert space $\mathcal E$ is defined as \begin{equation} (S_t^\mathcal E f)(x) = \begin{cases} f(x-t) &\text{if } x\ge t,\\ 0 & \text{otherwise,} \end{cases} \end{equation} for $f\in L^2(\mathbb R_+,\mathcal E).$ Cooper showed that the role of the unilateral shift in the Wold decomposition of an isometry is played by the right-shift-semigroup for a $C_0$-semigroup of isometries. The factorizations of the unilateral shift have been explored by BCL, we are interested in examining the factorizations of the right-shift-semigroup. Firstly, we shall discuss the contractive $C_0$-semigroups which commute with the right-shift-semigroup. Then, we give a complete description of the pairs $(\mathcal V_1,\mathcal V_2)$ of commuting $C_0$-semigroups of contractions which satisfy $\mathcal S^\mathcal E=\mathcal V_1\mathcal V_2$, (such a pair is called as a factorization of $\mathcal S^\mathcal E$), when $\mathcal E$ is a finite dimensional Hilbert space.

Next, we discuss the Taylor joint spectrum for a pair of commuting isometries $(V_1,V_2)$ using the defect operator $C(V_1,V_2)$ defined as \begin{equation} C(V_1,V_2)=I-V_1V_1^*-V_2V_2^*+ V_1V_2V_2^*V_1^*. \end{equation} We show that the joint spectrum of two commuting isometries can vary widely depending on various factors. It can range from being small (of measure zero or an analytic disc for example) to the full bidisc. En route, we discover a new model pair in the negative defect case.