##
**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.

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.

Reviewer: K.Seddighi (Shiraz)

### MSC:

47B38 | Linear operators on function spaces (general) |

47B20 | Subnormal operators, hyponormal operators, etc. |