Norm for Sums of Two Basic Elementary Operators

Abstract

We give necessary and sufficient conditions under which the norm of basic elementary operators attains its optimal value in terms of the numerical range.

1. Introduction

Let 𝐸 be a normed space over 𝕂 (ℝ or ℂ), 𝑆_𝐸 its unit sphere, and 𝐸^∗ its dual topological space. Let 𝐷 be the normalized duality mapping form 𝐸 to 𝐸^∗ given by 𝐷(𝑥)=𝜑∈𝐸^∗∶𝜑(𝑥)=‖𝑥‖²,𝜑=‖𝑥‖,∀𝑥∈𝐸. (1.1) Let 𝐵(𝐸) be the normed space of all bounded linear operators acting on 𝐸. For any operator 𝐴∈𝐵(𝐸) and 𝑥∈𝐸, 𝑊_𝑥(𝐴)=𝜑()(𝑥,𝑊(𝐴)𝑊𝐴𝑥∶𝜑∈𝐷)=∪_𝑥(𝐴)∶𝑥∈𝑆_𝐸 (1.2) is called the spatial numerical range of 𝐴, which may be defined as 𝑊(𝐴)=𝜑(𝐴𝑥)∶𝑥∈𝑆_𝐸;𝜑∈𝐷(𝑥). (1.3) This definition was extended to arbitrary elements of a normed algebra 𝒜 by Bonsall [1–3] who defined the numerical range of 𝑎∈𝒜 as 𝑉(𝑎)=𝑊𝐴_𝑎, (1.4) where 𝐴_𝑎 is the left regular representation of 𝐴 in 𝐵(𝒜), that is, 𝐴_𝑎=𝑎𝑏 for all 𝑏∈𝒜. 𝑉(𝑎) is known as the algebra numerical range of 𝑎∈𝒜, and, according to the above definitions, 𝑉(𝑎) is defined by 𝑉(𝑎)=𝜑(𝑎𝑏)∶𝑏∈𝑆_𝒜;𝜑∈𝐷(𝑏). (1.5) For an operator 𝐴∈𝐵(𝐸), Bachir and Segres [4] have extended the usual definitions of numerical range from one operator to two operators in different ways as follows.

The spatial numerical range 𝑊(𝐴)_𝐵 of 𝐴∈𝐵(𝐸) relative to 𝐵 is 𝑊(𝐴)_𝐵=𝜑(𝐴𝑥)∶𝑥∈𝑆_𝐸;𝜑∈𝐷(𝐵𝑥). (1.6) The spatial numerical range 𝐺(𝐴)_𝐵 of 𝐴∈𝐵(𝐸) relative to 𝐵 is 𝐺(𝐴)_𝐵=𝜑(𝐴𝑥)∶𝑥∈𝐸;‖𝐵𝑥‖=1,𝜑∈𝐷(𝐵𝑥). (1.7) The maximal spatial numerical range of 𝐴∈𝐵(𝐸) relative to 𝐵 is 𝑀(𝐴)_𝐵=𝜑(𝐴𝑥)∶𝑥∈𝑆_𝐸;‖𝐵𝑥‖=‖𝐵‖,𝜑∈𝐷(𝐵𝑥). (1.8) For 𝐴,𝐵∈𝐵(𝐸), let 𝑆_𝐸(𝐵)={(𝑥_𝑛)_𝑛∶𝑥_𝑛∈𝑆_𝐸,‖𝐵𝑥_𝑛‖→‖𝐵‖}, then the set ℳ(𝐴)_𝐵=lim𝜑_𝑛𝐴𝑥_𝑛∶𝑥_𝑛𝑛∈𝑆_𝐸(𝐵),𝜑_𝑛∈𝐷𝐵𝑥_𝑛 (1.9) is called the generalized maximal numerical range of 𝐴 relative to 𝐵. It is known that ℳ(𝐴)_𝐵 is a nonempty closed subset of 𝕂 and 𝑀(𝐴)_𝐵⊆ℳ(𝐴)_𝐵⊆𝑊(𝐴)_𝐵. The definition of ℳ(𝐴)_𝐵 can be rewritten, with respect to the semi-inner product [⋅,⋅] as ℳ(𝐴)_𝐵=lim𝐴𝑥_𝑛,𝐵𝑥_𝑛∶𝑥_𝑛𝑛∈𝑆_𝐸(𝐵), (1.10) with respect to an inner product (⋅,⋅) as ℳ(𝐴)_𝐵=lim𝐴𝑥_𝑛,𝐵𝑥_𝑛∶𝑥_𝑛𝑛∈𝑆_𝐸(𝐵). (1.11) We shall be concerned to estimate the norm of the elementary operator 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2, where 𝐴₁,𝐴₂,𝐵₁,𝐵₂ are bounded linear operators on a normed space 𝐸 and 𝑀_𝐴1,𝐵1 is the basic elementary operator defined on 𝐵(𝐸) by 𝑀_𝐴1,𝐵1(𝑋)=𝐴₁𝑋𝐵₁. (1.12) We also give necessary and sufficient conditions on the operators 𝐴₁,𝐴₂,𝐵₁,𝐵₂ under which 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2 attaints its optimal value ‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖.

