Research Article | Open Access

# A Formula for the Numerical Range of Elementary Operators

**Academic Editor:**M. Lindstrom

#### Abstract

Let be the algebra of bounded linear operators on a complex Hilbert space . For -tuples of elements of and , let denote the elementary operator on defined by . In this paper, we prove the following formula for the numerical range of : , where is the set of unitary operators.

#### 1. Introduction

Let be a complex Banach algebra with unit. For -tuples of elements of , and , let denote the elementary operator on defined by This is a bounded linear operator on . Some interesting cases are the generalized derivation and the multiplication for .

The numerical range of is defined by where is the set of normalized states in : See [1ā3]. It is well known that is convex and closed and contains the spectrum . For , the algebra of bounded linear operators on a normed space , and , in addition to , we have the spatial numerical range of , given by and it is known that , the closed convex hull of . In the case of a Hilbert space , then is convex but not closed in general and .

Many facts about the relation between the spectrum of and the spectrums of the coefficients and are known. This is not the case with the relation between the numerical range of and the numerical ranges of and . Apparently, the only elementary operator on a Hilbert space for which the numerical range is computed is the generalized derivations [4ā8]. It is Fong [4] who first gives the following formula: where is the inner derivation defined by . 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 .

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. *Let be a complex Hilbert space. Let and be two -tuples of elements in . Then, one has
**
In particular for multiplication and generalized derivation, one has :
*

From Fongās formula (see [4, 6, 10]), we can deduce the following.

Corollary 2. *For , one has
*

#### 3. Proof of the Main Result

One of the keys to the proof of our main result is the following lemma.

Lemma 3. *Let and be two -tuples of elements in . Then, one has
**
In particular, for , one has
*

*Proof. *Let ; by definition, there exists with such that
Here, is the linear form trace. Let be the linear form defined by
This is a bounded linear form on , with norm being equal to 1, because
Since
the form is a state on . So,
Hence, .

Let be a Banach space. We say that is an isometry if for all . If is an invertible isometry, then its inverse is also an isometry, and we have

In the case of a Hilbert space, an invertible isometry is unitary and its inverse is the adjoint.

Let and be two unitaries operators on ; then with being an invertible isometry and its inverse beingāāā.

From this result, we deduce that Now, using Lemma 3, we get

But, the numerical range is closed and the product of two unitaries is also an unitary, hence: 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]). *Let be Banach algebra. For , one has
*

The norm of an elementary operator is defined by Let be -algebra. An element is said to be unitary if . In the following, denote the set of unitaries in .

Theorem 5 (Russo-Dye [12]). *Let be algebra. Let and be two -tuples of elements in . Then, one has
*

We return now to the proof of the main theorem.

*Proof. *We need only to show the inclusion āā By Theorem 4, we have
And, by Theorem 5, we get
But for all and . Hence,

Hence, if , then, for all ,
Let fixed, there exists a unitary such that
Now, using Theorem 4, we have

So, there exists such that . But is arbitrary, . This finishes the proof of the main theorem.

#### 4. Some Applications

It is well known that, for the spectrum, if , then we have For the numerical range, this not true, but we can deduce the following corollary from the proof of Theorem 1.

Corollary 6. *For all , one has
*

The numerical radius of an operator is denoted by and defined by

Corollary 7. *Let and be two -tuples of elements in . Then, one has
**
In particular, for ,
*

Let be a nonempty subset of the plane and let From Corollary 7 (), one has So, the diameter of the numerical range is equal to the diameter of the -unitary orbit of the operator .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### References

- F. F. Bonsall and J. Duncan,
*Numerical Ranges Vol I*, Cambridge University Press, New York, NY, USA, 1973. View at: MathSciNet - F. F. Bonsall and J. Duncan,
*Numerical Rangesvol Vol II*, Cambridge University Press, New York, NY, USA, 1973. View at: MathSciNet - K. E. Gustafson and D. K. M. Rao,
*Numerical Range: The Field of Values of Linear Operators and Matrices*, Springer, New York, NY, USA, 1997. View at: MathSciNet - C. K. Fong,
*Some Aspects of Derivations on B(H)*, University of Toronto, Seminar Notes, 1978. - J. Kyle, āNumerical ranges of derivations,ā
*Proceedings of the Edinburgh Mathematical Society*, vol. 21, no. 1, pp. 33ā39, 1979. View at: Publisher Site | Google Scholar | MathSciNet - K. Mattila, āComplex strict and uniform convexity and hyponormal operators,ā
*Mathematical Proceedings of the Cambridge Philosophical Society*, vol. 96, no. 3, pp. 483ā493, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - S.-Y. Shaw, āOn numerical ranges of generalized derivations and related properties,ā
*Australian Mathematical Society Journal A*, vol. 36, no. 1, pp. 134ā142, 1984. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - A. Seddik, āThe numerical range of elementary operators. II,ā
*Linear Algebra and Its Applications*, vol. 338, pp. 239ā244, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - L. A. Fialkow, āStructural properties of elementary operators,ā in
*Elementary Operators and Applications (Blaubeuren, 1991)*, pp. 55ā113, World Scientific, River Edge, NJ, USA, 1992. View at: Google Scholar | MathSciNet - R. Schatten,
*Norm Ideals of Completely Continuous Operators*, Springer, Berlin, Germany, 1960. View at: MathSciNet - J. G. Stampfli and J. P. Williams, āGrowth conditions and the numerical range in a Banach algebra,ā
*The Tohoku Mathematical Journal*, vol. 20, pp. 417ā424, 1968. View at: Google Scholar | MathSciNet - B. Russo and H. A. Dye, āA note on unitary operators in ${C}^{*}$-algebras,ā
*Duke Mathematical Journal*, vol. 33, pp. 413ā416, 1966. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 M. Barraa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.