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. 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 [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 [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 Fong’s 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