2. Equality of Norms

Our next aim is to give necessary and sufficient conditions on the set {𝐴₁,𝐴₂,𝐵₁,𝐵₂} of operators for which the norm of 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2 equals ‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖.

Lemma 2.1. For any of the operators 𝐴,𝐵,𝐶∈𝐵(𝐸) and all 𝛼,𝛽∈𝕂, one has ℳ𝛼𝐴+𝛽𝐵_𝐵=𝛼ℳ(𝐴)_𝐵+𝛽‖𝐵‖²;ℳ𝛼𝐴+𝛽𝐶_𝐵⊆𝛼ℳ(𝐴)_𝐵+𝛽ℳ(𝐶)_𝐵. (2.1)

Proof. The proof is elementary.

Theorem 2.2. Let 𝐴₁,𝐴₂,𝐵₁,𝐵₂ be operators in 𝐵(𝐸).
If ‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴₁)_𝐴2∪ℳ(𝐴₂)_𝐴1 and ‖𝐵₁‖‖𝐵₂‖∈ℳ(𝐵₁)_𝐵2∪ℳ(𝐵₂)_𝐵1, then 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2=𝐴₁𝐵₁+𝐴₂𝐵₂. (2.2)

Proof. The proof will be done in four steps; we choose one and the others will be proved similarly. Suppose that ‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴₁)_𝐴2 and ‖𝐵₁‖‖𝐵₂‖∈ℳ(𝐵₁)_𝐵2, then there exist (𝑥_𝑛)_𝑛∈𝑆_𝐸(𝐴₂),𝜑_𝑛∈𝐷(𝐴₂𝑥_𝑛) such that ‖𝐴₁‖‖𝐴₂‖=lim_𝑛𝜑_𝑛(𝐴₁𝑥_𝑛) and there exist (𝑦_𝑛)_𝑛∈𝑆_𝐸(𝐵₂),𝜓_𝑛∈𝐷(𝐵₂𝑦_𝑛) such that ‖𝐵₁‖‖𝐵₂‖=lim_𝑛𝜓_𝑛(𝐵₁𝑦_𝑛). Define the operators 𝑋_𝑛∈𝐵(𝐸) as follows: 𝑋_𝑛𝑦_𝑛=𝜓_𝑛⊗𝑥_𝑛𝑦_𝑛=𝜓_𝑛𝑦_𝑛𝑥_𝑛,∀𝑛. (2.3) Then ‖𝑋_𝑛‖≤‖𝐵₂‖,forall𝑛≥1, and 𝑀_𝐴1+𝐵1+𝑀_𝐴2+𝐵2𝑋_𝑛𝑦_𝑛=𝐴₁𝑋_𝑛𝐵₁+𝐴₂𝑋_𝑛𝐵₂𝑦_𝑛=𝐴₁𝑋_𝑛𝐵₁𝑦_𝑛+𝐴₂𝑋_𝑛𝐵₂𝑦_𝑛=𝜑_𝑛𝜑_𝑛𝐴₁𝜓_𝑛𝐵₁𝑦_𝑛𝑥_𝑛+𝐴₂𝜓_𝑛𝐵₂𝑦_𝑛𝑥_𝑛≥1𝜑_𝑛𝜑_𝑛𝜓_𝑛𝐵₁𝑦_𝑛𝐴₁𝑥_𝑛+𝜓_𝑛𝐵₂𝑦_𝑛𝐴₂𝑥_𝑛=1𝜑_𝑛𝜓_𝑛𝐵₁𝑦_𝑛𝜑_𝑛𝐴₁𝑥_𝑛+𝐵₂𝑦_𝑛²𝐴₂𝑥_𝑛².𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≥𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2𝑋_𝑛𝑦_𝑛𝑋_𝑛,∀𝑛≥1. (2.4)(2.5) Hence 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≥𝜓_𝑛𝐵₁𝑦_𝑛𝜑_𝑛𝐴₁𝑥_𝑛+𝐵₂𝑦_𝑛²𝐴₂𝑥_𝑛²𝐴₂𝐵₂,∀𝑛≥1. (2.6) Letting 𝑛→∞, 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≥𝐴₁𝐵₁+𝐴₂𝐵₂. (2.7) Since 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≤𝐴₁𝐵₁+𝐴₂𝐵₂, (2.8) therefore 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2=𝐴₁𝐵₁+𝐴₂𝐵₂. (2.9)

