Remarks on A-skew-adjoint, A-almost Similarity Equivalence and Other Operators in Hilbert Space

In this paper, notions of A-almost similarity and the Lie algebra of A-skew-adjoint operators in Hilbert space are introduced. In this context, A is a self-adjoint and an invertible operator. It is shown that A-almost similarity is an equivalence relation. Conditions under which A-almost similarity implies similarity are outlined and in which case their spectra is located. Conditions under which an A-skew adjoint operator reduces to a skew adjoint operator are also given. By relaxing some conditions on normal and unitary operators, new results on A -normal, binormal and A-binormal operators are proved. Finally A-skew adjoint operators are characterized and the relationship between A-selfadjoint and A-skew adjoint operators is given.


Introduction
In this paper, Hilbert space(s) or subspace(s) will be denoted by capital letters, and respectively and , , etc denote bounded linear operators. In this context, an operator will mean a bounded linear transformation. ( ) will denote the Banach algebra of bounded linear operators on a Hilbert space and ( , ) denotes the set of bounded linear transformations from one Hilbert space to another one , which is equipped with the (induced uniform) norm. Hilbert space operators have been discussed by many others like [5], [16], [18] and [20] among other scholars.
In this research, we put more conditions on . In particular, if is a self adjoint and invertible operator, then we call such an − ./,0 -1 an − 2 .-1. Let be a linear operator on a Hilbert space . We define the − 34,.of to be an operator such that = * whose existence is not guaranteed. It may or may not exist. In fact a given ∈ ( ) may admit many A-adjoints and if such an − 34,.of exists, we denote it as [ * ] . Thus [ * ] = * . As it were before, is invertible and so [ * ] = 78 * . It is also clear that − 34,.of is the adjoint of if = . Earlier results proved by Kubrusly [5]  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).
The commutator of two operators and , denoted by It has to be noted that an -isometry whose range is dense in is an − .-1. which is an invertible operator from to such that P = P or equivalently = P 78 P or = P P 78 .

Basic Results
Two operators and in ( ) are said to be almost similar (a.s) (denoted by S .T ) if there exists an invertible operator P such that the following two conditions are satisfied: * = P 78 ( * )P * + = P 78 ( * + )P.
It has already been shown by many authors like [9], [12] and [13] that similarity, almost similarity and unitary equivalences are equivalence relations.
Remark 2.2: It has to be noted that almost similarity generally does not imply similarity. However, certain conditions can guarantee this preservation. These may include the following: Theorem 2.3 [13]: If ∈ ( ) and ∈ ( ) are almost similar projection operators, then ( ) = ( ).

A-Almost Similarity of Operators
and Pre-multiplication of (1) and (2) by P and post multiplication of (1 ) and (2) by P 78 and applying the adjoint operation gives where and P are invertible operators. Using (3) and (4) it is found out that   (2 ) b) Conditions imposed on operators S and T so that they have the same spectrum is that they should both be projections, that is = * ; = * and I = ; I = .

A-Skew Adjoint and A-Normal Operators
In this section, some properties of the Lie algebra    ). Therefore * is -skew-adjoint as required. From this, it is concluded that * and * commute, that is [ * , * ] = 0. Note also that if is -self-adjoint, then and * are similar and hence have the same spectrum. However, this is not always the case for an A-normal operator. This is illustrated in the example below:  . This clearly shows that - * = − is a skew-adjoint operator.  Remark 4.24: It is well known by earlier results that every skew-adjont operator is -skew-adjoint (see [10]). In view of this and the corollary above, it can also be deduced that every skew-adjoint operator ∈ ( ) is −binormal.

Some Results on A-self adjoint and A-skew-adjoint Operators
In what follows, the relationship between -self adjoint and -skew -adjoint operators is investigated. It is known that every normal operator is quasinormal and every quasinormal operator is binormal. Using results in [10,Theorem 3.9] and [10,Proposition 4.4], some common behaviour of -self adjoint and skew adjoint operators are established. It is also well known that every part of a skew adjoint is skew adjoint and so every part of a skew-adjoint operator is normal. Thus a skew adjoint operator has no completely non-normal part.
Proposition 5.1 [10]: Let be an -skew adjoint operator. Then • is -self adjoint for even values of ∈ P and • is (− )-skew adjoint for odd values of ∈ P.
Remark 5.2: This proposition is simply interpreted as follows: that if is -skew adjoint, then • is (−1) • -self adjoint. That is to say that • is -skew-adjoint for odd values of ∈ P and • is -self-adjoint for even values of , which can also be extended to polynomials. The Lie Algebra h i is closed under all odd degree polynomials over a field j while the Jordan Algebra s i is closed under all polynomials over j.
The following proposition now provides a characterization of an -skew adjoint operator: Proposition 5.3: Suppose _ = , where is invertible and self-adjoint, then is skew adjoint if and only if _ isskew adjoint.
Proof: Let be skew-adjoint and _ = with invertible and self-adjoint. Then _ 78 = 78 = = − = −_ * , that is _ isskew adjoint. Conversely, let _ be -skew adjoint with _ = , then = _ 78 and so * = 78 _ * = 78 (− _ 78 ) = −_ 78 = − that is a skew-adjoint operator and this completes the proof. Remark 5.4: The converse of the above proposition gives a more general property of an -skew adjoint operator. It is also worth noting that the Lie Algebra h i is a linear space but it is not closed under multiplication. Nonetheless, h i is closed under the Lie bracket [ 8 , I ] = 8 I − I 8 .
Question: Is there any relationship between the Lie Algebra h i and the Jordan Algebra, s i ? A possible answer to this question can be summarised in the following propositions: Proposition 5.5: Let 8 and I be commuting -skew adjoint linear operators. Then the product 8 I is -selfadjoint.
Since This operator, namely, the backward shift operator : H I → H I is an example of an operator that is neither in the class of the Jordan algebra of -self-adjoint nor the Lie algebra of the -skew adjoint operators. However we should also note that there exist non-zero operators that are skewadjoint and -self-adjoint. This is illustrated in the example that follow: Example 5.9: Let =˜− . 0 0 . ™ and =˜0 1 1 0 ™. Then a quick computation shows that is both -self-adjoint (that is * = 78 ) and skew-adjoint (i.e. * = − ). In view of this, it is seen that the only operator satisfying both conditions for -self-adjointness and -skew-adjointness is the zero operator.

Conclusion
From the preceding discussions and results above, it is clearly evident that -self-adjoint, -skew-adjoint andunitary operators are special casae of -normal operators. It has also been noted that the class of -self-adjoint operators contains some self-adjoint operators, some skew-adjoint operators and some which are neither of these categories. The backward shift operator as an example of such an operator as shown in the preceding section. That there exist operators which are skew-adjoint and -self-adjoint but not -skewadjoint.
There is no class inclusion between -self-adjoint andskew-adjoint operators. However, zero is the only operator that can satisfy this inclusion. The following class inclusions also hold: Symmetry ⊊ Unitary ⊊ Normal ⊊ -Normal and Symmetry ⊊ Self-adjoint ⊊ Normal ⊊ -Normal Skew -adjoint ⊊ Normal ⊊ -Normal. In addition the intersection of the class of self-adjoint and unitary operators yields a symmetry, i.e.
Finally, it has also been established that -almost similarity is an equivalence relation just like other equivalences like unitary and almost similarity on a Hilbert space. -almost similar operators have equal spectra if they are projection operators.