Essential g-Ascent and g-Descent of a Closed Linear Relation in Hilbert Spaces

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.

Keywords

• Range subspace
• Closed linear relation
• Spectrum
• Ascent
• Essential ascent
• Descent