Inequalities Involving <i>A</i>-Numerical Radius and Operator <i>A</i>-Norm for a Class of Operators Related to <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-4.5269pt" id="M1" height="16.7875pt" version="1.1" viewBox="-0.0657574 -12.2606 68.535 16.7875" width="68.535pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,5.937,0)"><path id="g113-223" d="M545 106L524 126C493 85 467 65 455 65C438 65 427 113 405 238C448 295 498 362 543 439L533 448L478 435C453 386 423 331 398 295H395C370 404 347 448 282 448C169 448 23 309 23 153C23 54 65 -12 128 -12C203 -12 283 70 339 155H341C360 29 380 -12 411 -12C444 -12 491 11 545 106ZM333 204C265 95 210 54 169 54C137 54 113 96 113 171C113 302 191 405 252 405C301 405 318 306 333 204Z"/></g><g transform="matrix(.017,0,0,-0.017,15.686,0)"><path id="g113-45" d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"/></g><g transform="matrix(.017,0,0,-0.017,22.474,0)"><path id="g113-224" d="M558 587C558 666 497 712 432 712C379 712 330 691 284 650C212 586 178 508 148 348L71 -65C49 -185 35 -229 23 -235L27 -261C49 -259 101 -251 124 -224C131 -216 138 -178 159 -24C171 66 197 200 227 356C264 550 295 668 393 668C443 668 479 632 479 575C479 516 446 458 383 418C361 404 344 398 318 397L296 350C395 338 460 281 460 192C460 110 411 40 300 40C258 40 215 55 192 69L181 51C191 18 222 -16 266 -16C308 -16 351 1 397 26C471 67 545 142 545 221C545 315 486 365 401 395C469 437 558 498 558 587Z"/></g><g transform="matrix(.017,0,0,-0.017,32.446,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g><g transform="matrix(.017,0,0,-0.017,42.219,0)"><path id="g117-33" d="M535 230V280H52V230H535Z"/></g><g transform="matrix(.017,0,0,-0.017,56.126,0)"><path id="g113-66" d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"/></g></svg>-</span>Normal Operators

Ould Ahmed Mahmoud, Sid Ahmed; Jah, Sidi Hamidou

doi:https://doi.org/10.1155/2022/1506330

Journal of Mathematics

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Abstract Introduction and Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 1506330 | https://doi.org/10.1155/2022/1506330

Inequalities Involving A-Numerical Radius and Operator A-Norm for a Class of Operators Related to -Normal Operators

Sid Ahmed Ould Ahmed Mahmoud¹and Sidi Hamidou Jah²

Academic Editor: Xiaolong Qin

Received23 Mar 2022

Revised12 Apr 2022

Accepted12 Apr 2022

Published20 May 2022

Abstract

In this article, we introduce and study a new class of operators, larger than -normal operators and different than -normal operators, named -quasi--normal operators. Considering the semi-inner product induced by a positive operator , the -quasi--normal operators turn into a generalization (for this new structure) of classical -quasi--normal operators. Several results concerning properties of this kind of operators are presented in the paper. Several inequalities for the -numerical radius and -operator norm for members of this class are established.

1. Introduction and Preliminaries

Let denote the -algebra of all bounded linear operators on an infinite dimension complex Hilbert space . denotes the cone of positive operators of . It is well known that .

For , we denote by the adjoint operator of , range by , and its null space by . If , then stands for its closure in the norm topology of . We denote by the orthogonal projection onto the closed linear subspace .

If is a positive operator on , the bilinear functional defines a positive semidefinite sesquilinear form given by .

Notice that

An operator is said to be -bounded if there exists such that

The set of all -bounded operators on will be denoted by .

For any , we take from [1],

In [1], the authors have introduced the concept of -self-adjoint for an operator . An is called an -adjoint operator of if and satisfy

We observe that

is an -self-adjoint operator if is an -adjoint of itself, that is, .

It was observed by Douglas Theorem [2] that an operator admits an -adjoint if and only if . In the following, the set of all operators which admit an -adjoint is denoted by . Throughout this paper, we consider that with . In [1], it was observed that if , the solution of the equation is a distinguished -adjoint operator of , which is denoted by . However, is given by where is the Moore–Penrose inverse of . The -adjoint operator satisfies the following axioms:

The authors in [1, 3, 4] have studied the following useful properties of for .

For , then the following axioms hold:(1).(2).(3)If , then .

In recent years, the theory of normal operators has known many extensions due to the work carried out by several authors ([5–8]). The study of operators in semi-Hilbertian spaces is of significant interest and is currently being done by a number of mathematicians around the world. Some developments toward this subject have been done in [1, 3, 4, 9–18].

Any operator is -positive if , that is,

For .

The classes isometry, unitary, normal, and -normal operators in Hilbert space have been generalized to the concepts of so called -isometry, -unary, and -normal and -normal operators by many authors. An operator is called(1)-isometry if ( [19]),(2)-unitary if ([1]),(3)-normal if ([12, 17]),(4)-normal, if , for ([11]),(5)-normal if , for some positive integers and [9].

