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The Putnam-Fuglede theorems on some non-normal operators. (Chinese) Zbl 0584.47020

Some results for non-normal operators relative to the Putnam-Fuglede theorem are obtained. For example, the author proves:
(1) Let T and S be contractions. If for some \(n,T^ n\) and \(S^{*_ n}\) are dominant (or para-normal), then \(TXS=X\) implies \(T^*XS^*=X;\)
(2) Let \(T_ i\) be M-hyponormal and \(N_ i\) be normal for which \(N_ iT_ i=T_ iN_ i\), \(i=1,2\). Then \(T_ 1XT_ 2=N_ 1XN_ 2\) \((T_ 1XN_ 2=N_ 1XT_ 2)\) implies \(T^*_ 1XT^*_ 2=N^*_ 1XN^*_ 2\) \((T^*_ 1XN^*_ 2=N^*_ 1XT^*_ 2);\)
(3) If T is hyponormal and \(S^*\) subnormal, then for every (bounded) operator X, the inequalities \(\| TX-XS\|_ 2\geq \| T^*X- XS^*\|_ 2\) and \(\| TXS-X\|_ 2\geq \| T^*XS^*- X\|_ 2\) hold, where \(\| \cdot \|_ 2\) is the Hilbert-Schmidt norm (this result has been generalized by the author to that if \(T_ i\) are hyponormal and \(S^*_ i\) subnormal, then \(\| T_ 1XS_ 1-T_ 2XS_ 2\|_ 2\geq \| T^*_ 1XS^*_ 1-T^*_ 2XS^*_ 2\|_ 2\) holds for all X).

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
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