Quasinormality and subscalarity of class p-wA(s,t) operators

Kotaro Tanahashi


Let $T$ be a bounded linear operator on a complex Hilbert space ${\mathcal H}$ and let $T=U|T|$ be the polar decomposition of $T$.  $T $ is called a class $p$-$wA(s,t)$ operator
 if $(|T^{*}|^{t}|T|^{2s}|T^{*}|^{t})^{\frac{tp}{s+t}}\geq |T^{*}|^{2tp}$ and
$(|T|^{s}|T^{*}|^{2t}|T|^{s})^{\frac{sp}{s+t}}\leq |T|^{2sp}$ where $0 <s, t $ and $0 < p \leq 1$.
We investigate quasinormality and
subscarity of class $p$-$wA(s,t)$ operators.

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bibitem{aa} A. Aluthge, textit{On p-hyponormal operators for $0 < p < 1$} , Integral Equations Operator Theory., textbf{13} (1990), 307 -- 315.

bibitem{ae} E. Albrecht and J. Eschmeier, textit{Analytic functional model and local spectral theory}, Proc. London Math. Soc., textbf{75} (1997), 323--348.

bibitem{bz} C. Benhida and E. H. Zerouali, textit{Local spectral theory of linear operators $RS$ and $SR$} , Integral Equations Operator Theory., textbf{54} (2006), 1--8.

bibitem{chen} L. Chen, R. Yingbin, and Y. Zikun, textit{$p$-Hyponormal operators are subscalar}, Proc. Amer. Math. Soc., textbf{131} (2003), 2753--2759.

bibitem{douglas} R. G. Douglas, textit{On majorization, factorization, and range inclusion of operators on Hilbert space}, Proc. Amer. Math. Soc., textbf{17} (1966), 413--415.

bibitem{ep} J. Eschmeier and M. Putinar, textit{Bishop's condition $(beta )$ and rich extensions of linear operators}, Indiana Univ. Math. J., textbf{37} (1988), 325--348.

bibitem{fjll} M. Fujii, D. Jung, S. H. Lee., M. Y. Lee., and R. Nakamoto, textit{Some classes of operators related to paranormal and log hyponormal operators}, Math. Japon., textbf{51} (2000), 395--402.


T. Furuta, {it $A geq B geq O$ assures $(B^{r}A^{p}B^{r})^{frac{1}{q}} geq B^{frac{p+2r}{q}} $ for $ rgeq 0, p geq 0, q geq 1$ with

$(1+2r)qgeq (p+2r)$,} Proc. Amer. Math. Soc., textbf{101} (1987), 85--88.

bibitem{ito} M. Ito, textit{ Some classes of operators with generalised Aluthege transformations}, SUT J. Math., textbf{35} (1999), 149--165.

bibitem{iy} M. Ito and T. Yamazaki, textit{Relations between two inequalities $(B^{frac{r}{2}}A^{p}B^{frac{r}{2}})^{frac{r}{P+r}}geq B^{r}$ and $A^{p}geq (A^{frac{p}{2}}B^{r}A^{frac{p}{2}})^{frac{r}{P+r}}$ and their applications}, Integral Equations Operator Theory, textbf{44} (2002), 442--450.

bibitem{jkp} I. B. Jung, E. Ko and C. Pearcy, textit{Aluthege transforms of operators}, Integral Equations Operator Theory, textbf{37} (2000), 437--448.

bibitem{ko} Eungil Ko, textit{ $w$-Hyponormal operators have scalar extensions}, Integral Equations Operator Theory, textbf{53} (2005), 363--372.

bibitem{ptuy} S. M. Patel, K. Tanahashi, A. Uchiyama and M. Yanagida,

textit{Quasinormality and Fuglede-Putnam theorem for class $A(s,t)$ operators}, Nihonkai Math. J., textbf{17} (2006), 49--67.

bibitem{pt} T. Prasad and K. Tanahashi, textit{On class $p$-$wA(s,t)$ operators}, Functional Analysis,

Approximation Computation, textbf{6} (2) (2014), 39--42.

bibitem{my} M. Yanagida, textit{Powers of class $wA(s,t)$ operators with generalised Aluthge transformation}, J. Inequal. Appl., textbf{7} (2002), 143--168.

bibitem{yy1} J. Yuan and C. Yang, textit{Spectrum of class $wF(p,r,q)$ operators for $p+r leq 1$ and $q geq 1$}, Acta Sci. Math. (Szeged), textbf{71} (2005), 767--779.

bibitem{yy2} J. Yuan and C. Yang, textit{Powers of class $wF(p,r,q)$ operators}, Journal Inequalities in Pure and Applied Math., textbf{7} (2006), Issue 1, article 32.

bibitem{yy3} C. Yang and J. Yuan, textit{On class $wF(p,r,q)$ operators}, Acta. Math. Sci., textbf{27} (2007), 769--780.


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