Hindawi Publishing Corporation ISRN Mathematical Analysis Volume 2014, Article ID 246301, 4 pages http://dx.doi.org/10.1155/2014/246301
Research Article A Formula for the Numerical Range of Elementary Operators M. Barraa Department of Mathematics, Faculty of Sciences Semlalia, Marrakech, Morocco Correspondence should be addressed to M. Barraa;
[email protected] Received 10 November 2013; Accepted 9 February 2014; Published 20 March 2014 Academic Editors: M. Lindstrom and C. Zhu 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 = 𝑘 (𝐴 1 , . . . , 𝐴 𝑘 ) and 𝐵 = (𝐵1 , . . . , 𝐵𝑘 ), let 𝑅𝐴,𝐵 denote the elementary operator on 𝐵(𝐻) defined by 𝑅𝐴,𝐵 (𝑋) = ∑𝑖=1 𝐴 𝑖 𝑋𝐵𝑖 . In this 𝑘 paper, we prove the following formula for the numerical range of 𝑅𝐴,𝐵 : 𝑉(𝑅𝐴,𝐵 , 𝐵(𝐵(𝐻))) = [∪𝑈∈𝑈(𝐻) 𝑊(∑𝑖=1 𝑈𝐴𝑖 𝑈∗ 𝐵𝑖 )− ]− , 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 = (𝑎1 , . . . , 𝑎𝑘 ) and 𝑏 = (𝑏1 , . . . , 𝑏𝑘 ), let 𝑅𝑎,𝑏 denote the elementary operator on A defined by
and it is known that 𝑉(𝑇, 𝐵(𝑋)) = co𝑊(𝑇), the closed convex hull of 𝑊(𝑇). In the case of a Hilbert space 𝑋 = 𝐻, then 𝑊 (𝑇) = {⟨𝑇𝑥, 𝑥⟩ : 𝑥 ∈ 𝐻, ‖𝑥‖ = 1}
(5)
𝑘
𝑅𝑎,𝑏 (𝑥) = ∑𝑎𝑖 𝑥𝑏𝑖 .
(1)
𝑖=1
This is a bounded linear operator on A. Some interesting cases are the generalized derivation 𝛿𝑎,𝑏 (𝑥) = 𝑎𝑥 − 𝑥𝑏 and the multiplication 𝑀𝑎,𝑏 (𝑥) = 𝑎𝑥𝑏 for 𝑎, 𝑏, 𝑥 ∈ 𝐴. The numerical range of 𝑎 ∈ A is defined by 𝑉 (𝑎, A) = {𝑓 (𝑎) : 𝑓 ∈ 𝑆} ,
(2)
where 𝑆 is the set of normalized states in A: 𝑆 = {𝑓 ∈ 𝐴∗ , 𝑓 = 1 = 𝑓 (𝑒)} .
(3)
See [1–3]. 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 𝑊 (𝑇) = {𝑓 (𝑇𝑥) : 𝑥 ∈ 𝑋, ‖𝑥‖ = 1, 𝑓 ∈ 𝑋∗ , 𝑓 = 𝑓 (𝑥) = 1} , (4)
is convex but not closed in general and 𝑉(𝑇, 𝐵(𝐻)) = 𝑊(𝑇)− . Many facts about the relation between the spectrum of 𝑅𝑎,𝑏 and the spectrums of the coefficients 𝑎𝑖 and 𝑏𝑖 are known. This is not the case with the relation between the numerical range of 𝑅𝑎,𝑏 and the numerical ranges of 𝑎𝑖 and 𝑏𝑖 . 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: 𝑊 (𝛿𝑇 ) = 𝑊(𝑇)− − 𝑊(𝑇)− ,
(6)
where 𝛿𝑇 is the inner derivation defined by 𝛿𝑇 (𝑋) = 𝑇𝑋 − 𝑋𝑇. 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 𝑅𝐴,𝐵 . In this paper, we give a formula that answers this problem.
