ISRN.MATHEMATICAL.ANALYSIS ISRN Mathematical Analysis 2090-4665 Hindawi Publishing Corporation 246301 10.1155/2014/246301 246301 Research Article A Formula for the Numerical Range of Elementary Operators Barraa M. Lindstrom M. Zhu C. Department of Mathematics Faculty of Sciences Semlalia, Marrakech Morocco 2014 2032014 2014 10 11 2013 09 02 2014 20 3 2014 2014 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.

Let B H be the algebra of bounded linear operators on a complex Hilbert space H . For k -tuples of elements of B H ,   A = A 1 , , A k and B = B 1 , , B k , let R A , B denote the elementary operator on B H defined by R A , B X = i = 1 k A i X B i . In this paper, we prove the following formula for the numerical range of R A , B : V R A , B , B B H = [ U U H W ( i = 1 k U A i U * B i ) - ] - , where U H is the set of unitary operators.

1. Introduction

Let 𝒜 be a complex Banach algebra with unit. For k -tuples of elements of 𝒜 , a = ( a 1 , , a k ) and b = ( b 1 , , b k ) , let R a , b denote the elementary operator on 𝒜 defined by (1) R a , b ( x ) = i = 1 k a i x b i . This is a bounded linear operator on 𝒜 . Some interesting cases are the generalized derivation δ a , b ( x ) = a x - x b and the multiplication M a , b ( x ) = a x b for a , b , x A .

The numerical range of a 𝒜 is defined by (2) V ( a , 𝒜 ) = { f ( a ) : f S } , where S is the set of normalized states in 𝒜 : (3) S = { f A * , f = 1 = f ( e ) } . See . It is well known that V ( a , 𝒜 ) is convex and closed and contains the spectrum σ ( a ) . For 𝒜 = B ( X ) , the algebra of bounded linear operators on a normed space X , and T B ( X ) , in addition to V ( T , B ( X ) ) , we have the spatial numerical range of T , given by (4) W ( T ) = { f ( T x ) : x X , x = 1 , f X * , f = f ( x ) = 1 } , and it is known that V ( T , B ( X ) ) = co ¯ W ( T ) , the closed convex hull of W ( T ) . In the case of a Hilbert space X = H , then (5) W ( T ) = { T x , x : x H , x = 1 } is convex but not closed in general and V ( 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 . It is Fong  who first gives the following formula: (6) W ( δ T ) = W ( T ) - - W ( T ) - , where δ T is the inner derivation defined by δ T ( X ) = T X - X T . 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 R A , B .

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 H be a complex Hilbert space. Let A = ( A 1 , , A k ) and B = ( B 1 , , B k ) be two k -tuples of elements in B ( H ) . Then, one has (7) V ( R A , B , B ( B ( H ) ) ) = [ U U ( H ) W ( i = 1 k U A i U * B i ) - ] - . In particular for multiplication and generalized derivation, one has ( A , B B ( H ) ) : (8) V ( M A , B , B ( B ( H ) ) ) = [ U U ( H ) W ( A U B U * ) - ] - , V ( δ A , B , B ( B ( H ) ) ) = [ U U ( H ) W ( U * A U - B ) - ] - .

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

Corollary 2.

For A , B B ( H ) , one has (9) W ( A ) - - W ( B ) - = [ U U ( H ) W ( U A U * - B ) - ] - = [ U U ( H ) W ( A - U B U * ) - ] - .

3. Proof of the Main Result

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

Lemma 3.

Let A = ( A 1 , , A k ) and B = ( B 1 , , B k ) be two k -tuples of elements in B ( H ) . Then, one has (10) W ( i = 1 k A i B i ) V ( R A , B , B ( B ( H ) ) ) . In particular, for A , B B ( H ) , one has (11) W ( A B ) V ( M A , B , B ( B ( H ) ) ) , W ( A - B ) V ( δ A , B , B ( B ( H ) ) ) .

Proof.

Let λ W ( i = 1 k A i B i ) ; by definition, there exists x H with x = 1 such that (12) λ = i = 1 k A i B i x , x = tr ( i = 1 k A i B i ( x x ) ) . Here, tr ( · ) is the linear form trace. Let ϕ x x be the linear form defined by (13) ϕ x x ( X ) = tr ( X ( x x ) ) = X x , x . This is a bounded linear form on B ( H ) , with norm being equal to 1, because (14) ϕ x x = x x = 1 . Since (15) ϕ x x ( I H ) = tr ( x x ) = x , x = 1 , the form ϕ x x is a state on B ( H ) . So, (16) ϕ x x ( R A , B ( I H ) ) V ( M A , B , B ( B ( H ) ) ) , ϕ x x ( R A , B ( I H ) ) = ϕ x x ( i = 1 k A i B i ) = tr ( i = 1 k A i B i ( x x ) ) = i = 1 k A i B i x , x = λ . Hence, W ( i = 1 k A i B i ) W ( R A , B ) V ( R A , B , B ( B ( H ) ) ) .

