Wandering subspace theorems (Q1774001)

Let \(H\) be a Hilbert space and \(L(H)\) be the space of all bounded linear operators on \(H\). For an operator \(T \in L(H)\) and a set \(M \subset H\), let \([M]_{T}\) denote the smallest closed \(T\)-invariant subspace of \(H\) containing the set \(M\). In this paper, the author deals with the relation \((\ast)\) \(H=[M]_{T}\), where \(M=H \circleddash T(H)\). A closed subspace \(M\) of \(H\) is said to be a wandering subspace for \(T\) if \(M \perp T^{k}(M)\) for \(k \geq 1\). A result satisfying the relation \((\ast)\) is called a wandering subspace theorem. An operator \(T \in L(H)\) is called expansive if \(\| Tx \| \geq \| x \|\) for \(x \in H\). In Section \(2\) and Section \(3\), the author studies expansive operators \(T \in L(H)\) such that \[ \| T^{k}x \|^{2} \leq c_{k}(\| Tx \|^{2}-\| x \|^{2})+c \| x \|^{2},\quad x \in H,\;k \geq 2 \] and \(\sum_{k \geq 2} {1 \over c_{k}}= \infty\) for some positive constants \(c_{k}\) \((k \geq 2)\) and \(c\). In Section \(4\), he investigates operators \(T \in L(H)\) such that \[ \| Tx + y \|^{2} \leq 2 (\| x \|^{2} + \| Ty \|^{2}),\quad x,y \in H. \]