Abstract
We introduce a new class of operators, which we will call the class of -quasi--symmetric operators that includes -symmetric operators and -quasi -symmetric operators. Some basic structural properties of this class of operators are established based on the operator matrix representation associated with such operators.
1. Introduction
Throughout this paper, stands for a complex Hilbert space of infinite-dimension with inner product . By , we denote the Banach algebra of all bounded linear operators on . For every , we denote by and the null space and the range of , respectively. Also, , and denote the point spectrum, the approximate spectrum, the spectrum, and the surjective spectrum of .
The authors in[1] introduced the class of Helton operators as followsan operator is said to be in the Helton class of if
We refer the interested reader to [1–6] for complete details.
The class of -symmetric operators or -self-adjoint operators on a Hilbert space has attracted much attention and has been the subject of intensive studies by several authors. -symmetric operators were introduced in [3, 4, 7] as followsan Hilbert space operator is said to be an -symmetric if satisfies the following identity:for some positive integer , where is the adjoint operator of . For , equation (2) is reduced to (A is symmetric or self-adjoint). If , equation (2) is reduced to ( is 2-symmetric). It has been proven that a power of -symmetric transformation is again -symmetric, and the product of two -symmetric transformations is also -symmetric under suitable conditions (see [8]).
Another extension of the relation in (2), we recall that if , for which is positive, is called an -symmetric ([9]) when
The authors Chō and Sid Ahmed in [10] generalized the concept of those operators on a Hilbert space. They introduced the -symmetric commuting tuple of operators.
Using the identity (2), the authors in [11] have introduced the concept of -quasi--symmetric operators as followsan operator is said to be -quasi--symmetric operator if satisfies the following identity:for some positive integers and . Obviously, every -symmetric operator is -quasi--symmetric operator. Many algebraic and spectral properties for -quasi--symmetric operators has been studied by the authors in [11] parallel to those obtained for the classes of -quasi--isometric operators and its variants studied intensively by many authors in the papers [12–16]. Recall from [17] that an operator is said to have the single-valued extension property (or SVEP) if for every open subset of and any -valued analytic function on such that on , we have on . admits Bishop’s property if, for every open subset of and every sequence of analytic functions with converges uniformly to 0 in norm on compact subsets of , converges uniformly to 0 in norm on compact subsets of .
For , we set . Note that
Hence, if is -symmetric operator, then is -symmetric operator for .
The aim of this paper is to introduce the class of -quasi--symmetric operators, where is a nonconstant complex polynomial and is a positive constant. This class of operators seems a natural generalization of -quasi--symmetric operators. We show that many results for -symmetric and -quasi -symmetric operators remain true for our new class.
2. Main Results
In this section, we study the concept of polynomial-quasi -symmetric operators.
Definition 1. An operator is said to be -quasi--symmetric for some nonconstant polynomial , if there exists a nonconstant complex polynomial such thatfor some positive integer .
Remark 1. (1)If for some positive integer , then is said to be -quasi--symmetric (see [11])(2)If , then is said to be quasisymmetric
Example 1. Consider . It is obvious that is -quasi-m-symmetric operator and with for all complex polynomial . By observing that for all , it follows from direct calculation thatMoreover,We deduce that is a -quasi--symmetric operator for all polynomial satisfying .
Theorem 1. Let be a nonconstant complex polynomial and let . Assume that is not dense. Then the following statements are equivalent:(1) is -quasi--symmetric operator(2) on , where is -symmetric and (i.e.; is algebraic operator). Furthermore, .
Proof. . Consider the matrix representation of with respect to the decomposition . Let Q be the orthogonal projection onto , then we have . Since is -quasi--symmetric operator, we haveThat is,Therefore, is an -symmetric operator.
Let , we haveSo that .
The proof of the statement is similar to the one given in [18], Lemma 3.9, so we omit it.
Assume that on where is -symmetry and . A simple computation shows thatwhere .
Moreover,Hence, , and therefore is -quasi--symmetric.
In the following corollary, we show the relationship between -quasi--symmetric and -quasi--symmetric operators.
Corollary 1. Let be a complex polynomial and let . If is -quasi--symmetric operator then is -quasi--symmetric operator for all positive integer .
Proof. Since is -quasi--symmetric operator, two cases can be distinguished.(1)If , then is -symmetric operator, and hence, is -symmetric operator for all by [8].(2)If , taking into account Theorem 1, we can write on , where is an -symmetric and (i.e.; is algebraic operator). From [8], is a -symmetric operator for all . Consequently, the desired results follow from the statement (2) of Theorem 1.
Corollary 2. Let be a complex polynomial, and let be a -quasi--symmetric operator such that on . If , then is similar to a direct sum of a -symmetric operator and an algebraic operator.
