×

Boundedness of closed linear operator T satisfying R(T)\(\subset D(T)\). (English) Zbl 0618.47002

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 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.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)

Citations:

Zbl 0537.47018
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] T. Furuta: On the class of paranormal operators. Proc. Japan Acad., 43, 594-598 (1967). · Zbl 0163.37706 · doi:10.3792/pja/1195521514
[2] S. Goldberg: Unbounded Linear Operators, Theory and Applications. McGraw Hill Comp. (1966). · Zbl 0925.47001
[3] S. Ota: Closed linear operators with domain containing their range. Proc. Edinburgh Math. Soc, 27, 229-233 (1984). · Zbl 0537.47018
[4] K. Yosida: Functional Analysis. Die Grundlehren der Math., vol. 123, Springer-Verlag (1968). · Zbl 0286.46002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.