On $m$-quasi class $\mathcal{A}(k^{*})$ and absolute-$(k^{*},m)$-paranormal operators

Year 2018, Volume: 47 Issue: 6, 1564 - 1577, 12.12.2018

İlmi Hoxha Naim L. Braha , Kotaro Tanahashi

Abstract

In this paper, we introduce a new class of operators, called $m$-quasi class $\mathcal{A}(k^{*})$ operators, which is a superclass of hyponormal operators and a subclass of absolute-$(k^{*},m)$-paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that if $T$ is $m$-quasi class $\mathcal{A}(k^{*})$, then $\sigma _{np}(T)\setminus \{0\}=\sigma _{p}(T)\setminus \{0\}$, $\sigma _{na}(T)\setminus \{0\}=\sigma _{a}(T)\setminus \{0\}$ and $T-\mu $ has finite ascent for all $\mu\in\mathbb{C}.$ Also, we consider the tensor product of $m$-quasi class $\mathcal{A}(k^{*})$ operators.

Keywords

$m$-quasi class $\mathcal{A}(k^{*})$, absolute-$(k^{*};m)$-paranormal

References

• Aiena,P., Semi-Fredholm operators, perturbations theory and localized SVEP, XX Escuela Venezolana de Matematicas, Merida, Venezuela 2007.
• Aluthge, A., Wang, D., The joint approximate point spectrum of an operator, Hokkaido Math. J., 31 (2002), 187-197.
• Ando, T., Operators with a norm condition, Acta Sci. Math.(Szeged), 33, 169-178, 1972.
• Arora, S. C., Thukral, J. K., On a class of operators, Glas. Math. Ser. III, 21(41) no.2, 381-386, 1986.
• Duggal, B. P., Jeon, I. H., Kim, I. H., On $*$-paranormal contractions and properties for $*$-class $\mathcal{A}$ operators, Linear Alg. Appl. 436, 954-962, 2012.
• Furuta, T., On the class of paranormal operators, Proc. Japan Acad. 43, 594-598, 1967.
• Furuta, T., Ito, M., Yamazaki, T., A subclass of paranormal operators including class of log-hyponormal and several related classes, Sci. Math. 1, no.3, 389-403, 1998.
• Han, J. K., Lee, H. Y., Lee, W. Y., Invertible completions of $2\times 2$ upper triangular operator matrices Proc. Amer. Math. Soc., 128, no.1, 119-123, 2000.
• Hansen, F., An operator inequality, Math. Ann. 246, 249-250, 1980.
• Kim, I. H., Weyl's theorem and tensor product for operators satisfying $T^{*k}|T^{2}|T^{k} \geq T^{*k}|T|^{2}T^{k}$, J. Korean Math. Soc. 47, No.2, 351-361, 2010.
• Kim, I. H., On spectral continuities and tensor products of operators, J. Chungcheong Math. Soc., 24, No.1, 113-119, 2011.
• McCarthy, C. A., $c_{p}$, Israel J. Math. 5, 249-271, 1967.
• Panayappan, S., Radharamani, A., A Note on $p$-$*$-paranormal Operators and Absolute-$k^{*}$-Paranormal Operators, Int. J. Math. Anal. 2, no.25-28, 1257-1261, 2008.
• Saito, T., Hyponormal operators and Related topics, Lecture notes in Math., Springer-Verlag, 247, 1971.
• Stochel, J., Seminormality of operators from their tensor products, Proc. Amer. Math. Soc., 124, 435-440, 1996.