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).

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)