Journal of Operators
Volume 2015 (2015), Article ID 804353, 3 pages
http://dx.doi.org/10.1155/2015/804353

Research Article

A Generalization of the Fuglede-Putnam Theorem to Unbounded Operators

Fotios C. Paliogiannis

Department of Mathematics, St. Francis College, 180 Remsen Street, Brooklyn Heights, NY 11201, USA

Received 8 September 2014; Accepted 8 March 2015

Academic Editor: El Hassan Youssfi

Copyright © 2015 Fotios C. Paliogiannis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be unbounded normal operators in a Hilbert space and let be a closed operator whose domain contains the domain of , and the domain contains the domain of . It is shown that if , then .

1. Introduction

In this note we prove a generalization of the classical Fuglede-Putnam theorem to unbounded operators. A special case of this generalization is given in [1]. We begin with some preliminary results.

Let be a complex Hilbert space and let be the algebra of bounded linear operators in . Let denote the set of unbounded densely defined linear operators in . For we denote the domain of by . Given , the operator is called an extension of , denoted by , if and for all . An operator is called closed if (the closure of ). A closed densely defined operator is said to commute with the bounded operator , if . This means that for each , we have and . Let . If this notion agrees with the usual notion of commutant. One sees is a strogly closed subalgebra of , and if and only if . Hence, is a von Neumann algebra.

Definition 1. Let be closed and a von Neumann algebra. If , the operator is said to be affiliated with , denoted by .
The algebra is the smallest von Neumann algebra with which is affiliated, and is referred to it as the von Neumann algebra generated by .

Definition 2. Let . A bounding sequence for is a non-decreasing sequence of projections on such that , and for all .

Lemma 3 (see [1]). If is an abelian von Neumann algebra and , then there is a bounding sequence for such that and for all .

A closed operator is normal if . This implies that and for every [2, page 51]. It turns out that the von Neumann algebra is abelian, and [3]. Hence, from Lemma 3, there is a bounding sequence for in . In fact, for each , where is the spectral family of the selfadjoint operator [1].

2. Results

The Fuglede-Putnam theorem [4] in its classical form states the following.

Theorem 4 (Fuglede-Putnam). Let and be normal operators in a Hilbert space. If is any bounded operator satisfying , then .

The following result from [2, page 97] is essential to our proof of the generalization of the Fuglede-Putnam theorem.

Lemma 5. Let be self-adjoint operators and let . Then if and only if for all , where and are the spectral families of and , respectively.

Theorem 6. Let be normal operators and let be a closed operator such that and . If , then .

Proof. Let and be the spectral families of the self-adjoint operators and , respectively. For , consider the bounding sequences and for and , respectively. Since , it follows . Since is closed, the closed graph theorem implies . Similarly, by the hypothesis on the domain of and the closed graph theorem, we see .
From the hypothesis , we have . Moreover, since , we also have . Hence,

Since is bounded, the Fuglede-Putnam theorem implies

From (1), (2), we have . That is,

Consequently, from Lemma 5,

Therefore

Taking adjoints in (5) we have

But

As , we get

Furthermore, since and commute,

that is, for every , we have and

Let and fix . Then since (strongly) as , it follows

Taking adjoints in (11) and using the closeness of ,

But . Hence,

Multiplying (2) by , we get . Since and , we obtain

Now let ; that is, and . Fix , and let . Then using (13) and the fact (strongly), we have

Moreover, from (14), the fact , and (13), we have

Since is closed, it follows and . Therefore, .

As a special case for , we obtain the following generalization of Fuglede’s theorem [5].

Corollary 7. Let be normal and let be a closed operator such that . If  , then .

Corollary 8. Let be normal operators. If , then .

Corollary 9. Let be normal operators. If , for , then .

Remark 10. Recently in the article “An All-Unbounded-Operator Version of the Fuglede-Putnam Theorem,” Complex Analysis and Operator Theory (2012) [6: 1269–1273], a similar result was offered, but its proof is incorrect. In fact, on the last page of this paper [page 1273] the proof is wrong; note that from the equality , the fact (strongly) gives ; however, (dealing with unbounded operators, as is the case here) the fact (alone) that (strongly) does not give the equality .

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

References

F. C. Paliogiannis, “A note on the Fuglede-Putnam theorem,” Proceedings—Mathematical Sciences, vol. 123, no. 2, pp. 253–256, 2013.

K. Schmündgen, Unbounded self-adjoint operators in Hilbert space, vol. 265 of Graduate Texts in Mathematics, Springer, 2012.

F. C. Paliogiannis, “On Fuglede's theorem for unbounded normal operators,” Ricerche di Matematica, vol. 51, no. 2, pp. 261–264 (2003), 2002.

C. R. Putnam, “On normal operators in Hilbert space,” American Journal of Mathematics, vol. 73, pp. 357–362, 1951.

B. Fuglede, “A commutativity theorem for normal operators,” Proceedings of the National Academy of Sciences of the United States of America, vol. 36, pp. 35–40, 1950.