Corollary 2.3. Let 𝐸 be a normed space and 𝐴,𝐵∈𝐵(𝐸). Then, the following assertions hold: (1)if ‖𝐴‖‖𝐵‖∈ℳ(𝐴)_𝐵, then ‖𝐴+𝐵‖=‖𝐴‖+‖𝐵‖; (2)if ‖𝐴‖∈ℳ(𝐼)_𝐴 and ‖𝐵‖∈ℳ(𝐼)_𝐵, then ‖𝑀_𝐴,𝐵+𝐼‖=1+‖𝐴‖‖𝐵‖.

Remark 2.4. In the previous corollary, if we set 𝐵=𝐼, then we obtain an important equation called the Daugavet equation: ‖𝐴+𝐼‖=1+‖𝐴‖. (2.10) It is well known that every compact operator on 𝐶[0,1] [5] or on 𝐿₁[0,1] [6] satisfies (2.10).
A Banach space 𝐸 is said to have the Daugavet property if every rank-one operator on 𝐸 satisfies (2.10). So that from our Corollary 2.3 if 1∈ℳ(𝐼)_𝐴 or 1∈ℳ(𝐴)_𝐼 for every rank-one operator 𝐴, then 𝐸 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 𝑐₀ be the classical space of sequences (𝑥_𝑛)_𝑛⊂ℂ∶𝑥_𝑛→0, equipped with the norm ‖(𝑥_𝑛)_𝑛‖=max_𝑛|𝑥_𝑛| and let 𝐿 be an infinite-dimensional Banach space. Taking the Banach space 𝐸=𝐿⊕𝑐₀ equipped with the norm, for 𝑥=(𝑥₁+𝑥₂)∈𝐸,‖𝑥‖=‖𝑥₁+𝑥₂‖=max{‖𝑥₁‖,‖𝑇𝑥₁‖+‖𝑥₂‖}, where 𝑇 is any norm-one operator from 𝐿 to 𝑐₀ which does not attain its norm (by Josefson-Nissenzweig’s theorem [7]), we can find a sequence (𝜑_𝑛)_𝑛⊂𝑆_𝐸^∗ such that 𝜑_𝑛 converges weakly to 0. Therefore we get the desired operator 𝑇∶𝐿→𝑐₀ defined by (𝑇𝑥)_𝑛=𝑛𝜑𝑛+1^𝑛(𝑥). (2.11) Let 𝐴₁,𝐴₂,𝐵₁,𝐵₂ be operators defined on 𝐸 as follows: 𝐴₁𝑥₁+𝑥₂=0+𝑇𝑥₁;𝐴₂𝑥=𝐴₂𝑥₁+𝑥₂=𝑥₁𝐵+0;₁𝑥₁+𝑥₂=𝑥₁−𝑥₂;𝐵₂=𝐼,∀𝑥=𝑥₁+𝑥₂∈𝐿×𝑐₀, (2.12) where 𝐼 is the identity operator on 𝐸. It easy to check that 𝐴₁,𝐴₂,𝐵₁ are linear bounded operators and ‖𝐴₁‖=‖𝐴₂‖=‖𝐵₁‖=‖𝐵₂‖=1. If we choose 𝑋₀=𝐼 and 𝑥₀=𝑥₁+0 such that 1=‖𝑇𝑥₁‖≥‖𝑥₁‖, then ‖𝑋₀‖=‖𝑥₀‖=1 and 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≥𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2𝑋₀𝑥₀=𝐴₁𝑋₀𝐵₁+𝐴₂𝑋₀𝐵₂𝑥₀=0+𝑇𝑥₁+𝑥₁𝑥+0=max₁,2𝑇𝑥₁=2, (2.13) and from 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2≤𝐴₁𝐴₂+𝐵₁𝐵₂=2 (2.14) we get 𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2=2=𝐴₁𝐴₂+𝐵₁𝐵₂. (2.15) It is clear from the definitions of ℳ(𝐴₁)_𝐴2 and 𝑊(𝐴₁)_𝐴2 that ℳ𝐴_1𝐴2⊆𝑊𝐴_1𝐴2 (2.16) (for details, see [4]).

