## Seminormal composition operators.(English)Zbl 0870.47020

Let $$(X,\Sigma,\mu)$$ be a complete $$\sigma$$-finite Lebesgue space. A transformation $$T:X\to Y$$ will be a measurable mapping with the property that $$\mu\circ T^{-1}\ll\mu$$. The composition operator $$C_T$$ is defined on the set of all measurable functions on $$X$$, by $$C_T(f)=f\circ T$$. A weighted composition operator $$W_{T,\omega}$$ is defined on the set of measurable functions on $$X$$ by $$W_{T,\omega}f(x)= \omega(x)f(Tx)$$, where $$\omega$$ is a non-negative finite complex valued measurable function. E. A. Nordgren [Lect. Notes Math. 693, 37-63 (1978; Zbl 0411.47022)] asked about the conditions on $$T$$ which would characterize the normality of $$C_T$$. R. Whitley [Proc. Am. Math. Soc. 70, 114-118 (1978; Zbl 0391.47018)] and R. K. Singh and A. A. Kumar [Bull. Austral. Math. Soc. 19, 81-95 (1978; Zbl 0385.47017)] characterized the normality of $$C_T$$.
The normality of $$W_T$$ was characterized by U. Krengel [Ergodic theorems (1985; Zbl 0575.28009)]. Also Whitley [loc. cit.] characterized the normality and quasinormality of $$C_T$$. The quasinormal $$W_T$$ was characterized by J. T. Campbell, M. Embry-Wardrop, R. J. Fleming, and S. K. Narayan [Glasg. Math. J. 33, No. 3, 275-279 (1991; Zbl 0818.47030)]. D. Harrington and R. Whitley [J. Oper. Theory 11, 125-135 (1984; Zbl 0534.47017)] obtained the characterization of cohyponormality and coquasinormality for $$C_T$$ and hyponormality of $$C_T$$ where $$\mu(X)<\infty$$. A. Lambert [Bull. Lond. Math. Soc. 18, 395-400 (1986; Zbl 0624.47014)] gave a characterization of $$W_T$$. In the present article the coquasinormality and cohyponormality of $$W_T$$ is determined.

### MSC:

 47B38 Linear operators on function spaces (general) 47B20 Subnormal operators, hyponormal operators, etc.