Essential g-Ascent and g-Descent of a Closed Linear Relation in Hilbert Spaces
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Abstract
We define and discuss for a closed linear relation in a Hilbert space the notions of essential g-ascent (resp. g-descent) and g-ascent (resp. g-descent) spectrums. We improve in the Hilbert space case some results given by E. Chafai in a Banach space [Acta Mathematica Sinica, 34 B, 1212-1224, 2014] and several results related to the ascent (resp. essential ascent) spectrum for a bounded linear operator on a Banach space [Studia Math, 187, 59-73, 2008] are extended to closed linear relations on Hilbert spaces. We prove also a decomposition theorem for closed linear relations with finite essential g-ascent or g-descent.
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How to Cite
Garbouj, Z., & Skhiri, H. (2017). Essential g-Ascent and g-Descent of a Closed Linear Relation in Hilbert Spaces. Extracta Mathematicae, 32(2), 125-161. Retrieved from https://publicaciones.unex.es/index.php/EM/article/view/2605-5686.32.2.125
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Banach Spaces and Operator Theory
References
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[10] A. Sandovici, H.S.V. De Snoo, H. Winkler, Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces, Linear Algebra Appl. 423 (2-3) (2007), 456 – 497.
[2] O. Bel Hadj Fredj, M. Burgos, M. Oudghiri, Ascent spectrum and essential ascent spectrum, Studia Math. 187 (2008), 59 – 73.
[3] E. Chafai, M. Mnif, Spectral mapping theorem for ascent, essential ascent, descent and essential descent spectrum of linear relations, Acta Mathematica Scientia, 34B (4) (2014), 1212 – 1224.
[4] R. Cross, “Multivalued Linear Operators”, Marcel Dekker, New York, 1998.
[5] Z. Garbouj, H. Skhiri, Essential ascent of closed operator and some decomposition theorems, Commun. Math. Anal. 16 (2) (2014), 19 – 47.
[6] Z. Garbouj, H. Skhiri, Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces, Acta Mathematica Hungarica 151 (2) (2017), 328 – 360.
[7] J.Ph. Labrousse, A. Sandovici, H.S.V. De Snoo, H. Winkler, The Kato decomposition of quasi-Fredholm relations, Operators and Matrices 4 (1) (2010), 1 – 51.
[8] A. Sandovici, Some basic properties of polynomials in a linear relation in linear spaces, in “Oper. Theory Adv. Appl. 175”, Birkhäuser, Basel, (2007), 231 – 240.
[9] A. Sandovici, H. De Snoo, An index formula for the product of linear relations, Linear Algebra Appl. 431 (11) (2009), 2160 – 2171.
[10] A. Sandovici, H.S.V. De Snoo, H. Winkler, Ascent, descent, nullity, defect, and related notions for linear relations in linear spaces, Linear Algebra Appl. 423 (2-3) (2007), 456 – 497.