# A Generalization of the Fuglede-Putnam Theorem to Unbounded Operators

Authors: Fotios C. Paliogiannis
Journal: Journal of Operators
 (2015)

---

## Abstract

LetN
,
M be unbounded normal operators in a Hilbert space and let T be a closed operator whose domain D
(
T
) contains the domain of N, and the domain D
(
T
*
) contains the domain of M. It is shown that if T
N
⊆
M
T, then T
N
*
⊆
M
*
T.


---

## Body

## 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.LetH be a complex Hilbert space and let B
(
H
) be the algebra of bounded linear operators in H. Let O
p
(
H
) denote the set of unbounded densely defined linear operators in H. For A
∈
O
p
(
H
) we denote the domain of A by D
(
A
). Given A
,
B
∈
O
p
(
H
), the operator B is called anextension of A, denoted by A
⊆
B, if D
(
A
)
⊆
D
(
B
) and A
x
=
B
x for all x
∈
D
(
A
). An operator A
∈
O
p
(
H
) is calledclosed if A
=
A
¯ (the closure of A). A closed densely defined operator A
∈
O
p
(
H
) is said tocommute with the bounded operator T
∈
B
(
H
), if T
A
⊆
A
T. This means that for each x
∈
D
(
A
), we have T
x
∈
D
(
A
) and T
A
x
=
A
T
x. Let A
′
=
T
∈
B
(
H
)
:
T
A
⊆
A
T. If A
∈
B
(
H
) this notion agrees with the usual notion ofcommutant. One sees A
′ is a strogly closed subalgebra of B
(
H
), and T
∈
A
′ if and only if T
*
∈
A
*
′. Hence, A
′
∩
A
*
′ is a von Neumann algebra.Definition 1.
LetA
∈
O
p
(
H
) be closed and A a von Neumann algebra. If A
′
⊆
A
′, the operator A is said to be affiliated with A, denoted by A
η
A.
The algebraW
*
(
A
)
=
A
′
∩
A
*
′
′ is the smallest von Neumann algebra with which A is affiliated, and is referred to it as thevon Neumann algebra generated by A.Definition 2.
LetA
∈
O
p
(
H
). A bounding sequence for A is a non-decreasing sequence F
n
n
∈
N of projections on H such that ⋁
n
=
1
∞
F
n
=
I, F
n
A
⊆
A
F
n and A
F
n
∈
B
(
H
) for all n
∈
N.Lemma 3 (see [1]).
IfA is an abelian von Neumann algebra and A
η
A, then there is a bounding sequence F
n for A such that F
n
∈
A and A
F
n
∈
A for all n
∈
N.A closed operatorN
∈
O
p
(
H
) isnormal if N
*
N
=
N
N
*. This implies that D
(
N
)
=
D
(
N
*
) and N
x
=
N
*
x for every x
∈
D
(
N
) [2, page 51]. It turns out that the von Neumann algebra W
*
(
N
) is abelian, and W
*
(
N
)
=
N
′′ [3]. Hence, from Lemma 3, there is a bounding sequence F
n for N in W
*
(
N
). In fact, F
n
=
E
n
-
E
-
n for each n
∈
N, where E
λ
λ
∈
R is the spectral family of the selfadjoint operator N
*
N [1].


