A Formula for the Numerical Range of Elementary Operators
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.
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 [1–3]. 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 [4–8]. It is Fong  who first gives the following formula: where is the inner derivation defined by . Shaw  (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 , 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 :
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 being .
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 ). 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 ). 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 “” 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.
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