Abstract
In this paper, we study the single-valued extension property of hyponormal operators on Banach spaces \({\mathcal {X}}\). In particular, we prove that if a bounded linear operator T on \({\mathcal {X}}\) has the property (II) or the property (I\(')\) (see Definition 2.3), then T has the single-valued extension property. Moreover, we show that for strictly convex (resp., smooth) \(\mathcal {{\mathcal {X}}}\), if \(T\in {{\mathcal {L}}}(\mathcal {{\mathcal {X}}})\) is hyponormal (resp., \(\,^*\)-hyponormal) on \(\mathcal {{\mathcal {X}}}\), then T has the single-valued extension property.
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References
Aiena, P.: Fredholm and Local Spectral Theory II, Lecture Notes in Mathematics, vol. 2235. Springer, New York (2018)
Berberian, S.K.: Approximate proper vectors. Proc. Am. Math. Soc. 13, 111–114 (1962)
Bonsall, F.F., Duncan, J.: Numerical ranges of operators on normed spaces and of elements of normed algebras, London Math. Soc. Lecture Note Series, vol. 2. Cambridge University Press, London, (1971)
Bonsall, F.F., Duncan, J.: Numerical Ranges II, London Math. Soc. Lecture Note Series, vol. 10. Cambridge University Press, London (1973)
Chō, Muneo: Hyponormal operators on uniformly convex spaces. Acta Sci. Math. (Szeged) 55, 141–147 (1991)
Chō, M., Tanahashi, K.: On conjugations for Banach spaces. Sci. Math. Jpn. 81(1), 37–45 (2018)
de Barra, G.: Some algebras of operators with closed convex numerical ranges. Proc. R. Irish Acad. Sect. A 72, 149–154 (1972)
de Barra, G.: Generalised limits and uniform convexity. Proc. R. Irish Acad. Sect. A 74, 73–77 (1974)
Garcia, S.R., Putinar, M.: Complex symmetric operators and applications. Trans. Am. Math. Soc. 358, 1285–1315 (2006)
Laursen, K., Neumann, M.: An Introduction to Local Spectral Theory. Clarendon Press, Oxford (2000)
Mattila, K.: Normal operators and proper boundary points of the spectra of operators on an Banach space. Math. Ph.D. Dissertations 19 (1978)
Mattila, K.: On proper boundary points of the spectrum and complemented eigenspace. Math. Scand. 43, 363–368 (1978)
Mattila, K.: Complex strict and uniform convexity and hyponormal operators. Math. Proc. Camb. Philos. Soc. 96, 483–493 (1984)
Mattila, K.: A class of hyponormal operators and weak $\,^*$-continuity of Hermitian operators. Arkiv Mat. 25, 265–274 (1987)
Motoyoshi, H.: Linear operators and conjugations on a Banach space. Acta Sci. Math. (Szeged) 85, 325–336 (2019)
Uchiyama, A., Tanahashi, K.: Bishop’s property $(\beta )$ for paranormal operators. Oper. Matrices 3(4), 517–524 (2009)
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This research is partially supported by Grant-in-Aid Scientific Research No. 15K04910. Injo Hur was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2019R1F1A1061300), and the Ji Eun Lee was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (2019R1A2C1002653)
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Chō, M., Hur, I. & Lee, J.E. On the Single-Valued Extension Property of Hyponormal Operators on Banach Spaces. Mediterr. J. Math. 18, 34 (2021). https://doi.org/10.1007/s00009-020-01666-5
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DOI: https://doi.org/10.1007/s00009-020-01666-5