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Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces

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Abstract

For a closed linear relation in a Hilbert space the notions of minimum modulus, essential g-ascent, essential ascent and essential descent are introduced and studied. We prove that some results of E. Chafai and M. Mnif [3] related to the stability of the essential descent and descent of a linear relation T everywhere defined such that \({T(0)\subseteq \mathsf{ker}(T)}\) by a finite rank operator F commuting with T, remain valid when F is an everywhere defined linear relation and without the assumption that \({T(0)\subseteq \mathsf{ker}(T)}\). We studied also the stability of the essential g-ascent and the essential ascent under a finite rank relation. Motivated by the recent work of T. Álvarez and A. Sandovici [1], we extend to a closed linear relation, the well known notion of minimum modulus of a linear operator (H. A. Gindler and A. E. Taylor [7]). Also, we introduce and study the new notion of minimum g-modulus for a linear relation.

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Authors and Affiliations

  1. Faculté des Sciences de Monastir, Département de Mathématiques, Avenue de l’environnement, 5019, Monastir, Tunisia

    Z. Garbouj & H. Skhiri

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  1. Z. Garbouj
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  2. H. Skhiri
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Correspondence to H. Skhiri.

Additional information

This work is supported by the Higher Education And Scientific Research In Tunisia, UR11ES52: Analyse, Géométrie et Applications.

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Garbouj, Z., Skhiri, H. Minimum modulus, perturbation for essential ascent and descent of a closed linear relation in Hilbert spaces. Acta Math. Hungar. 151, 328–360 (2017). https://doi.org/10.1007/s10474-016-0683-1

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  • DOI: https://doi.org/10.1007/s10474-016-0683-1

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