# Extension ofm-Symmetric Hilbert Space Operators

Authors: Hadi Obaid Alshammari
Journal: Journal of Mathematics
 (2022)

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## Abstract

We introduce a new class of operators, which we will call the class ofP-quasi-m-symmetric operators that includes m-symmetric operators and k-quasi m-symmetric operators. Some basic structural properties of this class of operators are established based on the operator matrix representation associated with such operators.


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## Body

## 1. Introduction

Throughout this paper,ℋ stands for a complex Hilbert space of infinite-dimension with inner product ⋅|⋅. By ℬℋ, we denote the Banach algebra of all bounded linear operators on ℋ. For every A∈ℬℋ, we denote by kerA and ℛA the null space and the range of A, respectively. Also, σpA,σapA,σT, and σsA denote the point spectrum, the approximate spectrum, the spectrum, and the surjective spectrum of A.The authors in[1] introduced the class of Helton operators as followsan operator A∈ℬℋ is said to be in the mth Helton class of B if(1)∑0≤j≤m−1m−jmjBjAm−j=0.We refer the interested reader to [1–6] for complete details.The class ofm-symmetric operators or m-self-adjoint operators on a Hilbert ℋ space has attracted much attention and has been the subject of intensive studies by several authors. m-symmetric operators were introduced in [3, 4, 7] as followsan Hilbert space operator A∈ℬℋ is said to be an m-symmetric if A satisfies the following identity:(2)∑0≤j≤m−1jmjA∗jAm−j=0,for some positive integer m, where A∗ is the adjoint operator of A. For m=1, equation (2) is reduced to A∗−A=0 (A is symmetric or self-adjoint). If m=2, equation (2) is reduced to A∗2−2A∗A+A2=0 (A is 2-symmetric). It has been proven that a power of m-symmetric transformation is again m-symmetric, and the product of two m-symmetric transformations is also m-symmetric under suitable conditions (see [8]).Another extension of the relation in (2), we recall that if T,A∈ℬℋ, for which A is positive, T is called an A,m-symmetric ([9]) when(3)∑0≤j≤m−1jmjT∗jATm−j=0.The authors Chō and Sid Ahmed in [10] generalized the concept of those operators on a Hilbert space. They introduced the A,m-symmetric commuting tuple of operators.Using the identity (2), the authors in [11] have introduced the concept of k-quasi-m-symmetric operators as followsan operator A∈ℬℋ is said to be k-quasi-m-symmetric operator if A satisfies the following identity:(4)A∗k∑0≤j≤m−1jmjA∗jAm−jAk=0,for some positive integers k and m. Obviously, every m-symmetric operator is k-quasi-m-symmetric operator. Many algebraic and spectral properties for k-quasi-m-symmetric operators has been studied by the authors in [11] parallel to those obtained for the classes of n-quasi-m-isometric operators and its variants studied intensively by many authors in the papers [12–16]. Recall from [17] that an operator A∈ℬℋ is said to have the single-valued extension property (or SVEP) if for every open subset W of ℂ and any ℋ-valued analytic function f on W such that A−λfλ≡0 on W, we have fλ≡0 on W. A∈ℬℋ admits Bishop’s property β if, for every open subset D of ℂ and every sequence gn:D⟶ℋ of analytic functions with T−zgnz converges uniformly to 0 in norm on compact subsets of D, gnz converges uniformly to 0 in norm on compact subsets of D.ForA∈ℬℋ, we set ΘkA:=∑0≤j≤k−1jkjA∗jAk−j. Note that(5)Θk+1A=A∗ΘkA−ΘkAA.Hence, ifA is m-symmetric operator, then A is p-symmetric operator for p≥m.The aim of this paper is to introduce the class ofP-quasi-m-symmetric operators, where P is a nonconstant complex polynomial and m is a positive constant. This class of operators seems a natural generalization of k-quasi-m-symmetric operators. We show that many results for m-symmetric and k-quasi m-symmetric operators remain true for our new class.