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 𝐸 be a normed space and 𝑥,𝑦∈𝐸. We say that 𝑥 is orthogonal to 𝑦 in the sense of Birkhoff-James ([8, 9]), in short 𝑥⟂_𝐵−𝐽𝑦, iff ∀𝜆∈𝕂∶𝑥+𝜆𝑦≥‖𝑥‖. (2.17)
If 𝐹,𝐺 are linear subspaces of 𝐸, we say that 𝐹 is orthogonal to 𝐺 in the sense of ⟂_𝐵−𝐽, written as 𝐹⟂_𝐵−𝐽𝐺 iff 𝑥⟂_𝐵−𝐽𝑦 for all 𝑥∈𝐹 and all 𝑦∈𝐺.
If 𝑇∈𝐵(𝐸), we will denote by Ran(𝑇) and 𝑇^† the range and the dual adjoint, respectively, of the operator 𝑇.

Theorem 2.7. Let 𝐴₁,𝐴₂,𝐵₁,𝐵₂ be operators in 𝐵(𝐸).
If ‖𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2‖=‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖, Ran𝐴^†₂⟂_𝐵−𝐽Ran𝐴^†₁−𝐴₁𝐴₂𝐴^†₂,Ran𝐵₂⟂_𝐵−𝐽Ran𝐵₁−𝐵₁𝐵₂𝐵₂, (2.18) then 𝐴₁𝐴₂∈ℳ𝐴^†_1𝐴†2,𝐵₁𝐵₂∈ℳ𝐵_1𝐵2. (2.19) Moreover, if Ran𝐴^†₁⟂_𝐵−𝐽Ran𝐴^†₂−𝐴₂𝐴₁𝐴^†₁,Ran𝐵₁⟂_𝐵−𝐽Ran𝐵₂−𝐵₂𝐵₁𝐵₁, (2.20) then 𝐴₁𝐴₂∈ℳ𝐴^†_1𝐴†2∩ℳ𝐴^†_2𝐴†1,𝐵₁𝐵₂∈ℳ𝐵_1𝐵2∩ℳ𝐵_2𝐵1. (2.21)

Proof. If ‖𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2‖=‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖, then we can find two normalized sequences (𝑋_𝑛)_𝑛⊆𝐵(𝐸) and (𝑥_𝑛)_𝑛⊆𝐸 such that lim_𝑛𝐴₁𝑋_𝑛𝐵₁𝑥_𝑛+𝐴₂𝑋_𝑛𝐵₂𝑥_𝑛=𝐴₁𝐵₁+𝐴₂𝐵₂. (2.22) We have for all 𝑛≥1𝐴₁𝑋_𝑛𝐵₁𝑥_𝑛≤𝐴₁𝐵₁𝑥_𝑛≤𝐴₁𝐵₁𝐴₂𝑋_𝑛𝐵₂𝑥_𝑛≤𝐴₂𝐵₂𝑥_𝑛≤𝐴₂𝐵₂, (2.23) so we can deduce from the above inequalities and (2.10) that lim_𝑛‖𝐵₁𝑥_𝑛‖=‖𝐵₁‖ and lim_𝑛‖𝐵₂𝑥_𝑛‖=‖𝐵₂‖. From the assumptions Ran(𝐵₂)⟂_𝐵−𝐽Ran(𝐵₁−(‖𝐵₁‖/‖𝐵₂‖)𝐵₂) we get Ran𝐵₁−𝐵₁𝐵₂𝐵₂∩Ran𝐵₂={0}. (2.24) Set 𝜒_𝑛=(𝐵₁−(‖𝐵₁‖/‖𝐵₂‖)𝐵₂)𝑥_𝑛 and 𝑦_𝑛=𝐵₂𝑥_𝑛 for all 𝑛 and define the function 𝜙_𝑛 on the closed subspace 𝐹 spanned by {𝑥_𝑛,𝑦_𝑛} for all 𝑛 as 𝜙_𝑛𝑎𝜒_𝑛+𝑏𝑦_𝑛=𝑏𝑦_𝑛²=𝑏𝐵₂𝑥_𝑛,∀𝑎,𝑏∈𝕂. (2.25) It is clear that 𝜙_𝑛 is linear for all 𝑛 and 𝜙_𝑛𝑎𝜒_𝑛+𝑏𝑦_𝑛=|𝑏|𝐵₂𝑥_𝑛²=𝑎𝜒_𝑛+𝑏𝑦_𝑛𝐵₂𝑥_𝑛𝑏𝑦_𝑛𝑎𝜒_𝑛+𝑏𝑦_𝑛. (2.26) From the assumptions Ran(𝐵₂)⟂_𝐵−𝐽Ran(𝐵₁−(‖𝐵₁‖/‖𝐵₂‖)𝐵₂) it follows that 𝜙𝑎𝜒_𝑛+𝑏𝑦_𝑛≤𝐵₂𝑥_𝑛𝑎𝜒_𝑛+𝑏𝑦_𝑛,∀𝑎,𝑏∈𝕂,∀𝑛. (2.27) This means that 𝜙_𝑛 is continuous for each 𝑛 on the subspace 𝐹 with ‖𝜙_𝑛‖=‖𝐵₂𝑥_𝑛‖ (by (2.27) and 𝜙_𝑛(𝑦_𝑛)=‖𝑦_𝑛‖‖𝐵₂𝑥_𝑛‖). Then by Hahn-Banach theorem there is 𝜙_𝑛∈𝐸^∗ with 𝜙_𝑛|_𝐹=𝜙_𝑛 and ‖𝜙_𝑛‖=‖𝜙_𝑛‖, for each 𝑛. So 𝜙_𝑛𝜒_𝑛=𝜙_𝑛𝐵₁−𝐵₁𝐵₂𝐵₂𝑥_𝑛=0, (2.28) hence lim_𝑛𝜙_𝑛𝜒_𝑛=𝜙_𝑛𝐵₁−𝐵₁𝐵₂𝐵₂𝑥_𝑛𝜙=0,_𝑛𝐵₂𝑥_𝑛=𝐵₂𝑥_𝑛²,𝜙_𝑛=𝐵₂𝑥_𝑛. (2.29) Thus, 0∈ℳ(𝐵₁−(‖𝐵₁‖/‖𝐵₂‖)𝐵₂)_𝐵2 and by Lemma 2.10∈ℳ𝐵_1𝐵2−𝐵₁𝐵₂𝐵₂²=ℳ𝐵_1𝐵2−𝐵₁𝐵₂. (2.30) Therefore, 𝐵₁𝐵₂∈ℳ𝐵_1𝐵2. (2.31)

