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 operator
such 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