## 2. Main Results

In this section, we study the concept of polynomial-quasim-symmetric operators.Definition 1.
An operatorA∈ℬℋ is said to be P-quasi-m-symmetric for some nonconstant polynomial P, if there exists a nonconstant complex polynomial P such that(6)PA∗∑0≤k≤m−1kmkA∗kAm−kPA=0,for some positive integer m.Remark 1.
(1)
IfPz=zn for some positive integer n, then A is said to be n-quasi-m-symmetric (see [11])(2)
IfPz=z, then A is said to be quasisymmetricExample 1.
ConsiderA=IℋIℋ00∈ℬℋ⊕ℋ. It is obvious that T=Iℋ is P-quasi-m-symmetric operator and PS=0 with S=0 for all complex polynomial P=∑0≤k≤makzk. By observing that Ak=A for all k≥1, it follows from direct calculation that(7)PA=∑0≤k≤makAk=a0Iℋ⊕ℋ+∑1≤k≤makA=∑0≤k≤makIℋ∑1≤k≤makIℋ0a0Iℋ,∑0≤k≤m−1kmkA∗kAm−k=Am+∑1≤k≤m−1kmkA∗A=A−A∗A=IℋIℋ00−IℋIℋIℋIℋ=00−Iℋ−Iℋ.
Moreover,(8)PA∗∑0≤k≤m−1kmkA∗kAm−kPA=∑0≤k≤mak¯Iℋ0∑1≤k≤mak¯Iℋa0¯Iℋ00−Iℋ−Iℋ∑0≤k≤makIℋ∑1≤k≤makIℋ0a0Iℋ=00−a0¯Iℋ−a0¯Iℋ∑0≤k≤makIℋ∑1≤k≤makIℋ0a0Iℋ=00−a¯0∑0≤k≤makIℋ−a0¯∑1≤k≤makIℋ−a02Iℋ.
We deduce thatA is a P-quasi-m-symmetric operator for all polynomial P satisfying P0=0.Theorem 1.
LetP be a nonconstant complex polynomial and let A∈ℬℋ. Assume that PAℋ¯ is not dense. Then the following statements are equivalent:(1)
A is P-quasi-m-symmetric operator(2)
A=A1A20A3 on ℋ=PAℋ¯⊕kerPA∗, where A1=A|PAℋ¯ is m-symmetric and PA3=0 (i.e.; A3 is algebraic operator). Furthermore, σA=σA1∪σA3.Proof.
1⇒2. Consider the matrix representation of A with respect to the decomposition ℋ=PAℋ¯⊕kerPA∗A=A1A20A3. Let Q be the orthogonal projection onto PAℋ¯, then we have A1000=AQ=QAQ. Since A is P-quasi-m-symmetric operator, we have(9)Q∑0≤k≤m−1kmkA∗kAm−kQ=0.
That is,(10)∑0≤k≤m−1kmkA1∗kA1m−k=0.
Therefore,A1 is an m-symmetric operator.
Letw=w1+w2∈PAℋ¯⊕kerPA∗, we have(11)PA3w2|w2=PAI−Qw|I−Qw=I−Qw|PA∗I−Qw=0.
So thatPA3=0.
The proof of the statementσA=σA1∪σA3 is similar to the one given in [18], Lemma 3.9, so we omit it.
2⇒1 Assume that A=A1A20A3 on ℋ=ℛPA¯⊕kerPA∗ where A1=A|ℛPA¯ is m-symmetry and PA3=0. A simple computation shows that(12)PA=PA1Z00andPAPA∗=PA1PA1∗+ZZ∗000=F000,where F=F∗=PA1PA1∗+ZZ∗.
Moreover,(13)PAPA∗∑0≤k≤m−1kmkA∗kAm−kPAPA∗=F∑0≤k≤m−1kmkA1∗kA1m−kF000=0000=0.
Hence,PA∗∑0≤k≤m−1kmkA∗kAm−kPA=0, and therefore A is P-quasi-m-symmetric.In the following corollary, we show the relationship betweenP-quasi-m-symmetric and P-quasi-m+1-symmetric operators.Corollary 1.
LetP be a complex polynomial and let A∈ℬℋ. If A is P-quasi-m-symmetric operator then A is P-quasi-q-symmetric operator for all positive integer q≥m.Proof.