From ‖𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2‖=‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖ we can find a normalized sequences (𝑋_𝑛)_𝑛⊆𝐵(𝐸) such that lim_𝑛𝐴₁𝑋_𝑛𝐵₁+𝐴₂𝑋_𝑛𝐵₂=𝐴₁𝐵₁+𝐴₂𝐵₂. (2.32) Since ‖𝐴₁𝑋_𝑛𝐵₁+𝐴₂𝑋_𝑛𝐵₂‖=‖𝐵^†₁𝑋^†_𝑛𝐴^†₁+𝐵^†₂𝑋^†_𝑛𝐴^†₂‖, for each 𝑛, then we can find a normalized 𝜙_𝑛𝑘∈𝐸^† such that lim_𝑘,𝑛𝐵^†₁𝑋^†_𝑛𝐴^†₁𝜙_𝑛𝑘+𝐵^†₂𝑋^†_𝑛𝐴^†₂𝜙_𝑛𝑘=𝐴₁𝐵₁+𝐴₂𝐵₂. (2.33) We argue similarly and get lim_𝑘,𝑛𝐴^†₁𝜙_𝑛𝑘=𝐴^†₁,lim_𝑘,𝑛𝐴^†₂𝜙_𝑛𝑘=𝐴^†₂. (2.34) Following the same steps as in the previous case we obtain ‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴^†₁)_𝐴^†₂.

Moreover, if we have Ran(𝐴^†₁)⟂_𝐵−𝐽Ran(𝐴^†₂−(‖𝐴₂‖/‖𝐴₁‖)𝐴^†₁) and Ran(𝐵₁)⟂_𝐵−𝐽Ran(𝐵₂−(‖𝐵₂‖/‖𝐵₁‖)𝐵₁), it suffices to reverse, in the proof of the previous case, the role of 𝐴^†₁ into 𝐴^†₂ and 𝐵₁ into 𝐵₂.

For the completeness of the previous theorem we need to prove the following result which is very interesting.

We recall that Phelps [10] has proved that, for a Banach space 𝐸, ∪{𝐷(𝑥)∶𝑥∈𝐸} is dense in 𝐸^∗; this property is called subreflexivity of the space 𝐸. Using this fact, Bonsall and Duncan [2] has proved that for any operator 𝑇∈𝐵(𝐸) we have 𝑊(𝑇)=𝑊(𝑇^†). The following result generalizes the Bollobas result in the case ℳ(𝐴)_𝐵, where 𝐴,𝐵∈𝐵(𝐸).

Proposition 2.8. Let 𝐸 be a Banach space with smooth dual and let 𝐴,𝐵∈𝐵(𝐸) such that 𝐵 is a surjective operator. Then ℳ(𝐴^†)_𝐵^†⊆ℳ(𝐴)_𝐵.

