Restriction of an operator to the range of its powers

Studia Mathematica (2000)

• Volume: 140, Issue: 2, page 163-175
• ISSN: 0039-3223

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Abstract

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Let T be a bounded linear operator acting on a Banach space X. For each integer n, define ${T}_{n}$ to be the restriction of T to $R\left({T}^{n}\right)$ viewed as a map from $R\left({T}^{n}\right)$ into $R\left({T}^{n}\right)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator ${T}_{n}$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where ${T}_{n}$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological uniform descent.

How to cite

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Berkani, M.. "Restriction of an operator to the range of its powers." Studia Mathematica 140.2 (2000): 163-175. <http://eudml.org/doc/216760>.

@article{Berkani2000,
abstract = {Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_n$ to be the restriction of T to $R(T^n)$ viewed as a map from $R(T^n)$ into $R(T^n)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_n$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_n$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological uniform descent.},
author = {Berkani, M.},
journal = {Studia Mathematica},
keywords = {range of powers; regularity; quasi-Fredholm operator},
language = {eng},
number = {2},
pages = {163-175},
title = {Restriction of an operator to the range of its powers},
url = {http://eudml.org/doc/216760},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Berkani, M.
TI - Restriction of an operator to the range of its powers
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 2
SP - 163
EP - 175
AB - Let T be a bounded linear operator acting on a Banach space X. For each integer n, define $T_n$ to be the restriction of T to $R(T^n)$ viewed as a map from $R(T^n)$ into $R(T^n)$. In [1] and [2] we have characterized operators T such that for a given integer n, the operator $T_n$ is a Fredholm or a semi-Fredholm operator. We continue those investigations and we study the cases where $T_n$ belongs to a given regularity in the sense defined by Kordula and Müller in[10]. We also consider the regularity of operators with topological uniform descent.
LA - eng
KW - range of powers; regularity; quasi-Fredholm operator
UR - http://eudml.org/doc/216760
ER -

References

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1. [1] M. Berkani, On a class of quasi-Fredholm operators, Integral Equations Oper. Theory 34 (1999), 244-249. Zbl0939.47010
2. [2] M. Berkani and M. Sarih, On semi-B-Fredholm operators, submitted.
3. [3] S. R. Caradus, Operator Theory of the Pseudo-Inverse, Queen's Papers in Pure and Appl. Math. 38 (1974), Queen's Univ., 1974. Zbl0286.47001
4. [4] S. Grabiner, Uniform ascent and descent of bounded operators, J. Math. Soc. Japan 34 (1982), 317-337. Zbl0477.47013
5. [5] R. Harte, On Kato non-singularity, Studia Math. 117 (1996), 107-114. Zbl0838.47005
6. [6] R. Harte and W. Y. Lee, A note on the punctured neighbourhood theorem, Glasgow Math. J. 39 (1997), 269-273. Zbl0894.47009
7. [7] M. A. Kaashoek, Ascent, descent, nullity and defect, a note on a paper by A. E. Taylor, Math. Ann. 172 (1967), 105-115. Zbl0152.33803
8. [8] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math. 6 (1958), 261-322. Zbl0090.09003
9. [9] J. J. Koliha, M. Mbekhta, V. Müller and P. W. Poon, Corrigendum and addendum: "On the axiomatic theory of spectrum II", Studia Math. 130 (1998), 193-198. Zbl0914.47016
10. [10] V. Kordula and V. Müller, On the axiomatic theory of spectrum, ibid. 119 (1996), 109-128. Zbl0857.47001
11. [11] J. P. Labrousse, Les opérateurs quasi-Fredholm: une généralisation des opérateurs semi-Fredholm, Rend. Circ. Mat. Palermo (2) 29 (1980), 161-258. Zbl0474.47008
12. [12] M. Mbekhta and M. Müller, On the axiomatic theory of spectrum II, Studia Math. 119 (1996), 129-147. Zbl0857.47002
13. [13] P. W. Poon, Spectral properties and structure theorems for bounded linear operators, thesis, Dept. of Math. and Statist., Univ. of Melbourne, 1997.
14. [14] C. Schmoeger, On a class of generalized Fredholm operators, I, Demonstratio Math. 30 (1997), 829-842. Zbl0903.47006
15. [15] C. Schmoeger, On a class of generalized Fredholm operators, V, ibid. 32 (1999), 595-604. Zbl0953.47010
16. [16] C. Schmoeger, On a generalized punctured neighborhood theorem in ℒ(X), Proc. Amer. Math. Soc. 123 (1995), 1237-1240.

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