Structure of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2723999pt" id="M1" height="12.533pt" version="1.1" viewBox="-0.0657574 -12.2606 8.79534 12.533" width="8.79534pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g113-108" d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"/></g></svg>-</span>Quasi-<span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.9939pt" id="M2" height="15.2545pt" version="1.1" viewBox="-0.0657574 -12.2606 41.091 15.2545" width="41.091pt"><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="g113-110" d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"/></g><g transform="matrix(.017,0,0,-0.017,19.477,0)"><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,-0.017,26.265,0)"><path id="g113-111" d="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"/></g><g transform="matrix(.017,0,0,-0.017,34.881,0)"><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></svg>-</span>Isosymmetric Operators

Mahmoud, Sid Ahmed Ould Ahmed; Beiba, El Moctar Ould; Hamidou Jah, Sidi; Sidi, Maawiya Ould

doi:https://doi.org/10.1155/2022/8377463

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Research Article | Open Access

Volume 2022 | Article ID 8377463 | https://doi.org/10.1155/2022/8377463

Structure of -Quasi--Isosymmetric Operators

Sid Ahmed Ould Ahmed Mahmoud,¹El Moctar Ould Beiba,²Sidi Hamidou Jah,³and Maawiya Ould Sidi¹

Academic Editor: Ali Jaballah

Received24 Apr 2022

Accepted03 Aug 2022

Published26 Sept 2022

Abstract

The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. In this study, we introduce and study a new class of operators called -quasi--isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the -isosymmetric operators. We give a characterization for any operator to be -quasi--isosymmetric operator. Using this characterization, we prove that any power of an -quasi--isosymmetric operator is also an -quasi--isosymmetric operator. Furthermore, we study the perturbation of an -quasi--isosymmetric operator with a nilpotent operator. The product and tensor products of two -quasi--isosymmetries are investigated.

1. Introduction

Let be the -algebra of bounded linear operators on a complex Hilbert space and let

For , we will write , , and the range, the kernel (or null space), and the adjoint of , respectively. Also, , and denote the point spectrum, the approximate spectrum, the spectrum, and the surjective spectrum .

The hereditary functional calculus defines

It is easy to check that for and , we have

Recall that an operator is called a hereditary root or simply root of if . For more details on the hereditary functional calculus, we refer the reader to [4, 5].

In recent years, the concepts of -isometric operators and the related classes of operators, namely, -quasi--isometries, -isometries, and -quasi--isometries have received substantial attention. Several authors have been introduced, and these classes of operators are studied intensively in the papers [12–18], [20–23, 28, 32]. It has been proved that some products of -isometries are again -isometry [8, 11], the powers of an -isometry are - isometries, and the perturbation of -isometries by nilpotent operators has been studied in [6, 9, 10]. The dynamics of -isometries has been explored in [7]. Almost all of these properties have been extended to -quasi--isometric operators, -isometries, and -quasi--isometries. The reader can refer to the papers [12–14, 17, 18, 20–23, 32] for more details.

Let and , , and be positive integers.(1) is called -isometry [1–3] if it is a root of , that is,(2) is called -symmetry [15, 27] if it is a root of , that is,(3) is called -quasi--isometry [19, 20, 31] if it is a root of , that is,(4) is called -quasi--symmetric [33] if it is a root of , that is,(5) is called -isosymmetric [29, 30] if it is a root ofthat is,

For , set , , and

For , we have

It is easy to see that

It is well known that a common way to prepare a scientific study is to introduce some new mathematical objects and then state several results related to them. The investigation of new operators belonging to some specific classes has been quite fashionable since the beginning of the century, and sometimes it is indeed relevant. The motivation of this study is to introduce and study the concept of -Quasi--isosymmetric operators on Hilbert spaces. This new class of operators emerges as a generalization of the -isosymmetric operators. It is proved that there is an operator which is a -quasi--isosymmetric operator, but not -isosymmetric, and thus, the proposed new class is larger than the class of -isosymmetric operators. In section two, we give a matrix characterization of -quasi--isosymmetric operators in terms of -isosymmetric operators. We give some results related to this class by using this matrix representation. In section three, we investigate some spectral properties of -quasi--isosymmetric operators; in particular, we explore conditions for -quasi--isosymmetric operators to be -quasi--isometric operators or -quasi--symmetric operators. Finally, in section forth, we study the sum of an -quasi--isosymmetric operator with a nilpotent operator. We also study the product and tensor product of two -quasi--isosymmetric operators.