SinceA is P-quasi-m-symmetric operator, two cases can be distinguished.(1)
IfℛP¯=ℋ, then A is m-symmetric operator, and hence, A is q-symmetric operator for all q≥m by [8].(2)
IfℛP¯≠ℋ, taking into account Theorem 1, we can write A=A1A20A3 on ℋ=PAℋ¯⊕kerPA∗, where A1=A|PAℋ¯ is an m-symmetric and PA3=0 (i.e.; A3 is algebraic operator). From [8], A1 is a q-symmetric operator for all q≥m. Consequently, the desired results follow from the statement (2) of Theorem 1.Corollary 2.
LetP be a complex polynomial, and let A∈ℬℋ be a P-quasi-m-symmetric operator such that A=A1A20A3 on ℋ=ℛPA¯⊕NPA∗. If σA1∩σA3=∅, then A is similar to a direct sum of a m-symmetric operator and an algebraic operator.Proof.
By Theorem1, we write the matrix representation of A on ℋ=ℛPA¯⊕NPA∗ as followsA=A1A20A3, where A1=A|ℛPA¯ is a m-symmetric operator and PA3=0. Since σA1∩σA3=∅, then there exists an operator C such that A1C−CA3=A2 by [19]. Hence,(14)A=A1A20A3=IC0I−1A100A3IC0I=IC0I−1A1⊕A2IC0I.
Consequently, the desired result follows from Theorem1.Theorem 2.
LetP be a nonconstant complex polynomial and A be a P-quasi-m-symmetric operator, and M is an closed invariant subspace for A . Then, the restriction A|M is also a P-quasi-m-symmetric operator.Proof.
Let us consider the following matrix representation ofA:(15)A=A1A20A3onℋ=M⊕M⊥.
SinceA is P-quasi m-symmetric, we have(16)0=PA1A20A3∗n∑0≤k≤mmkA1A20A3∗kA1A20A3m−kPA1A20A3=PA1∗0X∗PA3∗∑0≤k≤m−1kmkA1∗kA1m−kZYWPA1X0PA3=PA1∗∑0≤k≤m−1kmkA1∗kA1m−kPA1∗∗∗∗∗∗.
Therefore,(17)PA1∗∑0≤k≤mmkA1∗kA1m−kPA1=0.
Thus,A1 is an P-quasi m-symmetric operator.Proposition 1.
LetP be a nonconstant complex polynomial and let A∈ℬℋ. Suppose that A is a P-quasi-m-symmetric operator. If ℛPA is dense. Then, A is an m-symmetric operator.Proof.
We havePA∗∑0≤k≤m−1kmkA∗kAm−kPA=0, and therefore,(18)∑0≤k≤m−1kmkA∗kAm−k=0onℛPA¯.Proposition 2.
LetP be a nonconstant complex polynomial and A∈ℬℋ. If A is a P-quasi-m-symmetric operator, then An is also a P-quasi-m-symmetric operator for any positive integer n.Proof.
IfPA has a dense range, then A is an m-symmetric operator and so is An for any positive integer n (see [20], Theorem 2.4).
IfℛPA¯≠ℋ, taking into account Theorem 1, we can use the matrix representation as A=A1A20A3onℋ=ℛPA¯⊕NPA∗, where A1 is an m-symmetric operator and A3 is algebraic operator. As(19)An=A1n∑0≤j≤n−1A1jA2A3n−1−j0A3n.
SinceA1n is m-symmetric and A3n is algebraic, it follows that An is P-quasi-m-symmetric operator.Theorem 3.
LetP be a complex polynomial and consider A=TR0S∈ℬℋ⊕ℋ, such that T is P-quasi-m-symmetric operator and PS=0. If σsT∩σaS=∅, then A is similar to a P-quasi- m-symmetric operator.Proof.
