## Annals of Functional Analysis

### On $m$-generalized invertible operators on Banach spaces

Hamid Ezzahraoui

#### Abstract

A bounded linear operator $S$ on a Banach space $X$ is called an $m$-left generalized inverse of an operator $T$ for a positive integer $m$ if

$$T\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}S^{m-j}T^{m-j}=0,$$ and it is called an $m$-right generalized inverse of $T$ if

$$S\sum_{j=0}^{m}(-1)^{j}\binom{m}{j}T^{m-j}S^{m-j}=0.$$ If $T$ is both an $m$-left and an $m$-right generalized inverse of $T$, then it is said to be an $m$-generalized inverse of $T$.

This paper has two purposes. The first is to extend the notion of generalized inverse to $m$-generalized inverse of an operator on Banach spaces and to give some structure results. The second is to generalize some properties of $m$-partial isometries on Hilbert spaces to the class of $m$-left generalized invertible operators on Banach spaces. In particular, we study some cases in which a power of an $m$-left generalized invertible operator is again $m$-left generalized invertible.

#### Article information

Source
Ann. Funct. Anal., Volume 7, Number 4 (2016), 609-621.

Dates
Accepted: 3 May 2016
First available in Project Euclid: 23 September 2016

https://projecteuclid.org/euclid.afa/1474652185

Digital Object Identifier
doi:10.1215/20088752-3660801

Mathematical Reviews number (MathSciNet)
MR3550939

Zentralblatt MATH identifier
06667757

Subjects
Primary: 47B48: Operators on Banach algebras
Secondary: 47B99: None of the above, but in this section

#### Citation

Ezzahraoui, Hamid. On $m$ -generalized invertible operators on Banach spaces. Ann. Funct. Anal. 7 (2016), no. 4, 609--621. doi:10.1215/20088752-3660801. https://projecteuclid.org/euclid.afa/1474652185

#### References

• [1] J. Agler and M. Stankus, $m$-Isometric transformations of Hilbert space, I, Integral Equations Operator Theory 21 (1995), no. 4, 383–429.
• [2] J. Agler and M. Stankus, $m$-Isometric transformations of Hilbert space, II, Integral Equations Operator Theory 23 (1995), no. 1, 1–48.
• [3] J. Agler and M. Stankus, $m$-Isometric transformations of Hilbert space, III, Integral Equations Operator Theory 24 (1996), no. 4, 379–421.
• [4] C. Badea and M. Mbekhta, Operators similar to partial isometries, Acta Sci. Math. (Szeged) 71 (2005), 663–680.
• [5] A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, 2nd ed. CMS Books Math./Ouvrages Math SMC 15, Springer, New York, 2003.
• [6] S. R. Caradus, Generalized inverses and operator theory, Queen’s Papers in Pure and Appl. Math. 50, Queen’s Univ., Kingston, Ont., 1978.
• [7] S. H. Jah, Power of $m$-partial isometries on Hilbert spaces, Bull. Math. Anal. Appl. 5 (2013), no. 4, 79–89.
• [8] O. A. Mahmoud Sid Ahmed, Some properties of $m$-isometries and m-invertible operators on Banach spaces, Acta Math. Sci. Ser. B Engl. Ed. 32 (2012), no. 2, 520–530.
• [9] M. Mbekhta, Partial isometries and generalized inverses, Acta Sci. Math. (Szeged) 70 (2004), nos. 3–4, 767–781.
• [10] M. Mbekhta and L. Suciu, Generalized inverses and similarity to partial isometries, J. Math. Anal. Appl. 372 (2010), no. 2, 559–564.
• [11] S. M. Patel, $2$-isometric operators, Glas. Mat. Ser. III 37 (2002), no. 1, 141–145.
• [12] A. Saddi and O. A. Mahmoud Sid Ahmed, $m$-partial isometries on Hilbert spaces, Int. J. Funct. Anal. Oper. Theory Appl. 2 (2010), no. 1, 67–83.
• [13] S. Shimorin, Wold-type decompositions and wandering subspaces for operators close to isometries, J. Reine Angew. Math. 531 (2001), 147–189.