Proof. Let 𝑎∈ℳ(𝐴^†)_𝐵^†, then there are 𝜓_𝑛∈𝐷(𝐵^†𝜑_𝑛),(𝜑_𝑛)_𝑛∈𝑆_𝐸^∗(𝐵^†) such that 𝑎=lim_𝑛𝜓_𝑛(𝐴^†𝜑_𝑛).
By the subreflexivity of 𝐸 there exist sequences (𝜑_𝑛𝑘)_𝑛𝑘⊆𝐸^∗ and (𝑥_𝑛𝑘)⊆𝐸 such that 𝜑_𝑛𝑘∈𝐷(𝑥_𝑛𝑘) and ‖𝜑_𝑛𝑘−‖𝐵𝑥_𝑛𝑘‖𝜑_𝑛‖ to 0. It follows that the sequence (𝑥_𝑛𝑘)⊆𝐸^∗∗ has an 𝐸^∗∗-weak convergent subsequence (𝑥_𝑛𝑚)_𝑛𝑚, that is, 𝑥_𝑛𝑚𝑓→Ψ𝑓,∀𝑓∈𝐸^∗,Ψ∈𝐸^∗∗. (2.35) On the one hand, we have 𝐵𝑥_𝑛𝑚²=𝐵^†𝜑_𝑛𝑚−𝐵𝑥_𝑛𝑚𝜑_𝑛𝑥_𝑛𝑚+𝐵𝑥_𝑛𝑚𝐵^†𝜑_𝑛𝑥_𝑛𝑚. (2.36) Then 𝐵𝑥_𝑛𝑚²≤𝐵^†𝜑_𝑛𝑚−𝐵𝑥_𝑛𝑚𝜑_𝑛+𝐵𝑥_𝑛𝑚𝐵^†𝜑_𝑛. (2.37) Thus 𝐵𝑥_𝑛𝑚𝐵𝑥_𝑛𝑚−𝐵^†𝜑_𝑛≤𝐵^†𝜑_𝑛𝑚−𝐵𝑥_𝑛𝑚𝜑_𝑛. (2.38) On the other hand, 𝑥_𝑛𝑚𝐵^†𝜑_𝑛−𝐵^†𝜑_𝑛≤𝑥_𝑛𝑚𝐵^†𝜑_𝑛−𝑥_𝑛𝑚𝐵^†𝜑_𝑛𝑚𝐵𝑥_𝑛𝑚+1𝐵𝑥_𝑛𝑚𝑥_𝑛𝑚𝐵^†𝜑_𝑛𝑚−𝐵^†𝜑_𝑛=𝑥𝐵^†𝜑_𝑛−1𝐵𝑥_𝑛𝑚𝐵^†𝜑_𝑛𝑚+𝐵𝑥_𝑛𝑚−𝐵^†𝜑_𝑛→0as𝑚→∞. (2.39) So lim_𝑚𝑥_𝑛𝑚(𝐵^†𝜑_𝑛)=‖𝐵^†𝜑_𝑛‖ and ‖𝐵^†𝜑_𝑛‖Ψ_𝑛∈𝐷(𝐵^†𝜑_𝑛). Then by smoothness of the space 𝐸^∗ we get ‖𝐵^†𝜑_𝑛‖Ψ_𝑛=Ψ_𝑛,forall𝑛. Next, 𝑥_𝑛𝑚𝐴^†𝜑_𝑛𝑚−𝐵𝑥_𝑛𝑚𝐵^†𝜑_𝑛𝜓_𝑛𝐴^†𝜑_𝑛≤𝑥_𝑛𝑚𝐴^†𝜑_𝑛𝑚−𝑥_𝑛𝑚𝐵𝑥_𝑛𝑚𝐴^†𝜑_𝑛+𝐵𝑥_𝑛𝑚𝑥_𝑛𝑚𝐴^†𝜑_𝑛−1𝐵^†𝜑_𝑛𝜓_𝑛𝐴^†𝜑_𝑛=𝑥_𝑛𝑚𝐴^†𝜑_𝑛𝑚−𝐴^†𝜑_𝑛+𝐵𝑥_𝑛𝑚𝑥_𝑛𝑚𝐴^†𝜑_𝑛−𝜓_𝑛𝐴^†𝜑_𝑛→0as𝑚→∞. (2.40)
Then lim_𝑚𝑥_𝑛𝑚(𝐴^†𝜑_𝑛𝑚)=𝜓_𝑛(𝐴^†𝜑_𝑛) or lim_𝑚𝜑_𝑛𝑚(𝐴𝑥_𝑛𝑚=𝜓(𝐴^†𝜑_𝑛) and therefore lim_𝑛lim_𝑚𝜑_𝑛𝑚𝐴𝑥_𝑛𝑚=lim_𝑛𝜓_𝑛𝐴^†𝜑_𝑛=𝑎 (2.41) which means that 𝑎∈ℳ(𝐴)_𝐵.

