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Seminormality for measure-theoretic composition operators. (English) Zbl 0728.47022

Operator theory, operator algebras and applications, Proc. Summer Res. Inst., Durham/NH (USA) 1988, Proc. Symp. Pure Math. 51, Pt. 2, 183-186 (1990).
[For the entire collection see Zbl 0699.00028.]
The author indicates in notes of this paper that the results stated are considered in more detail and proved in other papers. If (X,\(\Sigma\),m) is a \(\sigma\)-finite measure space, and if T: \(X\to X\) is a measurable transformation mapping onto X, then the ‘composition operator’ C on \(L^ 2(X,\Sigma,m)=L^ 2_ m\) is defined by \(C(f)=f\circ T.\) In one of the results of the paper relating to hyponormality, it is stated that if T is invertible, C is hyponormal or ‘power hyponormal’ if and only if \(h\circ T\leq h\) m-a.e., where \(h=\frac{dm\circ T^{-1}}{dm}.\)
Results relating to cases in which the operator C is subnormal include a representation \(h_ n(x)=\int_{I}t^ nd\mu_ x(t)\), where \(h_ n=\frac{dm\circ T^{-n}}{dm}\), and are contained in papers by the author including: Proc. Amer. Math. Soc. 103, No.3, 750-754 (1988; Zbl 0669.47015), and Michigan Math. J. 35, 443-450 (1988; Zbl 0675.47012).

MSC:

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