# A Formula for the Numerical Range of Elementary Operators

Authors: M. Barraa
Journal: ISRN Mathematical Analysis
 (2014)

---

## Abstract

LetB
H be the algebra of bounded linear operators on a complex Hilbert space H. For k-tuples of elements of B
H
,
A
=
A
1
,
…
,
A
k and B
=
B
1
,
…
,
B
k, let R
A
,
B denote the elementary operator on B
H defined by R
A
,
B
X
=
∑
i
=
1
k
A
i
X
B
i. In this paper, we prove the following formula for the numerical range of
R
A
,
B:
V
R
A
,
B
,
B
B
H
=
[
∪
U
∈
U
H
W
(
∑
i
=
1
k
U
A
i
U
*
B
i
)
-
]
-, where U
H is the set of unitary operators.


---

## Body

## 1. Introduction

Let𝒜 be a complex Banach algebra with unit. For k-tuples of elements of 𝒜, a
=
(
a
1
,
…
,
a
k
) and b
=
(
b
1
,
…
,
b
k
), let R
a
,
b denote the elementary operator on 𝒜 defined by
(1)
R
a
,
b
(
x
)
=
∑
i
=
1
k
‍
a
i
x
b
i
.
This is a bounded linear operator on 𝒜. Some interesting cases are the generalized derivation δ
a
,
b
(
x
)
=
a
x
-
x
b and the multiplication M
a
,
b
(
x
)
=
a
x
b for a
,
b
,
x
∈
A.The numerical range ofa
∈
𝒜 is defined by
(2)
V
(
a
,
𝒜
)
=
{
f
(
a
)
:
f
∈
S
}
,
where S is the set of normalized states in 𝒜:
(3)
S
=
{
f
∈
A
*
,
∥
f
∥
=
1
=
f
(
e
)
}
.
See [1–3]. It is well known that V
(
a
,
𝒜
) is convex and closed and contains the spectrum σ
(
a
). For 𝒜
=
B
(
X
), the algebra of bounded linear operators on a normed space X, and T
∈
B
(
X
), in addition to V
(
T
,
B
(
X
)
), we have the spatial numerical range of T, given by
(4)
W
(
T
)
=
{
f
(
T
x
)
:
x
∈
X
,
∥
x
∥
=
1
,
f
∈
X
*
,
∥
f
∥
=
f
(
x
)
=
1
}
,
and it is known that V
(
T
,
B
(
X
)
)
=
co
⁡
¯
W
(
T
), the closed convex hull of W
(
T
). In the case of a Hilbert space X
=
H, then
(5)
W
(
T
)
=
{
〈
T
x
,
x
〉
:
x
∈
H
,
∥
x
∥
=
1
}
is convex but not closed in general and V
(
T
,
B
(
H
)
)
=
W
(
T
)
-.Many facts about the relation between the spectrum ofR
a
,
b and the spectrums of the coefficients a
i and b
i are known. This is not the case with the relation between the numerical range of R
a
,
b and the numerical ranges of a
i and b
i. Apparently, the only elementary operator on a Hilbert space for which the numerical range is computed is the generalized derivations [4–8]. It is Fong [4] who first gives the following formula:
(6)
W
(
δ
T
)
=
W
(
T
)
-
-
W
(
T
)
-
,
where δ
T is the inner derivation defined by δ
T
(
X
)
=
T
X
-
X
T. Shaw [7] (see also [5, 6]) extended this formula to generalized derivations in Banach spaces. For a good survey of the numerical range of elementary operators, you can see [9], where the following problem is posed.Problem. Determine the numerical range of the elementary operator R
A
,
B.In this paper, we give a formula that answers this problem.


## 2. Main Result

The following theorem is the main result in this paper.Theorem 1.
LetH be a complex Hilbert space. Let A
=
(
A
1
,
…
,
A
k
) and B
=
(
B
1
,
…
,
B
k
) be two k-tuples of elements in B
(
H
). Then, one has
(7)
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
=
[
⋃
U
∈
U
(
H
)
W
(
∑
i
=
1
k
‍
U
A
i
U
*
B
i
)
-
]
-
.
In particular for multiplication and generalized derivation, one has (
A
,
B
∈
B
(
H
)
):
(8)
V
(
M
A
,
B
,
B
(
B
(
H
)
)
)
=
[
⋃
U
∈
U
(
H
)
W
(
A
U
B
U
*
)
-
]
-
,
V
(
δ
A
,
B
,
B
(
B
(
H
)
)
)
=
[
⋃
U
∈
U
(
H
)
W
(
U
*
A
U
-
B
)
-
]
-
.From Fong’s formula (see [4, 6, 10]), we can deduce the following.Corollary 2.
ForA
,
B
∈
B
(
H
), one has
(9)
W
(
A
)
-
-
W
(
B
)
-
=
[
⋃
U
∈
U
(
H
)
W
(
U
A
U
*
-
B
)
-
]
-
=
[
⋃
U
∈
U
(
H
)
W
(
A
-
U
B
U
*
)
-
]
-
.