2. Structure of -Quasi--Isosymmetric Operators

In the present section, we give the definition and basic properties of -quasi--isosymmetric operators. The obtained results improve and generalize some works on -isometries, -quasi--isometric, -symmetries, and -quasi--symmetric operators.

Definition 1. An operator is said to be -quasi--isosymmetric operator if is a root of the polynomialfor some positive integers , , and , or equivalently,

Example 1. (i)Every -isometric operator is an -quasi--isosymmetric, and every -symmetric operator is an -quasi--isosymmetric operator(ii)Every -isosymmetric operator is an -quasi--isosymmetric operator(iii)Every -quasi--isometric operator is an -quasi--isosymmetric operator(iv)Every -quasi--symmetric operator is an -quasi--isosymmetric operator

Remark 1. Since , , and , the polynomial divides ; it follows that if is a -quasi--isosymmetric operator, then it is a -quasi--isosymmetric operator.

Remark 2. (1)If , 1-quasi--isosymmetric operator is a quasi-isosymmetric, i.e., an operator is quasi-isosymmetric if and only if(2)An operator is quasi--isosymmetric if and only if(3)An operator is quasi--isosymmetric if and only if

Remark 3. In the following example, we show that there is an operator which is -quasi--isosymmetric, but not -isosymmetric for some positive integers , and therefore, the proposed new class is large than the class of -isosymmetric operators.

Example 2. Let . The direct calculation shows thatThus, is a quasi-isosymmetric but not isosymmetric operator.

Example 3. (1)Let . A simple calculation shows that Thus, is a -quasi--isosymmetric operator; however, is neither -isometry nor -symmetry.(2)Let . A simple calculation shows that Thus, is a quasi-isosymmetric operator; however, is neither quasi-isometry nor quasi-symmetry.

Remark 4. The following inclusions hold:

Proposition 1. Let , then the following statements are equivalent:(1) is -quasi--isosymmetric operator(2)

Proof.

Corollary 1. Let with and are two nonnegative integers such that, then is -quasi--isosymmetric operator if and only if is -quasi--isosymmetric operator.

Proof. Straightforward from Proposition 1.

Theorem 1. Let be -quasi- -isosymmetric operator. If for some , then is -quasi- -isosymmetric.

Proof. From the assumptions and , it follows that . Thus, . Applying Corollary 1, we get the desired conclusion.

Proposition 2. Let be a closed subspace of which reduces . If is -quasi-(m,n)-isosymmetric, then is -quasi- -isosymmetric.

Proof. Let be the restriction of to . On the one hand, we haveOn the other hand, we have , where is the closer of in . Thus, since . Therefore, by statement (2) of Proposition 1, is -quasi--isosymmetric on .

The following theorem characterizes the members of -quasi--isosymmetric operators.

Theorem 2. Let such that . Then, the following properties are equivalent:(1)is-quasi--isosymmetric operator(2)on, whereis an-isosymmetric operator and

Proof. . By taking into account the matrix representation related to the decomposition as , we getwhere is the orthogonal projection onto .
From the condition that is -quasi--isosymmetric operator, we haveorFrom this, we deduce thatorTherefore, is -an isosymmetric operator.
On the other hand, let . The direct calculation shows thatSo that, .
Assume that onto , withand .
Direct calculation shows that and therefore, . Moreover,where .This means thatObviously, on , and consequently,on .
Therefore, is a -quasi--isosymmetric operator.

Corollary 2. If is an -quasi- -isosymmetric such that is dense, then is an -isosymmetric.

Proof. This is a direct consequence of Theorem 2.

Corollary 3. If is an invertible -quasi- -isosymmetric operator, then is a -quasi- -isosymmetric operator.

Proof. Under the assumption that is an invertible -quasi--isosymmetric operator, it follows that is an -isosymmetric operator, and so is by Theorem 2.4 in [24]. Therefore, is a -quasi--isosymmetric operator.

Corollary 4. Let be a -quasi- -isosymmetric operator such that . If the restriction is invertible, then is similar to a direct sum of an -isosymmetric operator and a nilpotent operator with an index of nilpotence less than or equal .

Proof. Consider the matrix decomposition of asThen, is an -isosymmetric operator by Theorem 2. Since is invertible, we have , which implies . By Rosenblum’s corollary [26], it follows that there exists for which . Therefore, we have

It was proved that power of -isometric (resp -symmetric) operator is again -isometric (resp -symmetric) operator. The following corollary shows that the same property holds for -quasi--isosymmetric operators.