SinceσsT∩σaS=∅, it follows from [17] (Theorem 3.5.1) that there exist some operator B∈ℬℋ for which TB−BS=R. Since(20)IB0ITR0S=T00SIB0I.
Therefore,A is similar to D=T00S.
SinceT is P-quasi-m-symmetric operator and PS=0, it follows(21)PD∗∑0≤k≤m−1kmkD∗kDm−kPD=PT∗00PS∗∑0≤k≤m−1kmkT∗kTm−k00∑0≤k≤m−1kmkS∗kSm−kPT00PS=PT∗∑0≤k≤m−1kmkT∗kTm−kPT000=0.
Consequently,A is similar to an P-quasi-m-symmetric operator.Question 1.
LetT,R,S∈ℬℋ. If T is P-quasi m-symmetric operator and PS=0 for some complex polynomial, then the operator matrix A=TR0S∈ℬℋ⊕ℋ is P-quasi-m-symmetric operator.Theorem 4 (see [21], Theorem 2.5).
Letℋ and K be infinite complex Hilbert spaces and let A∈ℬℋ⊕K the operator matrix of the form A=A1A20A3. Assume that A3 has Bishop’s property β. Then, the following assertions are equivalent:(i)
A has Bishop’s property β(ii)
A1 has Bishop’s property βTheorem 5.
LetP be a nonconstant complex polynomial and A∈ℬℋ. If A is P-quasi-3-symmetric operator, then A has Bishop’s property β.Proof.
IfℛPA is dense, then A is 3-symmetric operator, and therefore, A has Bishop’s property β by [11]. If ℛPA is not dense, we have by Theorem 1 the matrix representation A=A1A20A3, where A1=A|ℛPA is a 3-symmetric operator and A3 is algebraic operator.
Since every algebraic operator has Bishop’s propertyβ and A1 has Bishop´s property β from [11], the desired result follows from Theorem 4.Corollary 3.
LetP be a nonconstant complex polynomial and A∈ℬℋ. If A is P-quasi-3-symmetric operator, then A has SVEP.Definition 2.
LetP be a nonconstant complex polynomial and A∈ℬℋ. We say that A is a strict P-quasi m-symmetric operator if A is P-quasi m-symmetric but A is not P-quasi m−1-symmetric operator.The idea of the proof of the following theorem is inspired from [22].Theorem 6.
LetP be a complex polynomial and A∈ℬℋ. Assume that A is a strict P-quasi-m-symmetric operator, then the family of operators(22)PA∗ΘkAPA,k=0,1,⋯,m−1,m≥2.,is linearly independent.Proof.
SinceA is P-quasi m-symmetric operator, we have(23)PA∗ΘmAPA=0.
Note that from (5), we have(24)ΘkA=A∗Θk−1A−Θk−1AAfor allk≥1.
Now assume that for some complex numbersαk,(25)∑0≤k≤m−1αkPA∗ΘkAPA=0.
Multiplying the (25) on the left by A∗, we get(26)∑0≤k≤m−1αkPA∗A∗ΘkAPA=0.
Similarly, multiplying (25) on the right by A, we get(27)∑0≤k≤m−1αkPA∗ΘkAAPA=0.
Subtracting (26) and (27), we obtain(28)∑0≤k≤m−1αkPA∗A∗ΘkA−ΘkAAPA=0.
If we use (5), we see that(29)∑0≤k≤m−1αkPA∗θk+1APA=0.
The same procedure applied to (29) gives(30)∑0≤k≤m−1αkPAΘk+2APA=0.
