Geodesic
The Range Sequence of an Operator
Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 145-147

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Let T be a linear operator on a Banach space X and consider the sequence of ranges where the inclusions are not necessarily proper. The linear subspaces X n=Tn X (n>0) are, in general, not closed but they have some remarkable properties [1], [2]. Let X 0=X and denote by |x|0 (x∈X 0) the norm of X0. 
Vasilescu, F.-H. The Range Sequence of an Operator. Canadian mathematical bulletin, Tome 17 (1974) no. 1, pp. 145-147. doi: 10.4153/CMB-1974-032-7
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[1] 1. Fillmore, P. A. and Williams, J. P., On operator ranges, (preprint), Advances in Math. 7 (1971), 254-281. Google Scholar

[2] 2. Foias, C., Invariant para-closed subspaces, Indiana Univ. Math. J. 20 (1971), 897-900. Google Scholar

[3] 3. Grabiner, S., Ranges of quasinilpotent operators, Illinois J. Math. 15(1971), 150-152. Google Scholar

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