Of course, these extensions are not trivial since many difficulties arise.

Recently, the authors in [20] have introduced the concept of -quasi totally--normal operators as a generalization of totally--normal operators in Hilbert space. An element is called -quasi totally--normal if satisfies

In this paper, our goal is to introduce a new class of operators, larger than -normal operators, named -quasi--normal operators in semi-Hilbertian spaces. First of all, we introduce notations and consider a few preliminary results which are useful to prove the main result of the paper. We give sufficient conditions on two -quasi -normal operators defined on a semi-Hilbert space, which make their product and their tensor product -quasi -normal. The inspiration for our investigation comes from [11, 20]. Moreover, we established various inequalities between the -numerical radius and -operator norm of -quasi--normal operators.

2. -Quasi--Normal Operators

In this section, we define a class of operators in semi-Hilbertian spaces, i.e., spaces generated by positive semidefinite sesquilinear forms. This kind of spaces appears in many problems concerning linear and bounded operators on Hilbert spaces and is intensively studied in the present. The concept of operator theory is semi-Hilbertian spaces have attracted attention. Recently, many extensions of some concepts of Hilbert space operators to semi-Hilbertian space operators have attracted much attention of various authors in several papers [1, 3, 4, 9–18]. In this framework, we are interested to introducing a new concept of quasinormality in semi-Hilbertian spaces known as -quasi--normal operators. We investigate various structural properties of this class of operators and study some relations about it.

Definition 1. An operator is called -quasi--normal for iffor some positive integer , or equivalently, if

Remark 1. The reader will easily see from Definition 1 that(i)An -normal is an -quasi--normal operator.(ii)Every -quasi--normal operator is -quasi--normal operator.

Remark 2. Notice that the converse of the statement (i) in Remark 1 is not true in general, as shown in the following example:

Example 1. Let us consider and and . Direct calculations show that .This shows that is a quasi--normal, but it is not a -normal.

Remark 3. From Example 1, it is obvious that the class of -quasi--normal operators contained the class of -normal operators as a proper subset.

Theorem 1. Let be an -quasi--normal operator. If has dense range, then is -normal.

Proof. Since has a dense range, it follows that . Let , then there exists a sequence in such that as .
From the fact that is -quasi--normal operator, we havefor all .
In particular, we haveConsequently,Therefore, is -normal operator.

Corollary 1. Let be an -quasi--normal operator. If and if has no nontrivial -invariant closed subspace, then is -normal.

Proof. From the fact that has no nontrivial invariant closed subspace, it follows that has no nontrivial hyperinvariant subspace. But, and are hyperinvariant subspaces, and , hence and . Therefore, is -normal operator by [11], Definition 6.

The following theorem gives a characterization of -quasi--normal operators.

Theorem 2. Let , then is -quasi--normal operator if and only iffor all .

Proof. By using elementary properties of real quadratic forms, we inferSimilarly,Therefore, is the -quasi--normal operator.

In the following theorem, we show the relationship between -quasi--normal operator and it is -adjoint.

Theorem 3. Let such that and let . The following statements hold.(1)If is an -quasi--normal such that , then is -quasi--normal.(2)If is -quasi--normal such that and , then is an -quasi--normal.

Proof. (1) If is -quasi--normal, it follows thatTaking into account the assumption , we inferCombining these inequalities,So, is -quasi--normal as required.
(2) Since is -quasi--normal operator, we have for all ,After rearranging it further, we getFrom the assumption that , we observe that is a reducing subspace of and it follows that and . Thus, we haveFrom the assumption that , we inferThis givesthen is always -quasi--normal.

Corollary 2. Under the same conditions of Theorem 3, if , then is -quasi--normal if and only if is -quasi--normal.

Theorem 4. Let be -quasi--normal such that , then

Proof. By using the -quasi--normality of and given condition, we have the following:
is -quasi--normal if and only ifTherefore,Combining these inequalities, we getand so,This means thatHence, the result holds.

Theorem 5. Let be two -quasi--normal, then is also -quasi--normal, but tensor product is -quasi-- normal.

Proof. Since and are -quasi--normal, we haveMoreover,Therefore,Thus, is -quasi--normal operator.
To prove that is -quasi--normal operator, we observe thatWe getSimilarly,Therefore,Combining (36) and (38), we get

Theorem 6. The setand is Arcwise connected for .

Proof. Let and . We need to prove that is -quasi--normal. Now, for ,Otherwise,Hence,This means that the class of -quasi--normal operator is Arcwise connected.

Proposition 1. Let such that and . If is -quasi- -normal for and is an -isometry, then is -quasi--normal operator.

