Advances in Pure Mathematics
Vol.4 No.7(2014), Article
ID:47617,8
pages
DOI:10.4236/apm.2014.47041
On the Norm of Elementary Operator
Denis Njue Kingangi1, John Ogoji Agure2, Fredrick Oluoch Nyamwala3
1Department of Mathematics and Computer Science, University of Eldoret, Eldoret, Kenya
2Department of Pure and Applied Mathematics, Maseno University, Maseno, Kenya
3Department of Physics, Mathematics, Statistics and Computer Science, Moi University, Eldoret, Kenya
Email: dankingangi2003@yahoo.com, johnagure@yahoo.com, foluoch2000@yahoo.com
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 20 April 2014; revised 20 May 2014; accepted 3 June 2014
ABSTRACT
The norm of an elementary operator has been studied by many mathematicians. Varied results have been established especially on the lower bound of this norm. Here, we attempt the same problem for finite dimensional operators.
Keywords:Bounded Linear Operator, Elementary Operator
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
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:
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:
where,
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,
;
Therefore:
.
Letting, we obtain:
. (1)
On the other hand, we have:
with:
.
So, setting, and
, we have:
, with
fixed in
.
obtaining;
. (2)
Hence, from (1) and (2), we obtain
.
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
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;
That is.
Thus we have:
Hence
. (3)
Letting, we obtain:
and this is true for any
, and for any unit vector
.
Now, consider the set.
We have:
But.
Therefore:
and this completes the proof.
Corollary 2.3: Let be a complex Hilbert space and
be bounded linear operators on
. Let
. Then we have
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:
This is equivalent to:
, (4)
considering the sequenceTaking 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.
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:
Now, we have:
Therefore:
Letting we obtain:
That is and this implies that
.
Clearly, and therefore we obtain
.
We recall that an elementary operator, , is defined as
, for all
where
are fixed, for
. When
, we have
, for all
and
fixed, for
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
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
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,
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
Since, we have:
Now, since and
are all positive real numbers, we have
and
.
Thus and hence we have
.
That is,
. (6)
Now, (5) and (6) implies that:
and this completes the proof.
References
- Timoney, R.M. (2007) Some Formulae for Norms of Elementary Operators. The Journal of Operator Theory, 57, 121-145.
- Nyamwala, F.O. and Agure, J.O. (2008) Norms of Elementary Operators in Banach Algebras. Journal of Mathematical Analysis, 2, 411-424.
- Mathew, M. (1990) More Properties of the Product of Two Derivations of a C*-Algebras. Bulletin of the Australian Mathematical Society, 42, 115-120. http://dx.doi.org/10.1017/S0004972700028203
- Cabrera, M. and Rodriguez, A. (1994) Non-Degenerate Ultraprime Jordan-Banach Algebras: A Zelmano-Rian Treatment. Proceedings of the London Mathematical Society, 69, 576-604.
- Stacho, L.L. and Zalar, B. (1996) On the Norm of Jordan Elementary Operators in Standard Operator Algebras. Publicationes Mathematicae-Debrecen, 49, 127-134.
- Baraa, M. and Boumazgour, M. (2001) A Lower Bound of the Norm of the Operator . Extracta Mathematicae, 16, 223-227.
- Okelo, N. and Agure, J.O. (2011) A Two-Sided Multiplication Operator Norm. General Mathematics Notes, 2, 18-23.