On relaxation of normality in the Fuglede-Putnam theorem

Author:
Takayuki Furuta

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

MSC:
Primary 47B20

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

MathSciNet review:
545590

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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 , then . We shall relax the normality in the hypotheses on *A* and *B*.

Theorem 1. *If A and* *are subnormal and if X is an operator such that* , *then* .

Theorem 2. *Suppose A, B, X are operators in the Hilbert space H such that* . *Assume also that X is an operator of Hilbert-Schmidt class. Then* *under any one of the following hypotheses*:

(i) *A is k-quasihyponormal and* is invertible hyponormal,

(ii) *A is quasihyponormal and* is invertible hyponormal,

(iii) *A is nilpotent and* is invertible hyponormal.

**[1]**S. K. Berberian,*Extensions of a theorem of Fuglede and Putnam*, Proc. Amer. Math. Soc.**71**(1978), no. 1, 113–114. MR**0487554**, https://doi.org/10.1090/S0002-9939-1978-0487554-2**[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**0435914****[3]**Paul R. Halmos,*A Hilbert space problem book*, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0208368****[4]**C. R. Putnam,*On normal operators in Hilbert space*, Amer. J. Math.**73**(1951), 357–362. MR**0040585**, https://doi.org/10.2307/2372180**[5]**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**0410448**, https://doi.org/10.1512/iumj.1976.25.25031

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Additional Information

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

Keywords:
Quasinormal operators,
subnormal operators,
hyponormal operators,
quasihyponormal operators,
Hilbert-Schmidt operators

Article copyright:
© Copyright 1979
American Mathematical Society