A Formula for the Numerical Range of Elementary Operators

is convex but not closed in general andV(T, B(H)) = W(T)−. Many facts about the relation between the spectrum of R 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:


Introduction
Let A be a complex Banach algebra with unit.For -tuples of elements of A,  = ( 1 , . . .,   ) and  = ( 1 , . . .,   ), let  , denote the elementary operator on A defined by This is a bounded linear operator on A. Some interesting cases are the generalized derivation  , () =  −  and the multiplication  , () =  for , ,  ∈ .
The numerical range of  ∈ A is defined by where  is the set of normalized states in A:  = { ∈  * ,          = 1 =  ()} .
See [1][2][3].It is well known that (, A) is convex and closed and contains the spectrum ().For A = (), 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 (, ()) = co(), 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 [4][5][6][7][8].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.

Main Result
The following theorem is the main result in this paper.

Proof of the Main Result
One of the keys to the proof of our main result is the following lemma.
Let  and  be two unitaries operators on ; then (20) But, the numerical range is closed and the product of two unitaries is also an unitary, hence:

ISRN Mathematical Analysis 3
For the other inclusion, we will use the two following theorems.
Let A be  * -algebra.An element  ∈ A is said to be unitary if  *  =  * = .In the following, (A) denote the set of unitaries in A.
We return now to the proof of the main theorem.
Proof.We need only to show the inclusion "⊂."By Theorem 4, we have

Some Applications
It is well known that, for the spectrum, if ,  ∈ (), then we have  () ∪ {0} =  () ∪ {0} . (31) 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 V() and defined by So, the diameter of the numerical range () is equal to the diameter of the V-unitary orbit of the operator .
have proved the second inclusion of Theorem 1.