On A-Self-Adjoint, A-Unitary Operators and Quasiaffinities

Isaiah N. Sitati^*, Bernard M. Nzimbi, Stephen W. Luketero, Jairus M. Khalagai

School of Mathematics, College of Biological and Physical Sciences, University of Nairobi, Nairobi, Kenya

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To cite this article:

Isaiah N. Sitati, Bernard M. Nzimbi, Stephen W. Luketero, Jairus M. Khalagai. On A-Self-Adjoint, A-Unitary Operators and Quasiaffinities. Mathematics and Computer Science. Vol. 1, No. 3, 2016, pp. 56-60. doi: 10.11648/j.mcs.20160103.14

Received: August 8, 2016; Accepted: August 18, 2016; Published: September 7, 2016

Abstract: In this paper, we investigate properties of A-self-adjoint operators and other relations on Hilbert spaces. In this context, A is a self-adjoint and an invertible operator. More results on operator equivalences including similarity, unitary and metric equivalences are discussed. We also investigate conditions under which these classes of operators are self- adjoint and unitary. We finally locate their spectra.

Keywords: A-Self-Adjoint, A-Unitary, Hilbert Space, Metric Equivalence, Quasiaffinities

1. Introduction

Throughout this paper Hilbert spaces or subspaces will be denoted by capital letters, respectively and, etc denote bounded linear operators where an operator means a bounded linear transformation.  will denote the Banach algebra of bounded linear operators on.  denotes the set of bounded linear transformations from to, which is equipped with the (induced uniform) norm.

If , then  denotes the adjoint while ,,  and  stands for the kernel of , range of , closure of  and orthogonal complement of a closed subspace  of  respectively. For an operator, we also denote by , the spectrum and norm of  respectively.

A contraction on  is an operator  such that  (i.e. . A strict or proper contraction is an operator  with  (i.e.  .If, then T is called a non-strict contraction (or an isometry).Many authors like Kubrusly [7] have extensively studied this class of operators.

An operator  is said to be positive if  ∀ . Suppose that  is a positive operator, then an operator  is called an  on  if . If equality holds, that is, then  is called an , where is a self adjoint and invertible operator.

In this research, we put more conditions on. In particular, if is a self adjoint and invertible operator, then we call such an  an . Let  be a linear operator on a Hilbert space .

We define the  of  to be an operator S such that .The existence of such an operator is not guaranteed. It may or may not exist. In fact a given  may admit many  and if such an  of  exists, we denote it as . Thus .We are making an assumption that  is invertible and so
. It is also clear that  of  is the adjoint of  if. By [2],  admits an  if and only if . In this case the operator  is acting as a signature operator on .

Two operators  and  are similar (denoted  if there exists an operator  where  is a Banach subalgebra of  which is an invertible operator from  to  and are unitarily equivalent (denoted), if there exists a unitary operatorsuch that .

Two operators are considered the "same" if they are unitarily equivalent since they have the same, properties of invertibility, normality, spectral picture (norm, spectrum and spectral radius).

An operator  is quasi-invertible or a quasi-affinity if it is an injective operator with dense range (i.e. and ; equivalently,  and,  thus  is quasi-invertible if and only if  is quasi-invertible).

An operator  is a  quasi-affine transform of  if there exists a quasi-invertible  such that . is a quasiafiine transform of  if there exists a quasinvertible operator intertwining  to .

Two operators  are said to be almost similar (a.s) (denoted by  if there exists an invertible operator  such that the following two conditions are satisfied: and.

Two operators  are said to be metrically equivalent (denoted by  if (equivalently, for all ) or  if .This concept was introduced by Nzimbi et al ([8]).

Two linear operators  and  are said to be  equivalent (denoted), if there exists an  operator  such that

We shall also define the following classes of operators in this paper:

An operator   is said to be an involution if = .

An operator   is said to be self-adjoint or Hermitian if    equivalently, if   ∀  .

