Boundedness of closed linear operator T satisfying R(T)\(\subset D(T)\)
type: Item
instance of: scholarly article
title: Boundedness of closed linear operator T satisfying R(T)\(\subset D(T)\)
zbMATH Open document ID: 0618.47002
author: Yoshiaki Okazaki
published in: Proceedings of the Japan Academy. Series A
publication date: 1986-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 47A05
zbMATH DE Number: 4002846
zbMATH Keywords: paranormal
zbMATH Keywords: densely defined closed linear operator
zbMATH Keywords: domain
zbMATH Keywords: range
zbMATH Keywords: non- empty resolvent set
MaRDI profile type: Publication
full work available at URL: pjaa.62.294
OpenAlex ID: W1976536595
cites work: On the class of paranormal operators
cites work: Q4397001
cites work: Q3322696
cites work: Q4774550
DOI: 10.3792/PJAA.62.294
description: scientific article; zbMATH DE number 4002846