Operator properties of \(T\) and \(K(T)\)
Let \(\chi\) be an irreducible character. For linear characters \(\chi\), \textit{C.-K. Li} and \textit{A. Zaharia} [Trans. Am. Math. Soc. 354, No. 2, 807--836 (2002; Zbl 0984.15025)] gave necessary and sufficient conditions on \(\chi\), the operators \(T, S\), and the induced operator \(K(T)\) so that the following hold: (I) If \(K(T)\neq 0\) is normal and unitary, then \(T\) has the corresponding property. (II) If there exists \(\eta\in\mathbb C\) with \(|\eta|=1\) such that \(\eta K(T)\neq 0\) is Hermitian (respectively, positive definite or positive semi-definite), then \(\xi T\) also has the corresponding property for some \(\xi\in\mathbb C\) with \(\xi^m= \pm\eta\) (respectively, \(\xi^m= \eta\)). (III) Suppose \(K(T)\neq 0\). Then a linear operator \(S\) satisfies \(K(S)= K(T)\) if and only if \(S=\xi T\) for some \(\xi\in\mathbb C\) with \(\xi^m=1\). The paper is organized as follows. In Section 2, the authors give some prelimary results for induced operators. In Section 3 they present several lemmas. In Section 4 they divide \((X, n)\) into several classes, that will determine whether (I), (II), or (III) hold subsequently; some examples will be given to these classes. In Section 5 they determine the necessary and sufficient conditions on the irreducible character \(X\) on \(H\) and operators \(T\) and \(S\) on \(V\) for which (I) or (II) holds. In Section 6 they determine when (III) holds.
scientific article; zbMATH DE number 2173148
scientific article