An operator   is said to be unitary if   and normal if  (equivalently, if .

An operator   is said to be a partial isometry if   or equivalently, if   is a projection.

An operator   is said to be quasinormal if   or equivalently if  commutes with  that is .

Let  and  be Hilbert spaces. An operator is invertible if it is injective (one -to- one) and surjective (onto or has dense range); equivalently if  and  We denote the class of invertible linear operators by .The commutator of two operators  and, denoted by  is defined by. The self –commutator of an operator  is

Suppose  is a self-adjoint and invertible operator, not necessarily unique. An operator   is said to be equivalently, ,  equivalently, ,if or equivalently, T if  or equivalently, Clearly, an -isometry whose range is dense in  is an  .

2. Basic Results

We shall investigate operators in a Hilbert Space  that are not self-adjoint. It is well known that every self- adjoint operator has a real spectrum.

The following results will form a basis for our discussion throughout this paper.

Theorem 2.1 [7, Theorem 2.1]. An invertible operator  is a product of two self-adjoint operators if and only if  

Proof: [See 7].

Remark: The product of two self-adjoint operators need not to have real spectrum. To justify our claim, we consider self-adjoint operators  and . The product  has a purely imaginary spectrum. Denoting by  the set of all invertible products of self-adjoint operators and and by  the set of invertible operators that are similar to their adjoints, we see that  The above theorem asserts that  is also valid. By using the invariance of these two classes under similarity transformations, we notice that is strictly larger than the class of operators that are similar to their adjoints. We can give an example of a unilateral shift operator on
 in this context.

Theorem 2.2 [12]: is unitarily equivalent to its adjoint if and only if  is a product of a symmetry (self-adjoint or unitary involution) and a self-adjoint operator.

Theorem 2.3 [7, pp. 6]: Two normal operators that are similar are unitarily equivalent.

Remark: Any invertible normal operator which is similar to its adjoint can be expressed as a product of self-adjoint operators, that is, if  is normal and  then  .

Proposition 2.4 [17]: If  is self-adjoint and injective, then  is also self-adjoint.

Remark: Just like other bounded linear operators, the
-self adjoint operation satisfies the following properties which can easily be shown using the definition of an  :

(a). 

(b). 

(c).  

(d). 

3. A-Self-Adjoint Operators

Definition: A Jordan algebra  consists of a real vector space equipped with a bilinear product  satisfying the commutative law and the Jordan identity: and  . A Jordan algebra is formally real if   .

Remark: An associative algebra, over a real Hilbert space  gives rise to a Jordan algebra  under quasi-multiplication: the product  is commutative and satisfies the Jordan identity since

We say that a Jordan algebra  is  if it can be realized as a Jordan subalgebra of some Jordan algebra.

Example: If  is a set of Jordan operators, then the subspace of hermitian operator  is also closed under the Jordan product, since if  and , then  forms a special algebra . These hermitian algebras are the archetypes of all Jordan algebras. We can easily check that hermitian matrices over  form special Jordan algebras that are formally real.

We shall investigate the Jordan algebra
 of  Operators denoted by the set
 . Note that just like many other algebras like the Lie algebra  is an ℝ- linear subspace. That is, it is closed under real linear combinations.

We outline in the following results some conditions that guarantee an  to be self-adjoint.

Proposition 3.1: [7]. Every self –adjoint operator  is
.

Remark: The converse of the above proposition is not generally true. For consider the operators  and. A quick calculation reveals that  is  but it is not self-adjoint. We note that  coincides with self-adjointness when  is an identity operator.

We now answer the question: when is an  operator self-adjoint? The results below give us answer the question.

Lemma 3.2: Let   be  operator. Then  is self-adjoint if and only if   and  commute with an involution.

Proof: Suppose. Then  for some invertible and self-adjoint operator. Now suppose that the similarity transformation  is an involution. Then, clearly,. This assertion proves that    and so  is self-adjoint.

