Moore-Penrose inverse of linear operators in Hilbert space

In this paper, we investigate properties of with closed range satisfying the operator equations , , we investigate the invertibility of with closed range where the Moore-Penrose inverse of T turns out to be the usual inverse of T under some classes of operators. We also deduce the Moore-Penrose inverse of a perturbed linear operator with closed range where such that has closed ranges and S satisfying some given conditions. The relation between the ranges and null spaces of these operators is also shown.


INTRODUCTION
The invertibility of linear operators is useful in finding the solution to the operator equation where y is a given vector and x an unknown vector. The inverse of T exists and is unique if and only if T is bijective. If the operator equation has no solution. Also, if , then has many solutions. In such cases generalized inverses of T are used. It is known that the generalized inverse of exist if and only if the range of T is closed. However, there exists a unique generalized inverse called the Moore-Penrose inverse which gives the best approximate solution. That is such that for all is of minimum norm where is the Moore-Penrose inverse of T.
The Moore-Penrose inverse of operators with closed range has been considered by several authors among them: Moore (1920) and Penrose (1955) Campbell and Meyer (1991), Roger and Johnson (1985), Rao et al. (1971), James (1978) among others. Drazin (1958Drazin ( ), (2012, and (2016) studied Pseudo-inverses in associative rings and semigroups. The author gave a theory for a large class of uniquely-defined outer generalized inverses. Moreover, Drazin (2016) gave a way to define left and right versions of the large class of (b, c)-inverses. Wang et al. (2017) gave some characterizations of the (b, c)-inverse in terms of the direct sum decomposition, the annihilator and the invertible elements. Baksalary and Trenkler (2010) introduced the notion of the core inverse as an option to the group inverse and gave its properties. Mary (2011) studied generalized inverses on semigroups by means of green's relations. The author first defined an inverse along an element and studied its properties. Penrose et al. (1955) described a generalized inverse of a nonsingular matrix as a unique solution for some operator equations. Rakic et al. (2014) showed that the core generalized inverses are closely related and also gave several characterizations of these inverses. The reverse order of Moore-Penrose inverse, has been investigated by several authors among them: Israel and Greville (2003), Djordjevic and Dijana (2011). The authors gave conditions which imply . Brock (1990) gave a characterization of an EP operator in Hilbert spaces. In particular, the author showed equivalent conditions for a bounded linear operator with closed range. Koliha (2000) gave different conditions which imply that an operator is an EP operator. The author specifically gave equations which imply each other for an upper semi-Fredholm operator. That is, for an upper semi-Fredholm operator T, then, , , implies the other.
Several authors have discussed results on perturbation of operators in Hilbert spaces and Banach spaces among them: Chen and Xue (1997), Chen et al. (1996), Ding (2003, Ding and Huang (1997), Stewart (1977), Wei (2003), Wei and Chen (2001), Zhou and Wang (2007), Wei and Ding (2001) came up with a detailed formula for the generalized inverse of the perturbed operator under some conditions. Deng and Wei (2010) generalised the result of Wei and Ding (2001) under different conditions. Shani and Sivakumar (2013) discussed rank-one perturbations of closed range operators and obtained the Moore-Penrose inverse of the operators. Kulkarni and Ramesh (2015) gave an equation for Moore-Penrose inverse of a perturbed linear operator as where S is a perturbation of T and satisfies some given conditions.
They also gave the relation between the range of the operators. That is, conditions under which the closed range of T implies closeness of range of T+S. We contribute to this study by showing that the Moore-Penrose inverse of an EP operator can be the usual inverse of the operator under the given conditions. Under perturbation of linear operators, we give the Moore-Penrose inverse of where with closed ranges and S under some given conditions distinct from the ones used by Kulkarni and Ramesh (2015). We come up with corollaries relating to these theorems and also show that is closed.

NOTATIONS AND TERMINOLOGY
We will use H to denote a Hilbert space, of linear operators with closed range on H and B(H) the space of bounded linear operators on . The range of will be denoted by and its closure, its null space by ) and the orthogonal complement of its null space by . If has dense range then . If two operators T and S commute, then . We denote the commutants of T by ..

is bounded from below if for a scalar
we have for all x in H. Given an operator with closed range, we define the generalized inverse of T as the operator

Satisfying
. Also there exists a unique operator called the Moore-Penrose inverse of T which is a unique operator satisfying the following four conditions: such that is an orthogonal projection on and , We use W(T) for numerical range of T. The Drazin inverse of T is a unique element satisfying the following conditions: for some nonnegative integer . An operator T is said to have a spectral idempotent at 0 if is quasinilpotent, is invertible. An operator T is: (1) simply polar if where (2) upper semi-Fredholm if its range is closed and either its kernel or codimension of R(T) is finite.

Theorem 1
Let with closed range and T its generalized inverse. If: T is one to one, then T'T=I.

Proof
(i) The generalized inverse of satisfies . This implies and . Since T is one-to-one, then . Thus .
. If T has a closed range that is dense in H, then T its onto. This implies that Thus, implying . Since , then T≠ 0 and hence, . Implying.
. We note that since , , then is defined on H. It is known that an operator T has a right inverse if R(T)=H.

Corollary 1
Let with closed range and be the Moore-Penrose inverse of T. If T is a quasiaffinity, then .
Proof being a quasiaffinity implies that it has a dense range and is one-to-one. From Theorem 1, . Since is a unique generalized inverse of then

Remark 1
In Corollary 1, we have deduced that for the Mwanzia et al. 7 case of a quasiaffinity. In Theorem 2, we relax the condition of quasiaffinity in Corollary 1 to operator with either dense range or injective under the proviso that T is an EP operator.