## 3. Proof of the Main Result

One of the keys to the proof of our main result is the following lemma.Lemma 3.
LetA
=
(
A
1
,
…
,
A
k
) and B
=
(
B
1
,
…
,
B
k
) be two k-tuples of elements in B
(
H
). Then, one has
(10)
W
(
∑
i
=
1
k
‍
A
i
B
i
)
⊂
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
.
In particular, for A
,
B
∈
B
(
H
), one has
(11)
W
(
A
B
)
⊂
V
(
M
A
,
B
,
B
(
B
(
H
)
)
)
,
W
(
A
-
B
)
⊂
V
(
δ
A
,
B
,
B
(
B
(
H
)
)
)
.Proof.
Letλ
∈
W
(
∑
i
=
1
k
‍
A
i
B
i
); by definition, there exists x
∈
H with ∥
x
∥
=
1 such that
(12)
λ
=
〈
∑
i
=
1
k
‍
A
i
B
i
x
,
x
〉
=
tr
⁡
(
∑
i
=
1
k
‍
A
i
B
i
(
x
⊗
x
)
)
.
Here, tr
⁡
(
·
) is the linear form trace. Let ϕ
x
⊗
x be the linear form defined by
(13)
ϕ
x
⊗
x
(
X
)
=
tr
⁡
(
X
(
x
⊗
x
)
)
=
〈
X
x
,
x
〉
.
This is a bounded linear form on B
(
H
), with norm being equal to 1, because
(14)
∥
ϕ
x
⊗
x
∥
=
∥
x
⊗
x
∥
=
1
.
Since
(15)
ϕ
x
⊗
x
(
I
H
)
=
tr
⁡
(
x
⊗
x
)
=
〈
x
,
x
〉
=
1
,
the form ϕ
x
⊗
x is a state on B
(
H
). So,
(16)
ϕ
x
⊗
x
(
R
A
,
B
(
I
H
)
)
∈
V
(
M
A
,
B
,
B
(
B
(
H
)
)
)
,
ϕ
x
⊗
x
(
R
A
,
B
(
I
H
)
)
=
ϕ
x
⊗
x
(
∑
i
=
1
k
‍
A
i
B
i
)
=
tr
⁡
(
∑
i
=
1
k
‍
A
i
B
i
(
x
⊗
x
)
)
=
〈
∑
i
=
1
k
‍
A
i
B
i
x
,
x
〉
=
λ
.
Hence, W
(
∑
i
=
1
k
‍
A
i
B
i
)
⊂
W
(
R
A
,
B
)
⊂
V
(
R
A
,
B
,
B
(
B
(
H
)
)
).LetE be a Banach space. We say that S
∈
B
(
E
) is an isometry if ∥
S
x
∥
=
∥
x
∥ for all x
∈
E. If S is an invertible isometry, then its inverse S
-
1 is also an isometry, and we have
(17)
V
(
S
T
S
-
1
,
B
(
E
)
)
=
V
(
T
,
B
(
E
)
)
,
T
∈
B
(
E
)
.In the case of a Hilbert space, an invertible isometry is unitary and its inverse is the adjoint.LetU and V be two unitaries operators on H; then
(18)
R
U
A
U
*
,
V
B
V
*
=
M
U
,
V
*
R
A
,
B
M
U
*
,
V
,
with M
U
,
V
* being an invertible isometry and its inverse being   M
U
*
,
V.From this result, we deduce that(19)
V
(
R
U
A
U
*
,
V
B
V
*
,
B
(
B
(
H
)
)
)
=
V
(
M
U
,
V
*
R
A
,
B
M
U
*
,
V
,
B
(
B
(
H
)
)
)
=
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
.
Now, using Lemma 3, we get
(20)
W
(
∑
i
=
1
k
‍
U
A
i
U
*
V
B
i
V
*
)
⊂
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
,
⋃
U
,
V
∈
U
(
H
)
W
(
∑
i
=
1
k
‍
U
A
i
U
*
V
B
i
V
*
)
⊂
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
.But, the numerical range is closed and the product of two unitaries is also an unitary, hence:(21)
[
⋃
U
∈
U
(
H
)
W
(
∑
i
=
1
k
‍
U
A
i
U
*
B
i
)
-
]
-
=
[
⋃
V
∈
U
(
H
)
W
(
∑
i
=
1
k
‍
A
i
V
B
i
V
*
)
-
]
-
⊂
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
.
So, we have proved the second inclusion of Theorem 1.For the other inclusion, we will use the two following theorems.Theorem 4 (See [11]).
Let𝒜 be Banach algebra. For a
∈
𝒜, one has
(22)
V
(
a
,
𝒜
)
=
⋂
z
∈
C
‍
{
λ
:
|
λ
-
z
|
≤
∥
a
-
z
∥
}
.The norm of an elementary operator is defined by(23)
∥
R
A
,
B
∥
=
Sup
⁡
{
∥
R
A
,
B
(
X
)
∥
:
∥
X
∥
≤
1
}
.
Let 𝒜 be C
*-algebra. An element U
∈
𝒜 is said to be unitary if U
*
U
=
U
U
*
=
I. In the following, U
(
𝒜
) denote the set of unitaries in 𝒜.Theorem 5 (Russo-Dye [12]).
Let𝒜 be C
*
-algebra. Let A
=
(
a
1
,
…
,
a
k
) and B
=
(
b
1
,
…
,
b
k
) be two k-tuples of elements in 𝒜. Then, one has
(24)
∥
R
A
,
B
∥
=
Sup
⁡
{
∥
R
A
,
B
(
U
)
∥
:
U
∈
U
(
𝒜
)
}
.We return now to the proof of the main theorem.Proof.
We need only to show the inclusion “⊂.” By Theorem 4, we have
(25)
V
(
R
A
,
B
,
B
(
B
(
H
)
)
)
=
⋂
z
∈
C
‍
{
λ
:
|
λ
-
z
|
≤
∥
R
A
,
B
-
z
∥
}
.
And, by Theorem 5, we get
(26)
∥
R
A
,
B
-
z
∥
=
Sup
⁡
{
∥
∑
i
=
1
k
‍
A
i
U
B
i
-
z
U
∥
:
U
∈
U
(
H
)
}
.
But ∥
U
T
∥
=
∥
T
∥ for all T
∈
B
(
H
) and U
∈
U
(
H
). Hence,
(27)
∥
R
A
,
B
-
z
∥
=
Sup
⁡
{
∥
∑
i
=
1
k
‍
U
*
A
i
U
B
i
-
z
I
H
∥
:
U
∈
U
(
H
)
}
.
Hence, ifλ
∈
V
(
R
A
,
B
,
B
(
B
(
H
)
)
), then, for all z
∈
C,
(28)
λ
∈
{
|
μ
-
z
|
≤
∥
R
A
,
B
-
z
I
B
(
H
)
∥
}
.
Let ϵ
>
0 fixed, there exists a unitary U
ϵ such that
(29)
∥
R
A
,
B
-
z
I
B
(
H
)
∥
≤
∥
∑
i
=
1
k
‍
U
ϵ
*
A
i
U
ϵ
B
i
-
z
I
H
∥
+
ϵ
.
Now, using Theorem 4, we have
(30)
W
(
∑
i
=
1
k
‍
U
ϵ
*
A
i
U
ϵ
B
i
)
-
=
⋂
z
∈
C
{
λ
:
|
λ
-
z
|
≤
∥
∑
i
=
1
k
‍
U
ϵ
*
A
i
U
ϵ
B
i
-
z
I
H
∥
}
.
So, there existsμ
∈
W
(
∑
i
=
1
k
‍
U
ϵ
*
A
U
ϵ
B
) such that |
λ
-
μ
|
≤
ϵ. But ϵ is arbitrary, λ
∈
[
∪
U
∈
U
(
H
)
W
(
∑
i
=
1
k
‍
U
*
A
i
U
B
i
)
-
]
-. This finishes the proof of the main theorem.


