New class of operators where the distance between the identity operator and the generalized Jordan ∗-derivation range is maximal


               <jats:p>A new class of operators, larger than <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2021-0032_eq_001.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mo>∗</m:mo>
                        </m:math>
                        <jats:tex-math>\ast </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-finite operators, named generalized <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2021-0032_eq_002.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:mo>∗</m:mo>
                        </m:math>
                        <jats:tex-math>\ast </jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula>-finite operators and noted by <jats:inline-formula>
                     <jats:alternatives>
                        <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2021-0032_eq_003.png" />
                        <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
                           <m:msup>
                              <m:mrow>
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                              </m:mrow>
                           </m:msup>
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                              <m:mo>)</m:mo>
                           </m:mrow>
                        </m:math>
                        <jats:tex-math>{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})</jats:tex-math>
                     </jats:alternatives>
                  </jats:inline-formula> is introduced, where: <jats:disp-formula id="j_dema-2021-0032_eq_001">
                     <jats:alternatives>
                        <jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_dema-2021-0032_eq_004.png" />
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                           <m:mo>=</m:mo>
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                                       <m:mi>T</m:mi>
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                        <jats:tex-math>{{\mathcal{G {\mathcal F} }}}^{\ast }\left({\mathcal{ {\mathcal H} }})=\{(A,B)\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\times {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }}):\parallel TA-B{T}^{\ast }-\lambda I\parallel \ge | \lambda | ,\hspace{0.33em}\forall \lambda \in {\mathbb{C}},\hspace{0.33em}\forall T\in {\mathcal{ {\mathcal B} }}\left({\mathcal{ {\mathcal H} }})\}.</jats:tex-math>
                     </jats:alternatives>
                  </jats:disp-formula> Basic properties are given. Some examples are also presented.</jats:p>


Introduction
Let be a real or complex Banach * -algebra. Recently, in 1990, Šemrl [1] introduced a derivation called Jordan * -derivation. An additive mapping J : → is said to be Jordan * -derivation if: J a aJ a J a a , 2 ( ) ( ) ( ) = + * for all a ∈ , where a * stands for the adjoint of a. A Jordan * -derivation J on ( ) , where ( ) denotes the Banach algebra of all bounded linear operators acting on a complex and infinite dimensional separable Hilbert space , is called inner if there exists an operator A that satisfies: Šemrl [1] showed that every Jordan * -derivation on ( ) is inner. The motivation for the study of Jordan * -derivations is the well-known mapping called Derivation map. The properties of derivations on ( ) , their spectra, norms and ranges have been studied extensively by Williams [2], Stampfli [3] and others. In a similar way, some results were obtained for Jordan * -derivations, for instance, Molnár [4] proved that, just as derivations, the range of Jordan * -derivations cannot be dense in ( ) in the operator norm topology.
Another purpose of this study is the problem of representability of quadratic forms by sesquilinear ones. Šemrl showed that the structure of Jordan * -derivations arises as a "measure" of this representation problem.
This kind of mapping was studied by many authors like Molnár, Brešar, Battyanyi, Zalar and others. Some of them studied the structure of these mappings. They showed that a Jordan * -derivation defined on certain algebras is inner, see for example [5][6][7][8]. Others were interested in studying the range of Jordan * -derivations [4,9].
In [1] Šemrl treated a question in regard to generalized Jordan * -derivation as the concept of Jordan * -derivation pairs, which was introduced by Zalar [10]. It is shown that on a complex * -algebra Jordan * -derivation pairs are of the form: and its range is defined by: R TA BT T ; ) and the identity operator I is maximal, is called finite operator class and noted by ( ) . In other words, A ( ) ∈ is a finite operator if: AT TA I T 1; .
Based on the study of finite operators and inner Jordan * -derivation, Hamada [18] introduced a new class called the class of * -finite operators defined by: In [19], the author presented some properties of * -finite operators and proved that a paranormal operator under certain scalar perturbation is * -finite operator.
Depending on the researches of Williams [11] and Hamada [19] on finite and * -finite operators, we will introduce in this paper, a new class of operators named generalized * -finite operators denoted by ( ) * .

It is the class of operators A B
, where the distance between R A B , and the identity operator I is maximal, i.e.,

A B
TA BT λI λ λ T , : The aim of this paper is, first, to investigate ( ) * and give some basic algebraic properties of this class. The last part of the paper focuses on presenting some pairs of operators A B ,

the spectral radius of A) and log-hyponormal if: A is invertible and satisfies
It is known that: but the converse is not true [20].
is a class operator if: Class is a subclass of paranormal operators (see [21]).
The numerical range of an operator A is defined by: These are equivalent conditions on an operator A.

Main results
) is said to be a pair of generalized * -finite operators if: We note by ( ) * the class of generalized * -finite operators, i.e.: . These are equivalent conditions on A B , .
Proof. In [11] the author has shown that for any operator A, we have: = , which is a contradiction.
Remark 3.4. The condition (ii) above means that , for all λ ∈ .
In the following, we give some basic properties of generalized * -finite operator class.
Proof. Clearly, the pair of null operators is a generalized pair of * -finite operators. Let α ∈ * , given ε 0 > , then we have: . For ε 0 > , we have: and x ∈ . Hence, and all λ ∈ . By letting n → +∞, we obtain TA BT λI In what follows, we will present some pairs of generalized * -finite operators. Proof. For a finite rank operator T ( ) ∈ , TA BT − * is also of finite rank. Since 0 belongs to the spectrum of each finite rank operator, then . If there exist a normed sequence x n n 1 ( ) ⊂ ≥ and some scalar λ verifying: Proof. We have for all T ( ) ∈ : , then for all λ σ A a ( ) ∈ and for all C ( ) ∈ , . Then there exists a normed sequence x n n 1 ( ) ≥ in verifying: , then: The class of generalized * -finite operators is a generalization of * -finite operators class, and this work provides basic tools for further research on this subject.
First, a necessary and sufficient condition for belonging to ( ) * is given and some algebraic properties of ( ) * are presented and proved that the class of generalized * -finite operators is closed for uniform topology.
Second, we proved that ( ) * is invariant under unitary equivalence. Finally, we presented some pairs of operators A B , ( ) ( ) ∈ * .