On relaxation of normality in the Fuglede-Putnam theorem

Furuta, Takayuki

doi:10.1090/S0002-9939-1979-0545590-2

On relaxation of normality in the Fuglede-Putnam theorem
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by Takayuki Furuta

Proc. Amer. Math. Soc. 77 (1979), 324-328

DOI: https://doi.org/10.1090/S0002-9939-1979-0545590-2

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Abstract:

An operator means a bounded linear operator on a complex Hilbert space. The familiar Fuglede-Putnam theorem asserts that if A and B are normal operators and if X is an operator such that $AX = XB$, then ${A^ \ast }X = X{B^ \ast }$. We shall relax the normality in the hypotheses on A and B. Theorem 1. If A and ${B^\ast }$ are subnormal and if X is an operator such that $AX = XB$, then ${A^ \ast }X = X{B^ \ast }$. Theorem 2. Suppose A, B, X are operators in the Hilbert space H such that $AX = XB$. Assume also that X is an operator of Hilbert-Schmidt class. Then ${A^ \ast }X = X{B^ \ast }$ under any one of the following hypotheses: (i) A is k-quasihyponormal and ${B^ \ast }$ is invertible hyponormal, (ii) A is quasihyponormal and ${B^\ast }$ is invertible hyponormal, (iii) A is nilpotent and ${B^\ast }$ is invertible hyponormal.

References

S. K. Berberian, Extensions of a theorem of Fuglede and Putnam, Proc. Amer. Math. Soc. 71 (1978), no. 1, 113–114. MR 487554, DOI 10.1090/S0002-9939-1978-0487554-2
Takayuki Furuta, Kyoko Matsumoto, and Nobuhiro Moriya, A simple condition on hyponormal operators implying subnormality, Math. Japon. 21 (1976), no. 4, 399–400. MR 435914
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
C. R. Putnam, On normal operators in Hilbert space, Amer. J. Math. 73 (1951), 357–362. MR 40585, DOI 10.2307/2372180
Joseph G. Stampfli and Bhushan L. Wadhwa, An asymmetric Putnam-Fuglede theorem for dominant operators, Indiana Univ. Math. J. 25 (1976), no. 4, 359–365. MR 410448, DOI 10.1512/iumj.1976.25.25031