## 4. Some Applications

It is well known that, for the spectrum, ifA
,
B
∈
B
(
H
), then we have
(31)
σ
(
A
B
)
∪
{
0
}
=
σ
(
B
A
)
∪
{
0
}
.
For the numerical range, this not true, but we can deduce the following corollary from the proof of Theorem 1.Corollary 6.
For allA
,
B
∈
B
(
H
), one has
(32)
[
⋃
U
∈
U
(
H
)
W
(
A
U
B
U
*
)
-
]
-
=
[
⋃
U
∈
U
(
H
)
W
(
U
B
U
*
A
)
-
]
-
.The numerical radius of an operatorT
∈
B
(
E
) is denoted by v
(
T
) and defined by
(33)
v
(
T
)
=
sup
⁡
{
|
λ
|
:
λ
∈
V
(
T
,
B
(
E
)
)
}
.Corollary 7.
LetA
=
(
A
1
,
…
,
A
k
) and B
=
(
B
1
,
…
,
B
k
) be two k-tuples of elements in B
(
H
). Then, one has
(34)
v
(
R
A
,
B
)
=
Sup
⁡
{
v
(
∑
i
=
1
k
‍
U
*
A
i
U
B
i
)
:
U
∈
U
(
H
)
}
.
In particular, for A
,
B
∈
B
(
H
),
(35)
v
(
δ
A
,
B
)
=
sup
U
∈
U
(
H
)
⁡
v
(
U
A
U
*
-
B
)
=
sup
U
∈
U
(
H
)
⁡
v
(
A
-
U
B
U
*
)
=
sup
U
,
V
∈
U
(
H
)
⁡
v
(
U
A
U
*
-
V
B
V
*
)
.LetK be a nonempty subset of the plane and let
(36)
diam
⁡
(
K
)
=
sup
⁡
α
,
β
∈
K
⁡
|
α
-
β
|
.
From Corollary 7 (B
=
A), one has
(37)
diam
⁡
(
W
(
A
)
)
=
sup
⁡
{
v
(
U
A
U
*
-
A
)
:
U
∈
U
(
H
)
}
.
So, the diameter of the numerical range W
(
A
) is equal to the diameter of the v-unitary orbit of the operator A.



---

*Source: 246301-2014-03-20.xml*