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 impliesFrom (1), (2), we have . That is,Consequently, from Lemma 5,ThereforeTaking adjoints in (5) we have But As , we getFurthermore, since and commute, that is, for every , we have and Let and fix . Then since (strongly) as , it followsTaking adjoints in (11) and using the closeness of , But . Hence,Multiplying (2) by , we get . Since and , we obtainNow let ; that is, and . Fix , and let . Then using (13) and the fact (strongly), we have Moreover, from (14), the fact , and (13), we haveSince 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.