Corollary 2.9. Let 𝐸 be a Banach space with smooth dual and 𝐴₁,𝐴₂,𝐵₁,𝐵₂∈𝐵(𝐸).
If ‖𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2‖=‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖ and Ran(𝐴^†₂)⟂_𝐵−𝐽Ran(𝐴^†₁−(‖𝐴₁‖/‖𝐴₂‖)𝐴^†₂) with 𝐴₂ being surjective, and Ran(𝐵₂)⟂_𝐵−𝐽Ran(𝐵₁−(‖𝐵₁‖/‖𝐵₂‖)𝐵₂), then 𝐴₁𝐴₂∈ℳ𝐴_1𝐴2,𝐵₁𝐵₂∈ℳ𝐵_1𝐵2. (2.42) Moreover, if Ran(𝐴^†₁)⟂_𝐵−𝐽Ran(𝐴^†₂−(‖𝐴₂‖/‖𝐴₁‖)𝐴^†₁), 𝐴₂ is surjective, and Ran(𝐵₁)⟂_𝐵−𝐽Ran(𝐵₂−(‖𝐵₂‖/‖𝐵₁‖)𝐵₁), then 𝐴₁𝐴₂∈ℳ𝐴_1𝐴2∩ℳ𝐴_2𝐴1,𝐵₁𝐵₂∈ℳ𝐵_1𝐵2∩ℳ𝐵_2𝐵1. (2.43)

Corollary 2.10. Let 𝐸 be a Banach space with smooth dual and 𝐴₁,𝐴₂,𝐵₁,𝐵₂∈𝐵(𝐸) such that 𝐴₁,𝐴₂ are surjective operators. If Ran(𝐴^†_𝑖)⟂_𝐵−𝐽Ran(𝐴^†_𝑗−(‖𝐴_𝑗‖/‖𝐴_𝑖‖)𝐴^†_𝑖) and Ran(𝐵_𝑖)⟂_𝐵−𝐽Ran(𝐵_𝑗−(‖𝐵_𝑗‖/‖𝐵_𝑖‖)𝐵_𝑖),(𝑖,𝑗=1,2 such that 𝑖≠𝑗) then the following assertions are equivalent:(1)‖𝑀_𝐴1,𝐵1+𝑀_𝐴2,𝐵2‖=‖𝐴₁‖‖𝐵₁‖+‖𝐴₂‖‖𝐵₂‖; (2)‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴₁)_𝐴2∩ℳ(𝐴₂)_𝐴1and‖𝐵₁‖‖𝐵₂‖∈ℳ(𝐵₁)_𝐵2∩ℳ(𝐵₂)_𝐵1.

As a particular case, we obtain the following.

Corollary 2.11. Let 𝐸 be a Banach space with smooth dual and 𝐴,𝐵 are surjective operators in 𝐵(𝐻). If Ran𝐵^†⟂_𝐵−𝐽Ran𝐴^†−‖𝐴‖‖𝐵‖𝐵^†,Ran(𝐴)⟂_𝐵−𝐽‖𝐵‖Ran𝐵−‖𝐴‖𝐴, (2.44) then the following assertions are equivalent:(1)‖𝐴‖‖𝐵‖∈ℳ(𝐴)_𝐵∩ℳ(𝐵)_𝐴; (2)‖𝑀_𝐴,𝐵+𝑀_𝐵,𝐴‖=2‖𝐴‖‖𝐵‖.

3. Hilbert Space Case

Let 𝐸=ℋ be a complex Hilbert space and 𝐴∈𝐵(ℋ). The maximal numerical range of 𝐴 [11] denoted by 𝑊₀(𝐴) is defined by 𝜆∈ℂ∶∃𝑥_𝑛,𝑥_𝑛=1,suchthatlim𝐴𝑥_𝑛,𝑥_𝑛=𝜆andlim𝐴𝑥_𝑛=‖𝐴‖, (3.1) and its normalized maximal range, denoted by 𝑊_𝑁(𝐴), is given by 𝑊_𝑁(𝐴)=𝑊₀𝐴‖𝐴‖if𝐴≠00if𝐴=0. (3.2) The set 𝑊₀(𝐴) is nonempty, closed, convex, and contained in the closure of the numerical range of 𝐴.