2
ISRN Mathematical Analysis
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 = (𝐵1 , . . . , 𝐵𝑘 ) be two 𝑘-tuples of elements in 𝐵(𝐻). Then, one has 𝑘
𝜙𝑥⊗𝑥 (𝐼𝐻) = tr (𝑥 ⊗ 𝑥) = ⟨𝑥, 𝑥⟩ = 1,
𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) = [ ⋃ 𝑊(∑𝑈𝐴 𝑖 𝑈 𝐵𝑖 ) ] . (7) 𝑖=1 [𝑈∈𝑈(𝐻) ] ∗
In particular for multiplication and generalized derivation, one has (𝐴, 𝐵 ∈ 𝐵(𝐻)): −
𝜙𝑥⊗𝑥 (𝑅𝐴,𝐵 (𝐼𝐻)) ∈ 𝑉 (𝑀𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) , 𝑘
𝜙𝑥⊗𝑥 (𝑅𝐴,𝐵 (𝐼𝐻)) = 𝜙𝑥⊗𝑥 (∑𝐴 𝑖 𝐵𝑖 ) 𝑖=1
−
−
−
From Fong’s formula (see [4, 6, 10]), we can deduce the following. Corollary 2. For 𝐴, 𝐵 ∈ 𝐵(𝐻), one has 𝑊(𝐴)− − 𝑊(𝐵)− = [ ⋃ 𝑊(𝑈𝐴𝑈∗ − 𝐵) ] −
𝑖=1
(16)
Let 𝐸 be a Banach space. We say that 𝑆 ∈ 𝐵(𝐸) is an isometry if ‖𝑆𝑥‖ = ‖𝑥‖ for all 𝑥 ∈ 𝐸. If 𝑆 is an invertible isometry, then its inverse 𝑆−1 is also an isometry, and we have 𝑉 (𝑆𝑇𝑆−1 , 𝐵 (𝐸)) = 𝑉 (𝑇, 𝐵 (𝐸)) ,
−
−
𝑖=1
Hence, 𝑊(∑𝑘𝑖=1 𝐴 𝑖 𝐵𝑖 ) ⊂ 𝑊(𝑅𝐴,𝐵 ) ⊂ 𝑉(𝑅𝐴,𝐵 , 𝐵(𝐵(𝐻))).
𝑈∈𝑈(𝐻)
𝑈∈𝑈(𝐻)
𝑘
= 𝜆.
(8)
𝑉 (𝛿𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) = [ ⋃ 𝑊(𝑈∗ 𝐴𝑈 − 𝐵) ] .
−
𝑘
= tr (∑𝐴 𝑖 𝐵𝑖 (𝑥 ⊗ 𝑥)) = ⟨∑𝐴 𝑖 𝐵𝑖 𝑥, 𝑥⟩
𝑉 (𝑀𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) = [ ⋃ 𝑊(𝐴𝑈𝐵𝑈∗ ) ] , 𝑈∈𝑈(𝐻)
(15)
the form 𝜙𝑥⊗𝑥 is a state on 𝐵(𝐻). So,
−
−
This is a bounded linear form on 𝐵(𝐻), with norm being equal to 1, because (14) 𝜙𝑥⊗𝑥 = ‖𝑥 ⊗ 𝑥‖ = 1. Since
(9)
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 𝑅𝑈𝐴𝑈∗ ,𝑉𝐵𝑉∗ = 𝑀𝑈,𝑉∗ 𝑅𝐴,𝐵 𝑀𝑈∗ ,𝑉,
= [ ⋃ 𝑊(𝐴 − 𝑈𝐵𝑈∗ ) ] .
(17)
(18)
with 𝑀𝑈,𝑉∗ being an invertible isometry and its inverse being 𝑀𝑈∗ ,𝑉. From this result, we deduce that
𝑈∈𝑈(𝐻)
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 = (𝐵1 , . . . , 𝐵𝑘 ) be two 𝑘-tuples of elements in 𝐵(𝐻). Then, one has
𝑉 (𝑅𝑈𝐴𝑈∗ ,𝑉𝐵𝑉∗ , 𝐵 (𝐵 (𝐻))) = 𝑉 (𝑀𝑈,𝑉∗ 𝑅𝐴,𝐵 𝑀𝑈∗ ,𝑉, 𝐵 (𝐵 (𝐻))) = 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) . (19) Now, using Lemma 3, we get 𝑘
𝑘
𝑊 (∑𝐴 𝑖 𝐵𝑖 ) ⊂ 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) .
(10)
𝑊 (∑𝑈𝐴 𝑖 𝑈∗ 𝑉𝐵𝑖 𝑉∗ ) ⊂ 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) , 𝑖=1
𝑖=1
In particular, for 𝐴, 𝐵 ∈ 𝐵(𝐻), one has 𝑊 (𝐴𝐵) ⊂ 𝑉 (𝑀𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) , 𝑊 (𝐴 − 𝐵) ⊂ 𝑉 (𝛿𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) .
⋃
𝑘
𝑖=1
𝑖=1
𝜆 = ⟨∑𝐴 𝑖 𝐵𝑖 𝑥, 𝑥⟩ = tr (∑𝐴 𝑖 𝐵𝑖 (𝑥 ⊗ 𝑥)) .
