# Putnam-Fuglede theorems

*Fuglede–Putnam theorems, Berberian–Putnam–Fuglede theorems*

Let denote a Hilbert space, the algebra of operators on (i.e., bounded linear transformations; cf. Linear transformation; Operator), the derivation (; cf. also Derivation in a ring) and let . If , then is normal (cf. Normal operator; simply choose in ). The question whether the converse assertion, namely "Is kerdAkerdA* for normal A?" , also holds was raised by J. von Neumann in 1942, and answered in the affirmative in 1950 by B. Fuglede [a7], p. 349, 45. C.R. Putnam extended the Fuglede theorem to , for normal and [a7], p. 352, 109, and a beautiful proof of the Putnam–Fuglede theorem was given by M. Rosenblum [a7], p. 352, 118. Introducing the trick of considering the operators and on , S.K. Berberian [a7], p. 347, 9, showed that the Putnam–Fuglede theorem indeed follows from the Fuglede theorem. For this reason, Putnam–Fuglede theorems are sometimes also referred to as Berberian–Putnam–Fuglede theorems.

The Putnam–Fuglede theorem, namely "kerdA,BkerdA*,B* for normal A and B" , has since been considered in a large number of papers, and various generalizations of it have appeared over the past four decades. Broadly speaking, these generalizations fall into the following four types:

i) where the normality is replaced by a weaker requirement, such as subnormality or -hyponormality;

ii) asymptotic Putnam–Fuglede theorems;

iii) Putnam–Fuglede theorems modulo (proper, two-sided) ideals of ; and

iv) Putnam–Fuglede theorems in a -space setting. Before briefly examining some of these, note that there exist subnormal operators and for which [a7], p. 107. This implies that in any generalization of the Putnam–Fuglede theorem to a wider class of operators, the hypotheses on and are not symmetric (and that it is more appropriate to think of and as being normal in the Putnam–Fuglede theorem).

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## Asymmetric Putnam–Fuglede theorems.

If and are subnormal operators with normal extensions and on (say) and , then , and it follows that and . This asymmetric extension of the Putnam–Fuglede theorem was proved by T. Furuta [a6] (though an avatar of this result had already appeared in [a10]). Following a lot of activity during the 1970s and the 1980s ([a2], [a5], [a6], [a9] list some of the references), it is now (1998) known that for and belonging to a large number of suitably paired classes of operators, amongst them -hyponormal (), -hyponormal, dominant and -quasi-hyponormal classes [a5].

## Asymptotic Putnam–Fuglede theorems.

Given normal and , and a neighbourhood of in some topology (weak operator, strong operator or uniform), does there exist a neighbourhood of in the same topology such that ? The answer to this question is (in general) no, for there exists a normal and a (non-uniformly bounded) sequence of operators such that but for all [a8]. If, however, the sequence is uniformly bounded, then the answer is in the affirmative for normal (and subnormal) and [a2] (and indeed, if one limits oneself to the uniform topology, for a number of classes of operators [a5], [a9]).

## Putnam–Fuglede theorems modulo ideals.

Say that the Putnam–Fuglede theorem holds modulo an ideal if, given normal operators and , implies for all . The Putnam–Fuglede theorem holds modulo the compacts (simply consider the Putnam–Fuglede theorem in the Calkin algebra), and does not hold modulo the ideal of finite-rank operators. In a remarkable extension of the Putnam–Fuglede theorem to Schatten-von Neumann ideals , (cf. also Calderón couples), G. Weiss proved in [a12] that implies . It has since been proved that the Putnam–Fuglede theorem holds modulo for all [a1], [a12], and also with normal , replaced by subnormal , . It is not known if the Putnam–Fuglede theorem holds modulo .

## Banach space formulation of the Putnam–Fuglede theorem.

Letting and , where , , , are self-adjoint operators such that and (cf. also Self-adjoint operator), the Putnam–Fuglede theorem can be written as

or, equivalently,

for all . Defining and by and , it is seen that and are Hermitian (i. e., the one-parameter groups and , a real number, are groups of isometries on the Banach space ) which commute. The Putnam–Fuglede theorem now says that if and , then . This version of the Putnam–Fuglede theorem has been generalized to the Banach space setting as follows: if and are commuting Hermitian operators on a complex Banach space , then, given ,

(see [a3], [a4] for more general results).

#### References

[a1] | A. Abdessemed, E.B. Davies, "Some commutator estimates in the Schatten classes" J. London Math. Soc. , 39 (1989) pp. 299–308 |

[a2] | S.T.M. Ackermans, S.J.L. Eijndhoven, F.J.L. Martens, "On almost commuting operators" Nederl. Akad. Wetensch. Proc. Ser. A , 86 (1983) pp. 389–391 |

[a3] | K. Boydazhiev, "Commuting groups and the Fuglede–Putnam theorem" Studia Math. , 81 (1985) pp. 303–306 |

[a4] | M.J. Crabb, P.G. Spain, "Commutators and normal operators" Glasgow Math. J. , 18 (1977) pp. 197–198 |

[a5] | B.P. Duggal, "On generalised Putnam–Fuglede theorems" Monatsh. Math. , 107 (1989) pp. 309–332 (See also: On quasi-similar hyponormal operators, Integral Eq. Oper. Th. 26 (1996), 338-345) |

[a6] | T. Furuta, "On relaxation of normality in the Fuglede–Putnam theorem" Proc. Amer. Math. Soc. , 77 (1979) pp. 324–328 |

[a7] | P.R. Halmos, "A Hilbert space problem book" , Springer (1982) |

[a8] | B.E. Johnson, J.P. Williams, "The range of a normal derivation" Pacific J. Math. , 58 (1975) pp. 105–122 |

[a9] | M. Radjabalipour, "An extension of Putnam–Fuglede theorem for hyponormal operators" Math. Z. , 194 (1987) pp. 117–120 |

[a10] | H. Radjavi, P. Rosenthal, "On roots of normal operators" J. Math. Anal. Appl. , 34 (1971) pp. 653–664 |

[a11] | V. Shulman, "Some remarks on the Fuglede–Weiss Theorem" Bull. London Math. Soc. , 28 (1996) pp. 385–392 |

[a12] | G. Weiss, "The Fuglede commutativity theorem modulo the Hilbert–Schmidt class and generating functions I" Trans. Amer. Math. Soc. , 246 (1978) pp. 193–209 (See also: II, J. Operator Th. 5 (1981), 3-16) |

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Putnam-Fuglede theorems.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Putnam-Fuglede_theorems&oldid=22959