1. Introduction
Let
be a complex Hilbert space and
be the set of bounded operators on
. A basic elementary operator,
, is defined as:
for
and
fixed.
An elementary operator,
, is a finite sum of the basic elementary operators, defined as,
, for all
, where
are fixed, for
.
When
, we have
, for all
and
fixed, for ![](https://www.scirp.org/html/htmlimages\2-5300712x\073a197c-8c32-4204-9417-0029d4ceb05c.png)
Given the elementary operator
on
, the question on whether the equation
, holds remains an area of interest to many mathematicians. This paper attempts to answer this question for finite dimensional operators.
For a complex Hilbert space
, with dual
, we define a finite rank operator
by,
for all
, where
, and
is a unit vector, with:
![](https://www.scirp.org/html/htmlimages\2-5300712x\e85ea3c4-0ee5-40c8-bbbd-d93e56391cbe.png)
In this paper, we use finite rank operators to determine the norm of
. We first review some known results on the norm of the Jordan elementary operator
,
, for all
with
fixed. We will then proceed to show that for an operator
with
and
for all unit vectors
, then:
.
Some mathematicians have attempted to determine the norm of
. Timoney, used (matrix) numerical ranges and the tracial geometric mean to obtain an approximation of
[1] , while Nyamwala and Agure used the spectral resolution theorem to calculate the norm of
induced by normal operators in a finite dimensional Hilbert space [2] .
The study of the norm of the Jordan elementary operator has also attracted many researchers in operator theory. Mathieu [3] , in 1990, proved that in the case of a prime C*-algebra, the lower bound of the norm of
can be estimated by
In 1994, Cabrera and Rodriguez [4] , showed that
for prime JB*-algebras.
On their part, Stacho and Zalar [5] , in 1996 worked on the standard operator algebra which is a sub-algebra of
, that contains all finite rank operators. They first showed that the operator
actually represents a Jordan triple structure of a C*-algebra. They also showed that if
is a standard operator algebra acting on a Hilbert space
, and
, then
They later (1998), proved that
for the algebra of symmetric operators acting on a Hilbert space. They attached a family of Hilbert spaces to standard operator algebra, using the inner products on them to obtain their results.
In 2001, Barraa and Boumazguor [6] , used the concept of the maximal numerical range and finite rank operators to show that if
with
, then:
![](https://www.scirp.org/html/htmlimages\2-5300712x\1ac4f56e-cfb1-45e8-89c5-3a1e526993de.png)
where,
![](https://www.scirp.org/html/htmlimages\2-5300712x\8db138e3-b7e7-4302-87f8-3c2e51a9a464.png)
is the maximal numerical range of
relative to
, and
is the Hilbert adjoint of
.
Okelo and Agure [7] used the finite rank operators to determine the norm of the basic elementary operator. Their work forms the basis of the results in this paper.
2. The Norm of Elementary Operator
In this section, we present some of the known results on elementary operators and proceed to determine norm of the elementary operator
.
In the following theorem Okelo and Agure [7] , determined the norm of the basic elementary operator.
Theorem 2.1 [5] : Let
be a complex Hilbert space and
the algebra of bounded linear operators on
. Let
be defined by
for all
with
as fixed elements in
. If for all
with
, we have
for all unit vectors
, then;
.
Proof: Since
, we have,
;
![](https://www.scirp.org/html/htmlimages\2-5300712x\f9ff91bf-1843-4868-abee-bad802fc21ea.png)
Therefore:
.
Letting
, we obtain:
. (1)
On the other hand, we have:
with:
.
So, setting
, and
, we have:
, with
fixed in
.
![](https://www.scirp.org/html/htmlimages\2-5300712x\b0f71f56-40c0-4aae-9934-63c8318e5c27.png)
obtaining;
. (2)
Hence, from (1) and (2), we obtain
. ![](https://www.scirp.org/html/htmlimages\2-5300712x\3842d4ea-32bf-485e-8728-7bf1c61a8613.png)
For any vectors
, the rank one operator,
, is defined by
, for all
.
In the following three results Baraa and Boumazgour give three estimations to the lower bound of the norm of the Jordan elementary operator. See [6] . Recall that the Jordan elementary operator is the operator
, for all
with
fixed.
Theorem 2.2. Let
be the Jordan elementary operator with
fixed, and with
. Then
![](https://www.scirp.org/html/htmlimages\2-5300712x\02074b1e-38ff-4432-8827-089ca88371ed.png)
where,
is the maximal numerical range of
relative to
, as defined earlier.
Proof: Let
. Then there exists a sequence
of unit vectors in
such that
and
. Consider unit vectors
, and recall the rank one operator,
, defined as
, for all unit vectors
. For fixed operators
, we have;
![](https://www.scirp.org/html/htmlimages\2-5300712x\08966646-f0c4-4fa9-b868-42faec848204.png)
That is
.
Thus we have:
![](https://www.scirp.org/html/htmlimages\2-5300712x\97d37671-506f-4172-9564-44158c74d5e9.png)
Hence
. (3)
Letting
, we obtain:
and this is true for any
, and for any unit vector
.