Proof. In view of Definition 1 and the fact that is an -isometry ([1], Proposition 3.6), we haveMoreover, from the fact that is a reducing subspace for both and , it follows thatDirect calculations show thatThis impliesSimilarly, we haveThe conclusion holds.

Proposition 2. Let be commuting and such that is -quasi--normal operator. The following statements hold.(1)If is -self-adjoint and , then is -quasi--normal.(2)If is -unitary, and is invariant subspace for , then is -quasi--normal.

Proof. (1) Since is -quasi--normal, is -normal and , it follows that for ,On the other hand,Consequently, is the -quasi--normal operator.
(2) Since is invariant subspace for , we observe that and . In view of the fact that is -unitary, we have .
Now, direct calculations giveSince is -quasi--normal, it follows immediately from [11], Lemma 1 thatFrom the quantity in (1), we getSimilarly by (2), we haveThis means thatTherefore, is -quasi--normal operator.

Theorem 7. Let be commuting and . If is an -quasi--normal and is an -quasi--normal for , then is an -quasi--normal.

Proof. By using the -quasi--normality of and -quasi--normality of and given condition, we haveOtherwise,

Hence, the desired result is obtained.

3. Inequalities Involving -Numerical Radius and Operator -Norm

In this section, we are interested to study some inequalities concerning the -numerical radius and -norm of -quasi--normal operators. It is followed by several inequalities, refer to [21–29]. The authors in [20] have given various inequalities between the operator norm and the numerical radius of -quasi--normal operators in Hilbert spaces. Motivated by this work, we will extend some of these inequalities to -operator norm and -numerical radius for -quasi--normal operators in semi-Hilbertian spaces by using some known results for vectors in inner product spaces.

Definition 2 (see [17]). The -numerical radius of an operator is defined by

Theorem 8. If is -quasi- normal operator, then the following inequality holds:

Proof. From [28], we have inspired the following inequalities:for and with .
By replacing by and by for , we getfor and with and .
This meansfor and with and .
If we take , we obtain the following inequality:for any with and .
Hence,Applying over , of both sides of the above inequality, we inferThis proves the first inequality.
Analogously, by considering similar techniques, we get the second inequality for .

Theorem 9. If is -quasi-- normal operator and if , then

Proof. We give our proof depending on the following inequality inspired from [27],for .
By choosing and , we inferThis givesTaking the of both sides of the last inequality for , we infer the desired result.

Theorem 10. If is -quasi--normal operator and if , then

Proof. We give our proof depending on the following inequality inspired from [24]:provided and . By choosing , we inferThis givesTaking the of both sides of the last inequality for , the desired result will be obtained.

Theorem 11. If is -quasi--normal operator, then

Proof. In the following inequality [21],where and .
Choosing , , and with , it followsHence,Taking the of both sides of the last inequality for , the required inequality will be obtained.

Theorem 12. If is -quasi--normal operator and if , then

Proof. We use the following inequality [26]:for any and .
Now, if we choose , we getThat meansHence, since is -quasi- normal operator, we haveTaking the over all with in the above inequality, we get

Theorem 13. Let be an -quasi--normal operator and . Then,

Proof. Using the following elementary inequality,for and .
For any with , take , and in the above inequality to getThen,whenceSince is -quasi--normal, we haveHence, by Theorem 12, we obtainBut,Therefore,Taking the of both sides of the last inequality for , we infer the desired result.

Theorem 14. Let be an -quasi--normal operator. If with and , then

Proof. We use the following inequality inspired from [26],for any and .
Put and for to getSince is -quasi--normal operator, it follows thatBy observing that if , thenand thereforeHowever, if , thenand thereforeFrom either case, we getHence,Taking the of both sides over with , we will deduce the desired result.

Theorem 15. Let be an -quasi--normal operator and . Ifthen the following inequality holds:

Proof. We use the following inequality [22]:provided .
Setting and for , we getHence,Taking the on both sides of the last inequality on , , we get the desired result.

Theorem 16. Let be an -quasi--normal operator, .
If for , then the following inequality holds:

Proof. Using the following reverse of the Schwarz inequality inspired from [23],provided .Setting and for , we getFrom this, we obtainTaking the of both sides of the last inequality for , we infer the required result.

4. Conclusion

The study of classes of operators and related topics is one of the hottest areas in operator theory in Hilbert and semi-Hilbert spaces. In the work, we introduce a new class of operators known as -quasi--normal operators, which is a generalization of -normal and -quasi--normal operators. Several properties are proved by exploiting the special kind of structure associated with such operators. In the course of our investigation, we find some properties of -normal and -quasi--normal operators which are retained by -quasi--normal operators. Some inequalities between the -operator norm and -numerical radius of -quasi--normal operators are proved.

Data Availability

No data sets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was funded by the Deanship of Scientific Research at Jouf University under grant no. DSR-2021-03-0336.

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Copyright © 2022 Sid Ahmed Ould Ahmed Mahmoud and Sidi Hamidou Jah. 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.

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