Let E be a Banach space. We say that S B ( E ) is an isometry if S x = x for all x E . If S is an invertible isometry, then its inverse S - 1 is also an isometry, and we have (17) V ( S T S - 1 , B ( E ) ) = V ( T , B ( E ) ) , T B ( E ) .

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

Let U and V be two unitaries operators on H ; then (18) R U A U * , V B V * = M U , V * R A , B M U * , V , with M U , V * being an invertible isometry and its inverse being    M U * , V .

From this result, we deduce that (19) V ( R U A U * , V B V * , B ( B ( H ) ) ) = V ( M U , V * R A , B M U * , V , B ( B ( H ) ) ) = V ( R A , B , B ( B ( H ) ) ) . Now, using Lemma 3, we get (20) W ( i = 1 k U A i U * V B i V * ) V ( R A , B , B ( B ( H ) ) ) , U , V U ( H ) W ( i = 1 k U A i U * V B i V * ) V ( R A , B , B ( B ( H ) ) ) .

But, the numerical range is closed and the product of two unitaries is also an unitary, hence: (21) [ U U ( H ) W ( i = 1 k U A i U * B i ) - ] - = [ V U ( H ) W ( i = 1 k A i V B i V * ) - ] - V ( R A , B , B ( B ( H ) ) ) . So, we have proved the second inclusion of Theorem 1.

For the other inclusion, we will use the two following theorems.

Theorem 4 (See [<xref ref-type="bibr" rid="B12">11</xref>]).

Let 𝒜 be Banach algebra. For a 𝒜 , one has (22) V ( a , 𝒜 ) = z C { λ : | λ - z | a - z } .

The norm of an elementary operator is defined by (23) R A , B = Sup { R A , B ( X ) : X 1 } . Let 𝒜 be C * -algebra. An element U 𝒜 is said to be unitary if U * U = U U * = I . In the following, U ( 𝒜 ) denote the set of unitaries in 𝒜 .

Theorem 5 (Russo-Dye [<xref ref-type="bibr" rid="B13a">12</xref>]).

Let 𝒜 be C * - algebra. Let A = ( a 1 , , a k ) and B = ( b 1 , , b k ) be two k -tuples of elements in 𝒜 . Then, one has (24) R A , B = Sup { R A , B ( U ) : U U ( 𝒜 ) } .

We return now to the proof of the main theorem.

Proof.

We need only to show the inclusion “ ⊂. ” By Theorem 4, we have (25) V ( R A , B , B ( B ( H ) ) ) = z C { λ : | λ - z | R A , B - z } . And, by Theorem 5, we get (26) R A , B - z = Sup { i = 1 k A i U B i - z U : U U ( H ) } . But U T = T for all T B ( H ) and U U ( H ) . Hence, (27) R A , B - z = Sup { i = 1 k U * A i U B i - z I H : U U ( H ) } .

Hence, if λ V ( R A , B , B ( B ( H ) ) ) , then, for all z C , (28) λ { | μ - z | R A , B - z I B ( H ) } . Let ϵ > 0 fixed, there exists a unitary U ϵ such that (29) R A , B - z I B ( H ) i = 1 k U ϵ * A i U ϵ B i - z I H + ϵ . Now, using Theorem 4, we have (30) W ( i = 1 k U ϵ * A i U ϵ B i ) - = z C { λ : | λ - z | i = 1 k U ϵ * A i U ϵ B i - z I H } .

So, there exists μ W ( i = 1 k U ϵ * A U ϵ B ) such that | λ - μ | ϵ . But ϵ is arbitrary, λ [ U U ( H ) W ( i = 1 k U * A i U B i ) - ] - . This finishes the proof of the main theorem.

4. Some Applications

It is well known that, for the spectrum, if A , B B ( H ) , then we have (31) σ ( A B ) { 0 } = σ ( B A ) { 0 } . For the numerical range, this not true, but we can deduce the following corollary from the proof of Theorem 1.

Corollary 6.

For all A , B B ( H ) , one has (32) [ U U ( H ) W ( A U B U * ) - ] - = [ U U ( H ) W ( U B U * A ) - ] - .

The numerical radius of an operator T B ( E ) is denoted by v ( T ) and defined by (33) v ( T ) = sup { | λ | : λ V ( T , B ( E ) ) } .

Corollary 7.

Let A = ( A 1 , , A k ) and B = ( B 1 , , B k ) be two k -tuples of elements in B ( H ) . Then, one has (34) v ( R A , B ) = Sup { v ( i = 1 k U * A i U B i ) : U U ( H ) } . In particular, for A , B B ( H ) , (35) v ( δ A , B ) = sup U U ( H ) v ( U A U * - B ) = sup U U ( H ) v ( A - U B U * ) = sup U , V U ( H ) v ( U A U * - V B V * ) .

Let K be a nonempty subset of the plane and let (36) diam ( K ) = sup α , β K | α - β | . From Corollary 7 ( B = A ), one has (37) diam ( W ( A ) ) = sup { v ( U A U * - A ) : U U ( H ) } . So, the diameter of the numerical range W ( A ) is equal to the diameter of the v -unitary orbit of the operator A .

Conflict of Interests

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

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