Now, consider the set
.
We have:
![](https://www.scirp.org/html/htmlimages\2-5300712x\fac683f8-fe6e-426b-a4a3-469f84c04c9d.png)
But
.
Therefore:
and this completes the proof. ![](https://www.scirp.org/html/htmlimages\2-5300712x\210fc071-aeb3-4182-bbb8-59ecaaace47d.png)
Corollary 2.3: Let
be a complex Hilbert space and
be bounded linear operators on
. Let
. Then we have ![](https://www.scirp.org/html/htmlimages\2-5300712x\9144ffa5-f684-4059-9e8f-30746a8fe442.png)
Proof: Let
. Then ,
or
, and therefore, either there is a sequence
of unit vectors in
such that
and
or, there is a sequence
of unit vectors in
such that
and
.
Recall that in the previous theorem (Inequality (3)), we obtained:
![](https://www.scirp.org/html/htmlimages\2-5300712x\0cb834ee-4ac5-4289-8e2f-1af9b12c9610.png)
This is equivalent to:
, (4)
considering the sequence
Taking limits in either (3) or (4), we obtain
and this is true for any unit vector
.
Now, consider the set
.
We have:
.
But
.
Therefore:
and this completes the proof. ![](https://www.scirp.org/html/htmlimages\2-5300712x\7feb35ac-7443-4a44-9a04-e9cd1f156ab7.png)
Proposition 2.4: Let
be a complex Hilbert space and
be bounded linear operators on
. If
then:
.
Proof: Suppose
. Then
and
, and therefore we can find two sequences
and
of unit vectors in
such that:
,
and
,
.
Since
and
, then
and
.
For each
, we have:
![](https://www.scirp.org/html/htmlimages\2-5300712x\eb8c0bd4-8e44-466e-9a45-31a998d7e37b.png)
Now, we have:
![](https://www.scirp.org/html/htmlimages\2-5300712x\92594945-9619-47a8-98a3-475bee6c045e.png)
Therefore:
![](https://www.scirp.org/html/htmlimages\2-5300712x\3da9a654-22d1-4344-ac49-ccfa571fa9f4.png)
Letting
we obtain:
![](https://www.scirp.org/html/htmlimages\2-5300712x\b8aae992-de55-467e-b753-e9a4303146be.png)
That is
and this implies that
.
Clearly,
and therefore we obtain
. ![](https://www.scirp.org/html/htmlimages\2-5300712x\b14c12ef-6ac6-4b26-b3d1-54d1d7f7879d.png)
We recall that an elementary operator,
, is defined as
, for all
where
are fixed, for
. When
, we have
, for all
and
fixed, for ![](https://www.scirp.org/html/htmlimages\2-5300712x\1e3634c5-6bf5-40bf-9bb8-050f3fa8bbc8.png)
The following result gives the norm of
.
Theorem 2.5: Let
be a complex Hilbert space and
be the algebra of all bounded linear operators on
: Let
be the elementary operator on
defined above. If for an operator
with
, we have
for all unit vectors
, then:
.
Proof: Recall that
is defined as
, for all
and
fixed, for ![](https://www.scirp.org/html/htmlimages\2-5300712x\93dc8d83-3b45-4f21-ac7e-fb172741ba23.png)
We have:
.
Therefore,
for all
with
.
So, for all
,
for all
with
.
Therefore,
.
Letting
, we obtain:
. (5)
Next, we show that
.
Since
, then we have
for all
. But
.
Now, let
be functionals for ![](https://www.scirp.org/html/htmlimages\2-5300712x\a380170d-55ef-4c88-8e6e-af9a1f900559.png)
Choose unit vectors
and define finite rank operators
and
on
, for
by
for all
with
, for
, and
, for
with
, for
.
Observe that the norm of
for
is,
![](https://www.scirp.org/html/htmlimages\2-5300712x\eb308f26-4faa-4450-9313-034017463165.png)
That is
for any unit vector
with
, for
.
Likewise, the norm of
is
for any unit vector
with
, for
.
Therefore, for all
with
, we have
![](https://www.scirp.org/html/htmlimages\2-5300712x\a22caaf1-f21e-4ec9-b3c9-776ee24f9edf.png)
Since
, we have:
![](https://www.scirp.org/html/htmlimages\2-5300712x\4d5c7186-d647-482e-b9c1-9db751f5651f.png)
Now, since
and
are all positive real numbers, we have
![](https://www.scirp.org/html/htmlimages\2-5300712x\67aafd36-090e-4ed2-9bc3-02b15c5c6e15.png)
![](https://www.scirp.org/html/htmlimages\2-5300712x\83e7ab94-d865-46ca-a54c-da69b91e0ea1.png)
and
.
Thus
and hence we have
.
That is,
. (6)
Now, (5) and (6) implies that:
and this completes the proof. ![](https://www.scirp.org/html/htmlimages\2-5300712x\fd60a56a-265c-4aa8-a005-a64ad3e1095c.png)