Corollary 5. If is a -quasi- -isosymmetric operator, then is also a -quasi- -isosymmetric operator for any positive integer .

Proof. Let , then is an -isosymmetric and so is for any by Theorem 2.4 in [24]. If , we can use the decomposition of on , where is an -isosymmetric operator and so is . On the other hand, observing thatit follows that is -quasi--isosymmetric operator by applying Theorem 2.

Theorem 3. Let . If is a surjective -isosymmetric operator and , then is similar to an -quasi- -isosymmetric operator.

Proof. Under the assumptions on and , we have . From the axiom in Theorem 3.5.1 in [16], it follows that there exists some operator such that . Hence,From this, we deduce that is similar to .
By using the facts that is a -isosymmetric and , we can obtainConsequently, is similar to a -quasi--isosymmetric operator.

3. Spectral Properties

In this section, we investigate some spectral properties of -quasi--isosymmetric operators.

Let and . Let and be such that and , we get by a little calculation that

It follows from (40) that if is a root of and is a spectrum point of , then .

We have the following theorem.

Theorem 4. Let and . If and be two sequences of unit vectors such that and as , then

Proof. For , , and , we haveThus, we getLet . SetApplying (43) to and , respectively, we obtainThus, and as .
On the other hand, it is easy to verifySince the right side of the previous equality tends to , we obtainTaking linear combinations, we get

Corollary 6. Let and . Let and and such that , , and. Ifis a root of, then one of the following statements holds:(1)as(2)

Proof. Since is a root of , we have . This yields, from (41), the desired conclusion.

Corollary 7. Let and . Let be nonzero vectors and be such that and . If is a root of , then one of the following two statements holds:(1)(2)

Proof. Take and in Corollary 6.

As a consequence of Theorem 4, we have the following two results due to Stankus [30].

Corollary 8. (Proposition 21 [30]). Letand. Then,

Corollary 9. (Corollary 22 in [30]). Letand. Ifis a root of, then

Let and be two Hilbert spaces, and and

Consider the bounded linear transformationdefined by

We also define the maps by

and commute. Indeed, for , we have

Lemma 1. Let , , , , , , and be as above. Then,

Proof. It is easy to check that for , we haveThus, for , we haveThis yieldsTherefore, .

We need the following lemmas.

Lemma 2. (Lemma 0.11 in [25]). Ifand are commuting operators on the Banach space, thenfor every polynomial.

Lemma 3. (Lemma 27 in [30]). Let, , , then.

Remark 5. From (57), Lemma 3 is equivalent to

Remark 6. It is easy to verify that if is an isomorphism, then and are also isomorphisms and we have and . Since, for , , and , we get and .

Proposition 3. Let , , and, then

Proof. From (57), we have . Thus, by Lemma 2, we obtain . This yields, from Remark 6,

Corollary 10. Let and , then

Proof. Apply Proposition 3 by taking .

Proposition 4. Let and . If , then either or .

Proof. Suppose that . This means that is invertible. Thus, from (60), we obtain .

From the following, we give a sufficient condition for a -quasi--isosymmetric operator to be -quasi -isometric operator or -quasi--symmetric operator.

Theorem 5. Let be a -quasi- -isosymmetric operator. The following statements hold:(1)If, thenis a-quasi--isometry(2)If, thenis a-quasi--symmetric operator

Proof. (1)Set , where and . Since , we have . Thus, from (63), we get . This yields, by Proposition 4, . Therefore, is a -quasi--isometry.(2)Set , where and . Since , similarly, as in (1), we obtain that . Applying again Proposition 4, we obtain . Thus, is a -quasi--symmetric operator.

Theorem 6. Let be -quasi- -isosymmetric operator, then has the single-valued extension property (SEVP).

Proof. If , then is an -isosymmetric operator, and therefore has SVEP by Theorem 2.20 in [24]. If , we use the matrix decomposition of asFrom Theorem 2, is a -isosymmetric operator and is a nilpotent operator. Hence, and have SVEP; then, by simple calculations, we see that has SVEP as required.

Rachid [24] showed that if is -isosymmetric operator, then , where is the unit circle. Now, we extend this result to -quasi--isosymmetric operators.

Theorem 7. Let be -quasi- -isosymmetric operator, then .