## 2. Results

The Fuglede-Putnam theorem [4] in its classical form states the following.Theorem 4 (Fuglede-Putnam).
LetN and M be normal operators in a Hilbert space. If T is any bounded operator satisfying T
N
⊆
M
T, then T
N
*
⊆
M
*
T.The following result from [2, page 97] is essential to our proof of the generalization of the Fuglede-Putnam theorem.Lemma 5.
LetA
1
,
A
2
∈
O
p
(
H
) be self-adjoint operators and let T
∈
B
(
H
). Then T
A
1
⊆
A
2
T if and only if T
E
λ
=
P
λ
T for all λ
∈
R, where E
λ
λ
∈
R and P
λ
λ
∈
R are the spectral families of A
1 and A
2, respectively.Theorem 6.
LetN
,
M
∈
O
p
(
H
) be normal operators and let T
∈
O
p
(
H
) be a closed operator such that D
(
N
)
⊆
D
(
T
) and D
(
M
)
⊆
D
(
T
*
). If T
N
⊆
M
T, then T
N
*
⊆
M
*
T.Proof.
LetE
λ
λ
∈
R and P
λ
λ
∈
R be the spectral families of the self-adjoint operators N
*
N and M
*
M, respectively. For m
,
n
∈
N, consider the bounding sequences F
n
=
E
n
-
E
-
n and G
m
=
P
m
-
P
-
m for N and M, respectively. Since D
(
N
)
⊆
D
(
T
), it follows H
=
D
(
N
F
n
)
⊆
D
(
T
F
n
). Since T
F
n is closed, the closed graph theorem implies T
F
n
∈
B
(
H
). Similarly, by the hypothesis on the domain of M and the closed graph theorem, we see T
*
G
m
∈
B
(
H
).
From the hypothesisT
N
⊆
M
T, we have T
N
F
n
⊆
M
T
F
n. Moreover, since F
n
N
⊆
N
F
n, we also have T
F
n
N
⊆
T
N
F
n. Hence,(1)
(
T
F
n
)
N
⊆
M
T
F
n
,
∀
n
∈
N
.Since T
F
n is bounded, the Fuglede-Putnam theorem implies(2)
(
T
F
n
)
N
*
⊆
M
*
T
F
n
,
∀
n
∈
N
.From (1), (2), we have (
T
F
n
)
N
*
N
⊆
M
*
(
T
F
n
)
N
⊆
M
*
M
(
T
F
n
). That is,(3)
(
T
F
n
)
N
*
N
⊆
M
*
M
(
T
F
n
)
,
∀
n
∈
N
.Consequently, from Lemma 5,(4)
(
T
F
n
)
E
λ
=
P
λ
(
T
F
n
)
,
∀
λ
∈
R
.Therefore(5)
T
F
n
F
k
=
G
m
T
F
n
,
∀
k
,
n
,
m
∈
N
.Taking adjoints in (5) we have (6)
G
m
T
F
n
*
=
T
F
n
F
k
*
=
F
k
T
F
n
*
⊇
F
k
F
n
T
*
.But (7)
G
m
T
F
n
*
=
T
F
n
*
G
m
⊇
F
n
T
*
G
m
.As F
n
T
*
G
m
∈
B
(
H
), we get(8)
F
k
F
n
T
*
⊆
F
n
T
*
G
m
.Furthermore, since F
n and F
k commute, (9)
F
n
F
k
T
*
⊆
F
n
T
*
G
m
;that is, for every x
∈
D
(
T
*
), we have G
m
x
∈
D
(
T
*
) and (10)
F
n
F
k
T
*
x
=
F
n
T
*
G
m
x
.Let x
∈
D
(
T
*
) and fix k
,
m
<
n. Then since F
n
→
I (strongly) as n
→
∞, it follows(11)
F
k
T
*
⊆
T
*
G
m
,
∀
k
,
m
∈
N
.Taking adjoints in (11) and using the closeness of T, (12)
F
k
T
*
*
⊇
T
*
G
m
*
⊇
G
m
T
*
*
=
G
m
T
¯
=
G
m
T
.But (
F
k
T
*
)
*
=
T
*
*
F
k
=
T
¯
F
k
=
T
F
n. Hence,(13)
G
m
T
⊆
T
F
k
,
∀
k
,
m
∈
N
.
Multiplying (2) by F
n, we get (
T
F
n
)
N
*
F
n
⊆
M
*
(
T
F
n
)
F
n
=
M
*
T
F
n. Since (
T
F
n
)
(
N
*
F
n
)
=
T
N
*
F
n and (
T
F
n
)
(
N
*
F
n
)
∈
B
(
H
), we obtain(14)
T
N
*
F
n
=
M
*
T
F
n
∀
n
∈
N
.
Now letx
∈
D
(
T
N
*
); that is, x
∈
D
(
N
*
) and N
*
x
∈
D
(
T
). Fix m
>
k, and let m
→
∞. Then using (13) and the fact G
m
→
I (strongly), we have (15)
T
F
k
x
=
G
m
T
x
⟶
T
x
.Moreover, from (14), the fact F
n
N
*
⊆
N
*
F
n, and (13), we have(16)
M
*
T
F
k
x
=
T
N
*
F
k
x
=
T
F
k
N
*
x
=
G
m
T
N
*
x
⟶
T
N
*
x
.Since M
* is closed, it follows x
∈
D
(
M
*
T
) and M
*
T
x
=
T
N
*
x. Therefore, T
N
*
⊆
M
*
T.As a special case forM
=
N, we obtain the following generalization of Fuglede’s theorem [5].Corollary 7.
LetN
∈
O
p
(
H
) be normal and let T
∈
O
p
(
H
) be a closed operator such that D
N
⊆
D
T
∩
D
(
T
*
). If  T
N
⊆
N
T, then T
N
*
⊆
N
*
T.Corollary 8.
LetN
1
,
N
2
∈
O
p
(
H
) be normal operators. If D
(
N
1
)
⊆
D
(
N
2
), then N
2
N
1
⊆
N
1
N
2
⇔
N
2
N
1
*
⊆
N
1
*
N
2.Corollary 9.
LetN
,
N
1
,
N
2
∈
O
p
(
H
) be normal operators. If D
(
N
i
)
⊆
D
(
N
), for i
=
1,2, then N
N
1
⊆
N
2
N
⇔
N
N
1
*
⊆
N
2
*
N.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 equalityP
B
m
M
A
N
*
P
B
n
(
N
)
x
=
P
B
m
M
M
*
A
P
B
n
(
N
)
x, the fact P
B
m
M
→
I (strongly) gives A
N
*
P
B
n
(
N
)
x
=
M
*
A
P
B
n
(
N
)
x; however, (dealing with unbounded operators, as is the case here) the fact (alone) that P
B
n
N
→
I (strongly) does not give the equality A
N
*
x
=
M
*
A
x.



---

*Source: 804353-2015-03-25.xml*