×

Composition operators with weak hyponormality. (English) Zbl 1129.47023

Let \((X, \mathcal{F}, \mu)\) be a \(\sigma\) finite measure space and let \(T:X\to X\) be a transformation such that \(T^{-1}\mathcal{F}\subset\mathcal{F}\) and \(\mu\circ T^{-1}\ll\mu\). It is assumed that the Radon–Nikodym derivative \(h=d\mu\circ T^{-1}/d\mu\) is in \(L^{\infty}\). The composition operator \(C\) acting on \(L^{2}(X,\mathcal{F},\mu)\) is defined by \(Cf=f\circ T\). The authors obtain characterizations of the classes of \(p\)-quasihyponormal (i.e., \(C^{*}(C^{*}C)^{p}C\geq C^{*}(CC^{*})^{p}C\)), A(\(p\)) (i.e., \((C^{*}| C| ^{2p}C)^{\frac{1}{p+1}}\geq | C| ^{2}\)), absolute-\(p\)-paranormal (i.e., \(\| | C| ^{p}Cx\| \geq \| Cx\| ^{p+1}\) for all unit vectors \(x\)) and \(p\)-paranormal (i.e., \(\| | C| ^{p}| C^{*}|^{p}x\| \geq \| | C|^{p} x\|^{2}\) for all unit vectors \(x\)) composition operators for a fixed \(p>0\). Moreover, the authors show that these classes are coincide with each other in the composition operator case. Finally, the authors give some examples of composition operators which belong to the above classes.

MSC:

47B33 Linear composition operators
47B20 Subnormal operators, hyponormal operators, etc.
47A63 Linear operator inequalities

Citations:

Zbl 1114.47027
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Aluthge, A.; Wang, D., w-hyponormal operators, Integral equations operator theory, 36, 1-10, (2000) · Zbl 0938.47021
[2] Burnap, C.; Jung, I.; Lambert, A., Separating partial normality classes with composition operators, J. operator theory, 53, 381-397, (2005) · Zbl 1114.47027
[3] Campbell, J.; Hornor, W., Seminormal composition operators, J. operator theory, 29, 323-343, (1993) · Zbl 0870.47020
[4] Furuta, T., Invitation to linear operators, (2001), Taylor & Francis Inc.
[5] Harrington, D.; Whitley, R., Seminormal composition operators, J. operator theory, 11, 125-135, (1984) · Zbl 0534.47017
[6] Ito, M.; Yamazaki, T., Relations between two inequalities \((B^{r / 2} A^p B^{r / 2})^{r /(p + r)} \geqslant B^r\) and \(A^p \geqslant(A^{p / 2} B^r A^{p / 2})^{p /(p + r)}\) and their applications, Integral equations operator theory, 44, 442-450, (2002) · Zbl 1028.47013
[7] Lambert, A., Hyponormal composition operators, Bull. London math. soc., 18, 395-400, (1986) · Zbl 0624.47014
[8] Yamazaki, T.; Yanagida, M., A further generalization of paranormal operators, Sci. math., 3, 23-31, (2000) · Zbl 0958.47011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.