Theorem 3.3 [15]: Let and be Hilbert spaces and let
. Then

i.    

ii.  

iii.  =

iv. =

Remark: We note that if   is self-adjoint, then by iii above, = and so.

It has been proved in [7] that if  is an, then its adjoint  is injective. This result together with the corollary to Theorem 4.12 [13] enables us identify the relationship between operators and the quasi-affinity. (See Theorem 3.5 pp. 10, of [7]).

Evidently, if   is an operator, then  and its adjoint,  are quasi-affinities. In fact  and  are left invertible, that is if there exists an operator  such that  and .

We shall also give the relationship between metrically equivalent operators and unitarily equivalent operators for some given quasiaffinity:

Theorem 3.4: [10, Theorem 3.29 (ii)]: If  and  are metrically equivalent operators and  is self-adjoint, then .

Theorem 3.5 [9, Theorem 2.9 (Fuglede-Putnam-Rosenblum)]: Let  and .If holds for some operator , then .

Theorem 3.6: Let  . Suppose  and  are metrically equivalent operators, and
for some quasiaffinity  which is -unitary, then  and  are unitarily equivalent.

Proof: We first note that every unitary operator is . We show that if  and  are metrically equivalent then they are unitarily equivalent.

Suppose, and
 for some quasiaffinity  . Suppose  is the polar decomposition of  , where  is a partial isometry and  is positive.

Define  and  on  . Since is a quasiaffinity, so is . Using  we have that  and  which means that S and  are quasisimilar normal operators. By the Fuglede-Putnam-Rosenblum Theorem above, S and  are unitarily equivalent meaning that there exists a unitary operator such that  where  is a polar decomposition of  . That is  , which shows that .

Question: Is every part of an  operator  also -self adjoint? This question can be answered if we decompose  as a direct sum  by specifying certain conditions on the direct summands of.We summarize this in the following theorem:

Theorem 3.7: Every part of an  operator  is -self adjoint.

Proof: Suppose  where  has a certain property  while  is devoid of property. Then by definition of -self adjointness we have (. Thus,  and  as required.

Remark: It has been shown in [7] that if    is an  operator then  is unitary if  is an involution. In additional, the spectrum of  is either real or complex; if complex, then the eigen values come in complex conjugate pairs.(see [6]). This gives us a necessary and sufficient condition for -self adjointness.

In general, such operators have are symmetric with respect to the real axis. Equality of spectra is a necessary condition for -self adjointness. We summarize it in the following corollary:

Corollary 3.8: Let   is an . Then

a). 

b). 

c).  

Proof: Since  is an  then by definition. Thus,  and  are similar and hence have the same spectrum. Therefore the above claims follow since  is the disjoint union of  and .

Counter Example

The backward shift operator  defined by  is not -self adjoint. Its adjoint (called the unilateral shift) is defined by .We see (as an infinite matrix) that every  with   (open unit disc centred at the origin) is in  and that. Also, {. Hence  is not -self adjoint (for any  with the required properties) because the necessary condition for -self adjointness is not satisfied i.e.

Question: Given that  is -self adjoint, is self adjoint? We provide the solution in the following theorem.

Theorem 3.9:  is -self adjoint, if and only if is  self adjoint.

Proof:-self adjoint implies that . We then have that. Thus  (since  is self-adjoint).

Conversely, let  be -self adjoint. Then. Post multiplying both sides of this equation by  and using the definition we have . This completes the proof.

Remark: In view of the above theorem, we see that the mapping defined by  is an isomorphism i.e. it establishes a one-to-one correspondence between the class of self- adjoint and -self adjoint operators in the Hilbert space  In fact if we let  to be -self adjoint then we see that  is self-adjoint if  commutes with  i.e. . Here. Then  .

4. A-Self-Adjoint, Unitary Equivalence and A-Unitarily Equivalence of Operators

It is well known that unitary equivalence is an equivalence relation. We give a condition which shows that unitary equivalence preserves -self adjointness.

