Boundedness of closed linear operator T satisfying R(T)\(\subset D(T)\)
Let E be a Banach space and \(T: E\to E\) be a densely defined closed linear operator with the domain D(T) and the range R(T). The following problem was posed by \textit{S. Ôta} [Proc. Edinb. Math. Soc., II. Ser. 27, 229-233 (1984; Zbl 0537.47018)]. If it holds that R(T)\(\subset D(T)\), then is T bounded? In this paper, it is proved that if T satisfies R(T)\(\subset D(T)\) and one of the following conditions: (1) \(\| T^ 2x\| \geq \| Tx\|^ 2\) for every \(x\in D(T)\), or (2) T has a non-empty resolvent set, then T is bounded. Let \(T: E\to H\) be a densely defined closed linear operator and \(T^*:\) \(H\to E'\) be the adjoint operator, where H is a Hilbert space. It is shown that if \(R(T)\subset D(T^*)\) holds, then T is bounded.
scientific article; zbMATH DE number 4002846