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.
References