(20)
(11)
(12)
But, the numerical range is closed and the product of two unitaries is also an unitary, hence:
(13)
−
−
𝑘
[ ⋃ 𝑊(∑𝑈𝐴 𝑖 𝑈 𝐵𝑖 ) ] 𝑖=1 [𝑈∈𝑈(𝐻) ] ∗
𝑘
Here, tr(⋅) is the linear form trace. Let 𝜙𝑥⊗𝑥 be the linear form defined by 𝜙𝑥⊗𝑥 (𝑋) = tr (𝑋 (𝑥 ⊗ 𝑥)) = ⟨𝑋𝑥, 𝑥⟩ .
𝑖=1
𝑈,𝑉∈𝑈(𝐻)
Proof. Let 𝜆 ∈ 𝑊(∑𝑘𝑖=1 𝐴 𝑖 𝐵𝑖 ); by definition, there exists 𝑥 ∈ 𝐻 with ‖𝑥‖ = 1 such that 𝑘
𝑘
𝑊 (∑𝑈𝐴 𝑖 𝑈∗ 𝑉𝐵𝑖 𝑉∗ ) ⊂ 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) .
−
−
= [ ⋃ 𝑊(∑𝐴 𝑖 𝑉𝐵𝑖 𝑉 ) ] ⊂ 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) . 𝑖=1 [𝑉∈𝑈(𝐻) ] (21) ∗
So, we have proved the second inclusion of Theorem 1.
ISRN Mathematical Analysis
3
For the other inclusion, we will use the two following theorems. Theorem 4 (See [11]). Let A be Banach algebra. For 𝑎 ∈ A, one has 𝑉 (𝑎, A) = ⋂ {𝜆 : |𝜆 − 𝑧| ≤ ‖𝑎 − 𝑧‖} . 𝑧∈C
The norm of an elementary operator is defined by 𝑅𝐴,𝐵 = Sup {𝑅𝐴,𝐵 (𝑋) : ‖𝑋‖ ≤ 1} .
(22)
(23)
Let A be 𝐶∗ -algebra. An element 𝑈 ∈ A is said to be unitary if 𝑈∗ 𝑈 = 𝑈𝑈∗ = 𝐼. In the following, 𝑈(A) denote the set of unitaries in A. Theorem 5 (Russo-Dye [12]). Let A be 𝐶∗ -algebra. Let 𝐴 = (𝑎1 , . . . , 𝑎𝑘 ) and 𝐵 = (𝑏1 , . . . , 𝑏𝑘 ) be two 𝑘-tuples of elements in A. Then, one has (24) 𝑅𝐴,𝐵 = Sup {𝑅𝐴,𝐵 (𝑈) : 𝑈 ∈ 𝑈 (A)} . We return now to 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 𝜎 (𝐴𝐵) ∪ {0} = 𝜎 (𝐵𝐴) ∪ {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 −
𝑧∈C
𝑈∈𝑈(𝐻)
The numerical radius of an operator 𝑇 ∈ 𝐵(𝐸) is denoted by V(𝑇) and defined by V (𝑇) = sup {|𝜆| : 𝜆 ∈ 𝑉 (𝑇, 𝐵 (𝐸))} .
Corollary 7. Let 𝐴 = (𝐴 1 , . . . , 𝐴 𝑘 ) and 𝐵 = (𝐵1 , . . . , 𝐵𝑘 ) be two 𝑘-tuples of elements in 𝐵(𝐻). Then, one has V (𝑅𝐴,𝐵 ) = Sup {V (∑𝑈∗ 𝐴 𝑖 𝑈𝐵𝑖 ) : 𝑈 ∈ 𝑈 (𝐻)} .
(34)
𝑖=1
In particular, for 𝐴, 𝐵 ∈ 𝐵(𝐻),
= sup V (𝐴 − 𝑈𝐵𝑈∗ ) 𝑈∈𝑈(𝐻)
=
(35)
sup V (𝑈𝐴𝑈∗ − 𝑉𝐵𝑉∗ ) .
𝑈,𝑉∈𝑈(𝐻)
Let 𝐾 be a nonempty subset of the plane and let diam (𝐾) = sup 𝛼 − 𝛽 . 𝛼,𝛽∈𝐾
(36)
From Corollary 7 (𝐵 = 𝐴), one has
Hence, if 𝜆 ∈ 𝑉(𝑅𝐴,𝐵 , 𝐵(𝐵(𝐻))), then, for all 𝑧 ∈ C, (28)
diam (𝑊 (𝐴)) = sup {V (𝑈𝐴𝑈∗ − 𝐴) : 𝑈 ∈ 𝑈 (𝐻)} .
(37)
So, the diameter of the numerical range 𝑊(𝐴) is equal to the diameter of the V-unitary orbit of the operator 𝐴.
Let 𝜖 > 0 fixed, there exists a unitary 𝑈𝜖 such that (29)
Now, using Theorem 4, we have
Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper.
