Invariant subspaces for translation, dilation and multiplication semigroups (Q1042654)

Considering the right-shift semigroup \(\{S_\tau\}_{\tau\geq 0}\) and the composition operator \(C_{1/\lambda}\) defined by \(C_{1/\lambda}f(x)=f(x/\lambda)\) for \(f\in L^2(\mathbb R)\), a characterization of the common invariant subspaces is obtained. For any given hyperbolic automorphism \(\varphi\) of the unit disc \(\mathbb D\), the lattice of the invariant subspaces of the composition operator \(C_\varphi\) on \(H^2(\mathbb D)\) strictly contains \(\text{Lat}(S_\tau, C_{1/\lambda})\), with \(0< \lambda< 1\). In particular, it is shown that the elements of \(\text{Lat}(C_{1/\lambda})\) in \(L^2(0,\infty)\) are in one-one correspondence with those of the bilateral shift on a vector valued space. Also for \(M_a\), the multiplication by \(e^{iax}\), the lattice of jointly invariant subspaces \(\text{Lat}(S_\tau, M_a)\) is characterized. An application to the study of the lattice of invariant subspaces for composition operators induced by parabolic automorphism of the unit disc \(\mathbb D\) is provided.