Paranormal composition operators
Let \((X,\Sigma,\lambda)\) be a \(\sigma\)-finite measure space. Recall that a measurable transformation \(T: X\to X\) defines a composition operator \(C_ T\) on \(L^ 2(\lambda)\) when \(\lambda\circ T^{-1}\) is absolutely continuous with respect to \(\lambda\), and the Radon-Nikodym derivative \(f_ 0:= d\lambda \circ T^{-1}/d\lambda\) is essentially bounded. Let us denote \(h_ 0:= d\lambda(T\circ T)^{-1}/d\lambda\). An operator \(A\) on a Hilbert space is said to be \(M\)-paranormal if \(\| Tx\|^ 2\leq M\| T^ 2 x\|\) for all \(x\). In this paper, \(M\)- paranormal composition operators and its adjoints are characterized. The operator \(C_ T\) is \(M\)-paranormal if and only if \(M^ 2 h_ 0+ 2kf_ 0+ k^ 2\geq 0\) a.e. for all real numbers \(k\); and \(C_ T^*\) is \(M\)- paranormal if and only if \((f_ 0\circ T)^ 2 P_ 1^ 2\leq h_ 0\circ T^ 2 P_ 2\) a.e., where \(P_ 1\) and \(P_ 2\) are the orthogonal projections onto the closure of the ranges of \(C_ T\) and \(C_{T'}^ 2\) respectively.
scientific article; zbMATH DE number 202461