−
𝑊(∑𝑈𝜖∗ 𝐴 𝑖 𝑈𝜖 𝐵𝑖 ) 𝑘 ∗ = ⋂ {𝜆 : |𝜆 − 𝑧| ≤ ∑𝑈𝜖 𝐴 𝑖 𝑈𝜖 𝐵𝑖 − 𝑧𝐼𝐻} . 𝑖=1 𝑧∈C
(33)
𝑈∈𝑈(𝐻)
But ‖𝑈𝑇‖ = ‖𝑇‖ for all 𝑇 ∈ 𝐵(𝐻) and 𝑈 ∈ 𝑈(𝐻). Hence, 𝑘 ∗ 𝑅𝐴,𝐵 − 𝑧 = Sup {∑𝑈 𝐴 𝑖 𝑈𝐵𝑖 − 𝑧𝐼𝐻 : 𝑈 ∈ 𝑈 (𝐻)} . 𝑖=1 (27)
𝑖=1
−
𝑈∈𝑈(𝐻)
𝑘 𝑅𝐴,𝐵 − 𝑧 = Sup {∑𝐴 𝑖 𝑈𝐵𝑖 − 𝑧𝑈 : 𝑈 ∈ 𝑈 (𝐻)} . (26) 𝑖=1
𝑘
−
V (𝛿𝐴,𝐵 ) = sup V (𝑈𝐴𝑈∗ − 𝐵)
And, by Theorem 5, we get
𝑘 ∗ 𝑅𝐴,𝐵 − 𝑧𝐼𝐵(𝐻) ≤ ∑𝑈𝜖 𝐴 𝑖 𝑈𝜖 𝐵𝑖 − 𝑧𝐼𝐻 + 𝜖. 𝑖=1
−
[ ⋃ 𝑊(𝐴𝑈𝐵𝑈∗ ) ] = [ ⋃ 𝑊(𝑈𝐵𝑈∗ 𝐴) ] . (32)
𝑘
Proof. We need only to show the inclusion “⊂.” By Theorem 4, we have 𝑉 (𝑅𝐴,𝐵 , 𝐵 (𝐵 (𝐻))) = ⋂ {𝜆 : |𝜆 − 𝑧| ≤ 𝑅𝐴,𝐵 − 𝑧} . (25)
𝜆 ∈ {𝜇 − 𝑧 ≤ 𝑅𝐴,𝐵 − 𝑧𝐼𝐵(𝐻) } .
(31)
(30)
So, there exists 𝜇 ∈ 𝑊(∑𝑘𝑖=1 𝑈𝜖∗ 𝐴𝑈𝜖 𝐵) such that |𝜆−𝜇| ≤ 𝜖. But 𝜖 is arbitrary, 𝜆 ∈ [∪𝑈∈𝑈(𝐻) 𝑊(∑𝑘𝑖=1 𝑈∗ 𝐴 𝑖 𝑈𝐵𝑖 )− ]− . This finishes the proof of the main theorem.
References [1] F. F. Bonsall and J. Duncan, Numerical Ranges Vol I, Cambridge University Press, New York, NY, USA, 1973. [2] F. F. Bonsall and J. Duncan, Numerical Rangesvol Vol II, Cambridge University Press, New York, NY, USA, 1973. [3] 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.
4 [4] C. K. Fong, Some Aspects of Derivations on B(H), University of Toronto, Seminar Notes, 1978. [5] J. Kyle, “Numerical ranges of derivations,” Proceedings of the Edinburgh Mathematical Society, vol. 21, no. 1, pp. 33–39, 1979. [6] K. Mattila, “Complex strict and uniform convexity and hyponormal operators,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 96, no. 3, pp. 483–493, 1984. [7] S.-Y. Shaw, “On numerical ranges of generalized derivations and related properties,” Australian Mathematical Society Journal A, vol. 36, no. 1, pp. 134–142, 1984. [8] A. Seddik, “The numerical range of elementary operators. II,” Linear Algebra and Its Applications, vol. 338, pp. 239–244, 2001. [9] L. A. Fialkow, “Structural properties of elementary operators,” in Elementary Operators and Applications (Blaubeuren, 1991), pp. 55–113, World Scientific, River Edge, NJ, USA, 1992. [10] R. Schatten, Norm Ideals of Completely Continuous Operators, Springer, Berlin, Germany, 1960. [11] J. G. Stampfli and J. P. Williams, “Growth conditions and the numerical range in a Banach algebra,” The Tohoku Mathematical Journal, vol. 20, pp. 417–424, 1968. [12] B. Russo and H. A. Dye, “A note on unitary operators in 𝐶∗ algebras,” Duke Mathematical Journal, vol. 33, pp. 413–416, 1966.
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