Theorem 4.1: Let  and  be bounded linear operators on a Hilbert space . If   is -self adjoint and  is unitarily equivalent to , that is , where  is a unitary operator, then  is -self adjoint.

Proof: We have  and  for some unitary operator. Using these two equations we can simplify and re-write  in terms of operators,  and  only as:

  which establishes the claim.

Remark: The above theorem shows that unitary equivalence preserves -self adjointness if and only if . That is, if the unitary operator  is -unitary.

We see that unlike self-adjointness, unitary equivalence does not preserve -self adjointness.

The following results will enable us establish the relationship be -unitarily equivalence and -normal operators.

Definition 4.2: The automorphism group of -unitary operators is the set   :.

Theorem 4.3 [7]. Every unitary operator is -unitary.

Proof: [7, pp. 21].

Remark:  is a multiplicative group. If , then  . This follows from  (ST

Definition 4.4: Two linear operators  and  are said to be (denoted), if there exists an   operator such that  

In a real Hilbert space of dimension, an operator is called Lorentz if it is  where where 𝐍 and  . For instance if  , then  is Lorentz.

Definition 4.5: A conjugation is a conjugate-linear operator  which is both involutory (i.e.,) and isometric.

Remark: If we let  , then is a conjugation. Thus, this Jordan algebra  will contain the invertible normal operators, operators defined by Hankel matrices, Toeplitz and the Volterra integration operator  for a function  and

Remark: Every -unitary operator  is invertible. We note that if  is -unitary then  is also -unitary. This follows from the fact that = is -unitary.

Theorem 4.6 [8]: If  is a normal operator and is unitarily equivalent to, then  is normal.

Proof: [8].

Theorem 4.7 [7]: Every normal operator  is -normal.

Proof: [7, pp. 30-31].

Remark: Not all -normal operators are normal. For example, if and  a quick mathematical computation reveals that  and . Therefore,  is -normal but not normal.

We also see that -self adjoint and -unitary operators are special cases of -normal operators.

Corollary 4.8: If  is an - normal operator and
 is - unitarily equivalent to  then  is -normal.

Proof: From Theorem 4.7 above, every unitary operator (w.l.o.g, letting) is - unitary and using a similar argument, we see that every normal operator  is -normal. It suffices to show that S is normal.

Now, suppose that, that is  where  is unitary and  is -normal.

Then  (Since  (Since T is normal) (Since XT=SX and) (Since. That is  is normal.Since every normal operator  is -normal, it follows that  is -normal as required.

Finally, we discuss some conditions that guarantee a product of -self adjoint operators to be -self adjoint:

Theorem 4.9: [7, Theorem 3.19 (ii)] If and  are -self adjoint operators, then the product  is -self adjoint if and only if  .

By the above Theorem, we note that  is a linear space which is not closed under multiplication. However, it is closed with respect to the Jordan product given by the equation  .

Corollary 4.10: An invertible operator  is a product of -self adjoint operators  and  if and only if  is -self adjoint.

Proof: Suppose  is invertible with  and,. Invertibility of  implies that  and 0 implies that 0. Hence  and  are invertible and so is  . Clearly, (Since).That is  which shows that  is

Conversely, suppose  is invertible and is A-self adjoint. Since  is invertible, by the polar decomposition theorem,  has a unique polar decomposition , where  is unitary (and not necessarily self-adjoint) and ( is positive (hence self-adjoint) operator. We use -self adjointness of  to show that, must indeed, be self-adjoint. -self adjoint of  implies that , for some invertible operator -self adjoint of  (invertible) implies that  is self adjoint. But every self adjoint operator is -self adjoint. This completes the proof.

Potential Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

Acknowledgements

The author wishes to express his heartfelt gratitude to the referees for valuable comments and suggestions during the writing of this manuscript.

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