Generalized weakly demicompact and $S$-demicompact linear relations and their spectral properties

Majed Fakhfakh; Aref Jeribi

doi:10.3842/umzh.v75i8.7194

Majed Fakhfakh Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Tunisia
Aref Jeribi Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Tunisia

DOI: https://doi.org/10.3842/umzh.v75i8.7194

Keywords: Linear relation, weakly closed linear relation, weakly demicompact relation, Riesz relation, generalized weakly demicompact relation, generalized weakly S-demicompact relation, Fredholm and semi-Fredholm relations.

Abstract

UDC 517.9

We extend the concept of generalized weakly demicompact and relatively weakly demicompact operators on linear relations and present some outstanding results. Moreover, we address the theory of Fredholm and upper semi-Fredholm relations attempting to establish a connection with them.

References

T. Alvarez, D. Wilcox, Perturbation theory of multivalued atkinson operators in normed spaces, Bull. Austr. Math. Soc., 76, 195–204 (2007).

A. Ammar, H. Daoud, A. Jeribi, Demicompact and $k$-$D$-set-contractive multivalued linear operators, Mediterr. J. Math., 15 (2018).

A. Ammar, S. Fakhfakh, A. Jeribi, Shechter spectra and relatively demicompact linear relations, Commun. Korean Math. Soc., 35, (2020).

A. Ammar, S. Fakhfakh, A. Jeribi, Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations, J. Pseudo-Different. Oper. and Appl., 7, № 4, 493–509 (2016).

K. Astala, On measures of noncompactness and ideal variations in Banach spaces, Ann. Acad. Sci. Fenn. Math. Diss., 29 (1980).

J. Banas, A. Martinón, On measures of weak noncompactness in Banach sequence spaces, Port. Math., 52 (1995).

H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Springer, New York (2011).

E. A. Coddington, Multivalued operators and boundary value problems, Lect. Notes Math., 183, Springer-Verlag, Berlin (1971).

R. W. Cross, Multivalued lineaire operators, Monogr. and Textbooks in Pure and Appl. Math., vol. 213, Marcel Dekker, New York (1998).

F. S. De Blasi, On a property of the unit sphere in a Banach space, Bull. Math. Soc. Sci. Math. Roumanie (N.S.), 21 (1977).

G. Emmanuele, Measure of weak noncompactness and fixed point theorems, Boll. Math. Soc. Sci. Math. Roumanie, 25 (1981).

F. Fakhfakh, Perturbation theory of single valued and multivalued semi-Browder linear operators, PhD Thesis, University of Sfax (2010).

A. Favini, A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura ed Appl., 163, 353–384 (1993).

I. Ferjani, A. Jeribi, B. Krichen, Spectral properties involving generalized weakly demicompact operators, Mediterr. J. Math., 21, Article 30 (2019).

I. Ferjani, A. Jeribi, B. Krichen, Spectral properties for generalized weakly $S$-demicompact operators, Linear and Multilinear Algebra, 1–19 (2020).

A. Jeribi, B. Krichen, Nonlinear functional analysis in Banach spaces and Banach algebras: fixed point theory under weak topology for nonlinear operators and block operator matrices with applications, Monogr. and Res. Notes Math., CRC Press Taylor and Francis (2015).

T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Anal. Math., 6, 273–322 (1958).

B. Krichen, Relative essential spectra involving relative demicompact unbounded linear operators, Acta Math. Sci. Ser. B (Engl. Ed.), 34, 546–556 (2014).

J. V. Neumann, Functional operators, the geometry of orthogonal spaces, Ann. Math. Stud., Princeton Univ. Press., Princeton (1950).

W. V. Petryshyn, Construction of fixed points of demicompact mappings in Hilbert space, J. Math. Anal. and Appl., 14, 276–284 (1966).

A. Sandovici, H. de Snoo, An index formula for the product of linear relations, Linear Algebra and Appl., 431, № 11, 2160–2171 (2009).