In this section we prove that if 𝐸=ℋ, the conditions 𝐴₁𝐴₂∈ℳ𝐴_1𝐴2∩ℳ𝐴_2𝐴1,𝐵₁𝐵₂∈ℳ𝐵_1𝐵2∩ℳ𝐵_2𝐵1 (3.3) would imply that 𝐴^∗₂𝐴₁=𝐴₁𝐴₂,𝐵₂𝐵^∗₁=𝐵₁𝐵₂,𝑊_𝑁𝐴^∗₂𝐴₁∩𝑊_𝑁𝐵₂𝐵^∗₁≠∅. (3.4)

Proposition 3.1. Let ℋ be a complex Hilbert space, 𝐴₁,𝐴₂,𝐵₁,𝐵₂∈𝐵(ℋ).
If ‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴₁)_𝐴2∩ℳ(𝐴₂)_𝐴1and‖𝐵₁‖‖𝐵₂‖∈ℳ(𝐵₁)_𝐵2∩ℳ(𝐵₂)_𝐵1, then ‖𝐴^∗₂𝐴₁‖=‖𝐴₁‖‖𝐴₂‖ and ‖𝐵₂𝐵^∗₁‖=‖𝐵₁‖‖𝐵₂‖ and 𝑊_𝑁(𝐴^∗₂𝐴₁)∩𝑊_𝑁(𝐵₂𝐵^∗₁)≠∅.

Proof. If 𝐴₁=0 or 𝐴₂=0 and 𝐵₁=0 or 𝐵₂=0, the result is obvious.
The proof will be done in four steps, we choose one and the others will be proved similarly. Suppose that 𝐴₁≠0 and 𝐴₂≠0, if ‖𝐴₁‖‖𝐴₂‖∈ℳ(𝐴₁)_𝐴2, then there exists a sequence (𝑥_𝑛)_𝑛∈𝑆_ℋ(𝐴₂) such that 𝐴₁𝐴₂=lim⟨𝐴₁𝑥_𝑛,𝐴₂𝑥_𝑛⟩. (3.5) We have |⟨𝐴^∗₂𝐴₁𝑥_𝑛,𝑥_𝑛⟩|≤‖𝐴^∗₂𝐴₁‖≤‖𝐴₁‖‖𝐴₂‖; this yields lim𝐴^∗₂𝐴₁𝑥_𝑛=𝐴^∗₂𝐴₁=𝐴₁𝐴₂. (3.6) From (3.5) and (3.6) we get 𝐴^∗₂𝐴₁=𝐴₁𝐴₂,1∈𝑊₀𝐴^∗₂𝐴₁𝐴^∗₂𝐴₁. (3.7) Suppose now that 𝐵₁≠0 and 𝐵₂≠0, if ‖𝐵₁‖‖𝐵₂‖∈ℳ(𝐵₁)_𝐵2, then there exists a sequence (𝑦_𝑛)_𝑛∈𝑆_ℋ(𝐵₂) such that 𝐵₁𝐵₂=lim⟨𝐵₁𝑦_𝑛,𝐵₂𝑦_𝑛⟩. (3.8) Since lim_𝑛‖𝐵₁𝑦_𝑛‖=‖𝐵₁‖, then lim_𝑛(𝐵^∗₁𝐵₁𝑦_𝑛−‖𝐵₁‖²𝑦_𝑛)=0.
Suppose that 𝑤_𝑛=𝐵₁𝑦_𝑛/‖𝐵₁‖, then 𝑦_𝑛=𝐵^∗₁𝑤_𝑛/‖𝐵₁‖+𝑧_𝑛 such that lim_𝑛𝑧_𝑛=0.
Hence 𝐵₂𝑦_𝑛,𝐵₁𝑦_𝑛=𝐵₂𝐵^∗₁𝑤_𝑛𝐵₁,𝐵₁𝑤_𝑛=𝐵₂𝐵^∗₁𝑤_𝑛,𝑤_𝑛+𝐵₂𝑧_𝑛,𝐵₁𝑤_𝑛. (3.9) From this, we derive that lim𝐵₂𝐵^∗₁𝑤_𝑛=𝐵₂𝐵^∗₁=𝐵₁𝐵₂. (3.10) From (3.8) and (3.10) we have 𝐵₂𝐵^∗₁=𝐵₁𝐵₂,1∈𝑊₀𝐵₂𝐵^∗₁𝐵₂𝐵^∗₁. (3.11) From (3.7) and (3.11) we get ‖𝐴^∗₂𝐴₁‖=‖𝐴₁‖‖𝐴₂‖ and ‖𝐵₂𝐵^∗₁‖=‖𝐵₁‖‖𝐵₂‖ and 𝑊_𝑁(𝐴^∗₂𝐴₁)∩𝑊_𝑁(𝐵₂𝐵^∗₁)≠∅.

Remark 3.2. We remark that in the case 𝐸=ℋ we obtain an implication given by Boumazgour [12].

Acknowledgments

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).