Operator inequalities related to Beurling type theorem (Q2795685)

Let \(\mathcal H\) be a Hilbert space and \(\mathcal L(\mathcal H)\) denote the space of all bounded linear operators on \(\mathcal H\). For \(T\in\mathcal B(\mathcal H)\), define \(R^\infty(T):=\cap_{k=0}^\infty R(T^k)\). \(T\) is said to be \textit{pure} if \(R^\infty(T)=\{0\}\). For \(\mathcal E\subseteq\mathcal H\), the subspace \(\mathcal E_T\) denotes the smallest closed \(T\)-invariant subspace of \(\mathcal H\) containing \(\mathcal E\). A subset \(\mathcal E\) is said to be \textit{wandering relative to \(T\)} if \(\mathcal E\perp T^k(\mathcal E)\) for each \(k\geq 1\). \(T\) is said to have \textit{the wandering subspace property} if \(\mathcal H=\mathcal E_T\) for some wandering subspace \(\mathcal H\). \(T\in\mathcal B(\mathcal H)\) is said to admit a \textit{Wold-type decomposition} if \(R^\infty(T)\) is a closed reducing subspace of \(T\), the restriction of \(T\) onto \(R^\infty(T)\) is a unitary map and \(\mathcal H\) has the decomposition \(\mathcal H=\mathcal E_T\oplus R^\infty(T)\) for a wandering subspace \(\mathcal E\). Let \(H^2\) denote the Hardy space of analytic functions \(f(z)=\sum_{n=0}^\infty a_nz^n\) such that the sequence \(\{a_n\}\) is square summable. Here, \(z\) is a complex variable. Let \(M_z\) denote the multiplication operator in \(H^2\). Beurling's theorem states that a subspace \(E\) of \(H^2\) is invariant under \(M_z\) if and only if there exists an inner function \(\theta\) such that \(E=\theta H^2\). Further, \(\mathcal E=\mathbb C\theta\) is a wandering subspace of \(M_z\) and \(E=\mathcal E_{M_z}\). Let \(T \in\mathcal B(\mathcal H)\) be a bounded below operator satisfying the inequality: \(\| T(x) + y\|^2\leq 2(\| x \|^2+\| T(y)\|^2)\) for every \(x,y \in\mathcal H\). The authors show that such an operator must admit a Wold-type decomposition. They use this to show that, if \(T\) in addition is a pure operator, then \(T\) satisfies a Beurling-type theorem.