Restriction of an operator to the range of its powers

M. Berkani

Studia Mathematica (2000)

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

Abstract

top
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.

How to cite

top

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

top
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.