Norm for Sums of Two Basic Elementary Operators
We give necessary and sufficient conditions under which the norm of basic elementary operators attains its optimal value in terms of the numerical range.
Let be a normed space over ( or ), its unit sphere, and its dual topological space. Let be the normalized duality mapping form to given by Let be the normed space of all bounded linear operators acting on . For any operator and , is called the spatial numerical range of , which may be defined as This definition was extended to arbitrary elements of a normed algebra by Bonsall [1–3] who defined the numerical range of as where is the left regular representation of in , that is, for all . is known as the algebra numerical range of , and, according to the above definitions, is defined by For an operator , Bachir and Segres  have extended the usual definitions of numerical range from one operator to two operators in different ways as follows.
The spatial numerical range of relative to is The spatial numerical range of relative to is The maximal spatial numerical range of relative to is For , let , then the set is called the generalized maximal numerical range of relative to . It is known that is a nonempty closed subset of and . The definition of can be rewritten, with respect to the semi-inner product as with respect to an inner product as We shall be concerned to estimate the norm of the elementary operator , where are bounded linear operators on a normed space and is the basic elementary operator defined on by We also give necessary and sufficient conditions on the operators under which attaints its optimal value .
2. Equality of Norms
Our next aim is to give necessary and sufficient conditions on the set of operators for which the norm of equals .
Lemma 2.1. For any of the operators and all , one has
Proof. The proof is elementary.
Theorem 2.2. Let be operators in .
If and , then
Proof. The proof will be done in four steps; we choose one and the others will be proved similarly. Suppose that and , then there exist such that and there exist such that . Define the operators as follows: Then , and Hence Letting , Since therefore
Corollary 2.3. Let be a normed space and . Then, the following assertions hold: (1)if , then ; (2)if and , then .
Remark 2.4. In the previous corollary, if we set , then we obtain an important equation called the Daugavet equation:
It is well known that every compact operator on  or on  satisfies (2.10).
A Banach space is said to have the Daugavet property if every rank-one operator on satisfies (2.10). So that from our Corollary 2.3 if or for every rank-one operator , then has the Daugavet property.
The reverse implication in the previous theorem is not true, in general, as shown in the following example which is a modification of that given by the authors Bachir and Segres [4, Example 3.17].
Example 2.5. Let be the classical space of sequences , equipped with the norm and let be an infinite-dimensional Banach space. Taking the Banach space equipped with the norm, for , where is any norm-one operator from to which does not attain its norm (by Josefson-Nissenzweig’s theorem ), we can find a sequence such that converges weakly to . Therefore we get the desired operator defined by Let be operators defined on as follows: where is the identity operator on . It easy to check that are linear bounded operators and . If we choose and such that , then and and from we get It is clear from the definitions of and that (for details, see ).
The next result shows that the reverse is true under certain conditions, before that we recall the definition of Birkhoff-James orthogonality in normed spaces.
Definition 2.6. Let be a normed space and . We say that is orthogonal to in the sense of Birkhoff-James ([8, 9]), in short , iff
If are linear subspaces of , we say that is orthogonal to in the sense of , written as iff for all and all .
If , we will denote by and the range and the dual adjoint, respectively, of the operator .
Theorem 2.7. Let be operators in .
If , then Moreover, if then
Proof. If , then we can find two normalized sequences and such that We have for all so we can deduce from the above inequalities and (2.10) that and . From the assumptions we get Set and for all and define the function on the closed subspace spanned by for all as It is clear that is linear for all and From the assumptions it follows that This means that is continuous for each on the subspace with (by (2.27) and ). Then by Hahn-Banach theorem there is with and , for each . So hence Thus, and by Lemma 2.1 Therefore,
From we can find a normalized sequences such that Since , for each , then we can find a normalized such that We argue similarly and get Following the same steps as in the previous case we obtain .
Moreover, if we have and , it suffices to reverse, in the proof of the previous case, the role of into and into .
For the completeness of the previous theorem we need to prove the following result which is very interesting.
We recall that Phelps  has proved that, for a Banach space , is dense in ; this property is called subreflexivity of the space . Using this fact, Bonsall and Duncan  has proved that for any operator we have . The following result generalizes the Bollobas result in the case , where .
Proposition 2.8. Let be a Banach space with smooth dual and let such that is a surjective operator. Then .
Proof. Let , then there are such that .
By the subreflexivity of there exist sequences and such that and to . It follows that the sequence has an weak convergent subsequence , that is, On the one hand, we have Then Thus On the other hand, So and . Then by smoothness of the space we get . Next,
Then or and therefore which means that .
Corollary 2.9. Let be a Banach space with smooth dual and .
If and with being surjective, and , then Moreover, if , is surjective, and , then
Corollary 2.10. Let be a Banach space with smooth dual and such that are surjective operators. If and such that then the following assertions are equivalent:(1); (2).
As a particular case, we obtain the following.
Corollary 2.11. Let be a Banach space with smooth dual and are surjective operators in . If then the following assertions are equivalent:(1); (2).
3. Hilbert Space Case
Let be a complex Hilbert space and . The maximal numerical range of  denoted by is defined by and its normalized maximal range, denoted by , is given by The set is nonempty, closed, convex, and contained in the closure of the numerical range of .
In this section we prove that if , the conditions would imply that
Proposition 3.1. Let be a complex Hilbert space, .
If , then and and .
Proof. If or and or , the result is obvious.
The proof will be done in four steps, we choose one and the others will be proved similarly. Suppose that and , if , then there exists a sequence such that We have ; this yields From (3.5) and (3.6) we get Suppose now that and , if , then there exists a sequence such that Since , then .
Suppose that , then such that .
Hence From this, we derive that From (3.8) and (3.10) we have From (3.7) and (3.11) we get and and .
Remark 3.2. We remark that in the case we obtain an implication given by Boumazgour .
The authors would like to sincerely thank the anonymous referees for their valuable comments which improved the paper. This research was supported by a grant from King Khalid University (no. KKU_ S130_ 33).
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