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Document Type |
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Article In Journal |
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Document Title |
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On the numerical range and norm of elementary operators On the numerical range and norm of elementary operators |
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Document Language |
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Arabic |
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Abstract |
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Let B(E) be the complex Banach algebra of all bounded linear operators on a complex Banach space E. For n-tuples A = (A(1),...,A(n)) and B = (B-1,...B-n) of operators on E , let R-A,R-B denote the operator on B(E) defined by R-A,R-B(X) = Sigma(i=1)(n)A(i)XB(i).
For A, B is an element of B(E), we put U-A,U-B = R-(A,R-B),R- (B,R-A). In this note, we prove that
[GRAPHICS]
where V(.) is the joint spatial numerical range, W-0(.) is the algebraic numerical range and J is a norm ideal of B(E). We shall show that this inclusion becomes an equality when R-A,R-B is taken to be a derivation. Also, we deduce that w(U-A,U-B\J) greater than or equal to 2(root2 -1)w(A)w(B) is an element of B(E) and J is a norm ideal of B(E), where w(.) is the numerical radius.
On the other hand, in the particular case when E is a Hilbert space, we shall prove that the lower estimate bound parallel toU(A,B)\Jparallel to greater than or equal to 2(root2 -1) parallel toAparallel toparallel toBparallel to holds, if one of the following two conditions is satisfied:
(i) J is a standard operator algebra of B(E) and A,B is an element of J.
(ii) J is a norm ideal of B(E) and A,B is an element of B(E).
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Journal Name |
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LINEAR & MULTILINEAR ALGEBRA |
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Volume |
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52 |
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Issue Number |
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3 |
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Publishing Year |
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2004 AH
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Added Date |
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Monday, June 23, 2008 |
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