Proof. Let , then there exists a sequence , with such that as . We have as for all positive integers. From the condition that is an -quasi--isosymmetric operator, one hasConsequently, or or . This completes the proof.

Proposition 5. Let be -quasi- -isosymmetric operator. If with is an approximate eigenvalue of , then is an approximate eigenvalue of .

Proof. Assume that with , then there exists : such that as . By taking into account that is a -quasi--isosymmetric operator, it follows thatBy observing thatand for all positive integers , we get from the above relations thatIf is bounded from below, then so is and therefore there exists a positive constant such thatFrom this, we deduce thatConsequently, . So, we haveThus, or , which are a contradiction. Hence, is not bounded from below. In view of Theorem 7, we have and so , which implies that is not bounded from below. This proves the statement of the proposition.

4. Products and Perturbation of -Quasi-(N, m)-Isosymmetric Operators

In this section, we study the perturbation of an -quasi--isosymmetric operator by a nilpotent operator and we study the product and tensor product of two -quasi--isosymmetric operators.

Lemma 4. For , the following identity holds:whereandare some constants.

Proof. See the proof of statement (ii) in Theorem 3.3 in [14].

The following theorem shows that the power of a -quasi--isosymmetric operator is again a -quasi--isosymmetric which is similar to that of Corollary 5 but with another proof.

Theorem 8. If is -quasi- -isosymmetric, then is -quasi - -isosymmetric for any .

Proof. Two different proofs of this statement will be given.
First Proof. We need to prove that .
In fact, from Lemma 4, we obtain thatSince is -quasi--isosymmetric, we have and thereforeThis means that is -quasi--isosymmetric for any positive integer .
Second Proof. Let . It is easy to verify that . A little calculation shows that there exists a polynomial such that . Thus, by (60), we getwhich yields since . Therefore, is -quasi--isosymmetric operator.

Lemma 5. Let such that . Then, the following identity holds:

Proof. In view of the following identity (see [14]),it follows by taking , and with and that

Theorem 9. Let and such that . If is an -quasi- -isosymmetric and is an -quasi- -isometric and -symmetric, then is an -quasi- -isosymmetric.

Proof. We prove that .
In fact, by taking into account Lemma 5, we getWe have the following observations:(i)If or , then .(ii)If and , then and . So that Therefore, Hence, is an -quasi--isosymmetric.

Corollary 11. Let and such that . If is an -quasi- -isosymmetric and is an -quasi- -isometric and -symmetric, then is an -quasi- -isosymmetric for any positive integers and .

Proof. In view of Theorem 8, we have as -quasi--isosymmetric for any positive integer . Similarly, from Theorem 12 in [18] and corollary 3.1 in [27], we have is -quasi--isometric and -symmetric for any positive integer .
Applying Theorem 9, we get is an -quasi--isosymmetric for any positive integers and .

Corollary 12. Let and . If is an -quasi- -isosymmetric and is an -quasi- -isometric and -symmetric, then is -quasi- -isosymmetric.

Proof. It is well known that and moreoverOn the other hand, is an -quasi--isosymmetric if and only if is an -quasi--isosymmetric and is an -quasi--isometric and -symmetric if and only if is an -quasi--isometric and -symmetric. From Theorem 9, it follows that is a -quasi--isosymmetric.

Lemma 6. Let such that , then the following identity holds:

Proof. By the equation (see [14]),If , , , and with , we get

Theorem 10. Let be doubly commuting. If is an -quasi- -isosymmetric and is a nilpotent operator of order , then is a -quasi- -isosymmetric operator.

Proof. We need to show that .
Note that by Lemma 6, we haveHowever,Now, observe that if or or , then or or and henceHowever, if , , and , we obtainFrom the fact that is a -quasi---isosymmetric, we getConsequently, we obtain .
Therefore, is -quasi--isosymmetric operator.

Corollary 13. Let such that . If is an -quasi- -isosymmetric, then the operator is -quasi- -isosymmetric.

Proof. Consider and . Clearly, is -quasi--isosymmetric and (i.e., is 2-nilpotent). On the other hand, sinceit follows that . In view of Theorem 10, we deduce that is -quasi--isosymmetric.

Corollary 14. Let be -quasi- -isosymmetric and be -nilpotent. Then, is-quasi--isosymmetric.

Proof. Observe that is -quasi--isosymmetric and is -nilpotent. Moreover,Applying Theorem 10, we deduce that is -quasi--isosymmetric.

Data Availability

Data sharing is not applicable to this study as 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-2021-03-0334.

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