On dominant operators. (English) Zbl 0624.47015
The following Putnam-Fuglede-type theorem is proven: For two Hilbert spaces \(H_ 1\) and \(H_ 2\) let \(A:H_ 1\to H_ 1\), \(B:H_ 2\to H_ 2\), and \(X:H_ 2\to H_ 1\) be bounded linear operators such that \(AX=XB\). Then \(A^*X=XB^*\) if A is dominant (i.e. for each complex number \(\lambda\) there exists a constant \(M_{\lambda}\) such that \(\| (A-\lambda)^*x\| \leq M_{\lambda}\| (A-\lambda)x\|\) for any \(x\in H_ 1)\) and if \(B^*\) is M-hyponormal (i.e. \(B^*\) is dominant and all corresponding constants \(M_{\lambda}\) are bounded by a constant M). As application a number of related results in the setting of dominant and M-hyponormal operators are derived - some of them generalizing more special known results - and conditions implying normality are developed.
Reviewer: G.Garske
MSC:
| 47B20 | Subnormal operators, hyponormal operators, etc. |
References:
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