## 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

Zbl 1114.47027
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### References:

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