A note on infinitesimal generators of semigroups on \(H^ 2/\phi H^ 2\) (Q2367618)

Let \(\varphi\) be an inner function in the Hardy space \(H^ 2\). For a function \(k\in H^ \infty\) let \(T=T(k+\varphi H^ \infty)\) be the ``multiplication'' operator in \(H^ 2/\varphi H^ 2\) defined by \[ g+\varphi H^ 2@>T>> kg+\varphi H^ 2 \qquad (g\in H^ 2). \] Finally, let \(A\) denote the infinitesimal generator of the semigroup \[ {\mathfrak g}=\{T(e^{tC}+ \varphi H^ \infty):\;t>0\} \] of operators of the form above, where \(C\) is an analytic function in the unit disk \(\mathbb{D}\) with \(\text{Re }C\leq M\). The author describes the resolvent \(R(\lambda,A)\) of \(A\) and proves that \(\lambda\) lies in the resolvent set \(\rho(A)\) if and only if the (not necessarily) bounded function \(C- \lambda\) and \(\varphi\) satisfy the corona condition \[ \inf \{| C(z)- \lambda|+| \varphi(z)|:\;z\in\mathbb{D}\}>0. \]