Proof. By Theorem 1, we write the matrix representation of on as follows, where is a -symmetric operator and . Since , then there exists an operator such that by [19]. Hence,Consequently, the desired result follows from Theorem 1.
Theorem 2. Let be a nonconstant complex polynomial and be a -quasi--symmetric operator, and is an closed invariant subspace for . Then, the restriction is also a -quasi--symmetric operator.
Proof. Let us consider the following matrix representation of :Since is -quasi -symmetric, we haveTherefore,Thus, is an -quasi -symmetric operator.
Proposition 1. Let be a nonconstant complex polynomial and let . Suppose that is a -quasi--symmetric operator. If is dense. Then, is an -symmetric operator.
Proof. We have , and therefore,
Proposition 2. Let be a nonconstant complex polynomial and . If is a -quasi--symmetric operator, then is also a -quasi--symmetric operator for any positive integer .
Proof. If has a dense range, then is an -symmetric operator and so is for any positive integer (see [20], Theorem 2.4).
If , taking into account Theorem 1, we can use the matrix representation as , where is an -symmetric operator and is algebraic operator. AsSince is -symmetric and is algebraic, it follows that is -quasi--symmetric operator.
Theorem 3. Let be a complex polynomial and consider , such that is -quasi--symmetric operator and . If , then is similar to a -quasi- -symmetric operator.
Proof. Since , it follows from [17] (Theorem 3.5.1) that there exist some operator for which . SinceTherefore, is similar to .
Since is -quasi--symmetric operator and , it followsConsequently, is similar to an -quasi--symmetric operator.
Question 1. Let . If is -quasi -symmetric operator and for some complex polynomial, then the operator matrix is -quasi--symmetric operator.
Theorem 4 (see [21], Theorem 2.5). Let and be infinite complex Hilbert spaces and let the operator matrix of the form . Assume that has Bishop’s property . Then, the following assertions are equivalent:(i) has Bishop’s property (ii) has Bishop’s property
Theorem 5. Let be a nonconstant complex polynomial and . If is -quasi-3-symmetric operator, then has Bishop’s property .
Proof. If is dense, then is 3-symmetric operator, and therefore, has Bishop’s property by [11]. If is not dense, we have by Theorem 1 the matrix representation , where is a 3-symmetric operator and is algebraic operator.
Since every algebraic operator has Bishop’s property and has Bishop´s property from [11], the desired result follows from Theorem 4.
Corollary 3. Let be a nonconstant complex polynomial and . If is -quasi-3-symmetric operator, then has SVEP.
Definition 2. Let be a nonconstant complex polynomial and . We say that is a strict -quasi -symmetric operator if is -quasi -symmetric but is not -quasi -symmetric operator.
The idea of the proof of the following theorem is inspired from [22].
Theorem 6. Let be a complex polynomial and . Assume that is a strict -quasi--symmetric operator, then the family of operatorsis linearly independent.
Proof. Since is -quasi -symmetric operator, we haveNote that from (5), we haveNow assume that for some complex numbers ,Multiplying the (25) on the left by , we getSimilarly, multiplying (25) on the right by , we getSubtracting (26) and (27), we obtainIf we use (5), we see thatThe same procedure applied to (29) givesApplying the above process we obtain,From Corollary 1, it is well known that if is -quasi -symmetric operator, then is -quasi--symmetric operator for all . This implies the following implications:
For ,so from the fact that is a strict -quasi -symmetric operator.
For ,so for the same reason.
Doing this iteratively, we can find for and that for .
Then, the following implication is true:
The set of -quasi -symmetric operators is closed in norm as shown in the following theorem:
Theorem 7. Let be a nonconstant complex polynomial and , then the set of -quasi--symmetric operators is closed in norm in .
Proof. Suppose that is a sequence of -quasi--symmetric operators such thatSince for every positive integer , is -quasi--symmetric operator, we getHowever,Since we get by taking , that , and therefore belongs to the of -quasi--symmetric operators.
Lemma 1. Let be a nonconstant complex polynomial and let , the following statements are true:(1) is a -quasi -symmetric operator if and only if is -quasi -symmetric operator(2) is a -quasi -symmetric operator if and only if is -quasi -symmetric operator
Proof. Set , we have
Proposition 3. Let be a nonconstant complex polynomial and . If is -quasi--symmetric for some positive integer and is an isometry, then is an -quasi--symmetric operator.
Proof. Let such that is -quasi--symmetric operator. Since is an isometry, then . A direct calculation shows thatand moreoverWe haveTherefore, is a -quasi--symmetric operator [23].
Data Availability
Data sharing not applicable to this paper as no data sets were generated or analyzed during the current study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Authors’ Contributions
The author wrote and reviewed the manuscript.
Acknowledgments
This research was funded by the Deanship of Scientific Research at Jouf University under grant no. DSR-2021-03-03189.