Applying the above process we obtain,(31)∑0≤k≤m−1αkPAΘk+rAPA=0for allr∈ℕ.
From Corollary1, it is well known that if A is P-quasi m-symmetric operator, then A is P-quasi-q-symmetric operator for all q≥m. This implies the following implications:
Forr=m−1,(32)∑0≤k≤m−1αkPAΘk+m−1APA=0⇒α0PAΘm−1APA=0,so α0=0 from the fact that A is a strict P-quasi m-symmetric operator.
Forr=m−2,(33)∑0≤k≤m−1αkPAΘk+m−2APA=0⇒α1PAΘm−1APA=0,so α1=0 for the same reason.
Doing this iteratively, we can find forr=m−3,…,r=1 and r=0 that αk=0 for k=2,⋯,m−1.
Then, the following implication is true:(34)∑0≤k≤m−1αkPAΘkAPA=0⇒α0⇒α1⇒⋯⇒αm−1⇒0.The set ofP-quasi m-symmetric operators is closed in norm as shown in the following theorem:Theorem 7.
LetP be a nonconstant complex polynomial and A∈ℬℋ, then the set of P-quasi-m-symmetric operators is closed in norm in ℬℋ.Proof.
Suppose thatAqq is a sequence of P-quasi-m-symmetric operators such that(35)limq⟶∞Am−A=0.
Since for every positive integerq, Aq is P-quasi-m-symmetric operator, we get(36)PAq∗ΘAqPAq=0.
However,(37)PAq∗ΘqAqPAq=PAq∗ΘmAqPAq−PA∗ΘmAPA=PAq∗∑0≤j≤m−1jmjA∗jAqm−jpAq−PA∗∑0≤j≤m−1jmjA∗jAm−jPA≤PAq∗∑0≤j≤m−1jmjA∗jAqm−jPAq−PAq∗∑0≤j≤m−1jmjA∗jAm−jPA+PAq∑0≤j≤m−1jmjA∗jAm−jPA−PA∗∑0≤j≤m−1jmjAjAm−jPA≤PAq∑0≤j≤m−1jmjA∗jAqm−jPAq−Am−jPA+∑0≤j≤m−1jmjPAq∗A∗jAm−j−PA∗A∗jAm−jPAq−Am−jPA+PA∑0≤j≤m−1jmjA∗jAqm−jPAq−Am−jPA+∑0≤j≤m−1jmjPAq∗A∗jAm−j−PA∗A∗jAm−jPA.
SincePAq∗ΘmAqPAq=0 we get by taking q⟶∞, that PA∗ΘqAPA=0, and therefore A belongs to the of P-quasi-m-symmetric operators.Lemma 1.
LetP be a nonconstant complex polynomial and let A∈ℬℋ, the following statements are true:(1)
A is a P-quasi m-symmetric operator if and only if A⊗I is P-quasi m-symmetric operator(2)
A is a P-quasi m-symmetric operator if and only if I⊗A is P-quasi m-symmetric operatorProof.
SetP=∑0≤k≤makzk, we have(38)PA⊗I=∑0≤k≤makA⊗Ik=∑0≤k≤makAk⊗=PA⊗I,PA⊗I∗ΘmA⊗IPA⊗I=PA⊗I∗∑0≤j≤m−1jmjA⊗I∗jbigA⊗Im−jPA⊗I=PA∗∑0≤j≤m−1jmjA∗jAm−jPA⊗I=PA∗ΘmAPA⊗I.Proposition 3.
LetP be a nonconstant complex polynomial and A,V∈ℬℋ. If A is P-quasi-m-symmetric for some positive integer m and V is an isometry, then VAV∗ is an P-quasi-m-symmetric operator.Proof.
LetPz=∑0≤k≤nakzk such that A is P-quasi-m-symmetric operator. Since V is an isometry, then V∗V=I. A direct calculation shows that(39)VAV∗l=VAV∗VAAV∗⋯VAV∗l−times=VA2V∗⋯VAV∗=VAlV∗,and moreover(40)PVAV∗=∑0≤k≤nakVAV∗k=∑0≤k≤nakVAkV∗=V∑0≤k≤nakAkV∗=VPAV∗.
We have(41)PVAV∗∗∑0≤k≤m−1kmkVAV∗∗kVAV∗m−kPVAV∗=VPAV∗∗V∑0≤k≤m−1kmkA∗kAm−kV∗VPAV∗=VPA∗∑0≤k≤m−1kmkA∗kAm−kpAV∗=0=0.
Therefore,VAV∗ is a P-quasi-m-symmetric operator [23].



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