Theorem 2
If with closed range is an EP operator and its Moore-Penrose inverse, then in each of these cases: .

Proof
The  Brock (1990) gave a characterization of EP operator as follows.

Corollary A (Brock (1990))
The following statements are equivalent for with a closed range.

Corollary 2
If it is either one-to-one or has a dense range, then in each of the cases listed:

Proof
From Corollary A above, each of conditions (i) -(iii) implies T is an EP operator. Thus, the proof of Theorem 2 carries through.

Corollary B (Koliha (2000))
The following statements are equivalent for T, an upper semi-Fredholm operator on H.

Remark 2
In Corollary 3, we use the result of Koliha (2000) in Corollary B which uses another special inverse called the Drazin inverse and the spectral idempotent of at 0 to show that the Moore-Penrose inverse of an operator is the same as inverse of an operator under some conditions.

Corollary 3
If is an upper semi-Fredholm operator on H with a closed range and either 0 or . Then in each of the cases: (i) T is one to one. (ii) .

Proof
From Corollary B, T is an EP operator hence the proof of Theorem 2 carries through.

Theorem C (Wong ((1986))
if and only if its Moore-Penrose inverse is a polynomial of T provided that H has a finite dimension.

Remark 3
From Theorem C we note that every subspace of a finite dimensional space is closed hence if where H is a finite dimensional space, then it has a closed range and if its range is dense in H, then it's surjective. Again, if T is one to one, then it follows that T is invertible. Hence,

Theorem D (Khalagai and Sheth (1987))
If satisfy , then in each of the cases listed.
(i) for some positive integer m. (ii) T is normal and either or (iii) T is normal and either or .

Corollary 4
If has a closed range and it's either one-to-one or , then in each of the statements listed.

Proof
From Theorem D, each of the conditions implies that is an EP operator. Since is either injective or has closed range which is dense in H, then the proof of Theorem 2 carries through.

Lemma E (Anderson [2011])
If T is bounded from below, then: and R(T) is closed.

Remark 4
From Lemma E, it is worth noting that if an operator is bounded from below then the operator is one-to-one and has a closed range. Also, if it is normal with closed range, it is an EP operator.

Corollary 5
If is normal and bounded from below, then .

Proof
By Lemma E, T is one to one with a closed range and if normal, then it is an EP operator. Thus, the proof of Theorem 2 carries through.

Proposition F (Israel and Greville (2003))
The statements that follow implies the other for with dense domain.
is closed

Corollary 6
If is a densely defined normal operator with then if either R or are closed or is bounded.

Proof
The aforementioned conditions imply that T has a closed range and if its normal, then it is an EP operator. Again, if , then the proof of Theorem 2 carries through.

Theorem 3
If R-quasi -EP operator bounded from below, then .

Proof
If is R-quasi-EP operator then from definition T commutes with and . This implies . Also, and . Since T is linear, then and . Since T bounded from below by Lemma E, implying and .

Corollary 7
If is a partial isometry which is bounded from Mwanzia et al. 9 below and is quasinormal, then T is unitary.

Proof
From definition of a partial isometry . If is quasinormal, we have Substituting we have Thus, the proof of Theorem 3 carries through.

Theorem 4
If is L-quasi -EP operator with a closed and dense range, then . In view of Theorem 4, we extend the result of Mwanzia et al. (2021) in Corollary G for the case of partial isometries.

Corollary G (Mwanzia et al. (2021))
Let be partial isometry such that . If T is quasinormal, then is unitary.

Corollary 8
Let be a partial isometry with closed range. If T is quasinormal with , then .

Proof
From definition of partial isometry, and . Substituting we have, . Since T has a closed range with , the proof of Theorem 4 carries through.

Remark 5
In the sequel, we study the relation between the range and null spaces of operators and derive the Moore-Penrose inverse of (T+S) where and or with being a perturbation of . Frigyes and Bela (1955) gave the following result which helps in showing the relation between the ranges of operators in Hilbert space.

Proposition H (Frigyes and Bela (1955))
If are densely defined and , then

Remark 6
From results of Theorem 5, and under the said conditions.

Corollary 9
If is densely defined and is bounded from below, then .

Proof
If then by Theorem 5, is a projection on . That is . Thus, implying and Since is bounded from below then by Lemma E, P has a closed range and . Thus, and . This implies is an orthogonal projection on

If
) and then ( is bijective.

Corollary 10
Let where P and Q are densely defined operators with closed ranges. If is surjective and satisfies , and hen .
( Thus . Since is an orthogonal projection on , then we have: . . .

Thus
This implies: (ii) Lastly, from Proposition H, ) is densely defined operator then (iii) Equations (i), (ii) and (iii) justify the results.

Let
where P and Q are densely defined operators with closed ranges. If Q is surjective and satisfies , and , then .

From
Theorem 7, we have . Since is onto then, and multiplying both sides from the left by we have, . Following the steps of Theorem 7, satisfies the properties of Moore-Penrose inverse.

Let
where P and Q are densely defined operators with closed ranges. If P is bounded from below, Q is surjective and satisfies , and then: .
Since and are injective then multiplying both sides of the preceding equation by we have .
Following the steps of theorem 7, we have: satisfies the properties of Moore-Penrose inverse.