Abstract

New characterizations of partial isometries and EP elements in Banach algebra are presented.

1. Introduction

Generalized inverses of matrices have important roles in theoretical and numerical methods of linear algebra. The most significant fact is that we can use generalized inverses of matrices, in the case when ordinary inverses do not exist, in order to solve some matrix equations. Similar reasoning can be applied to linear (bounded or unbounded) operators on Banach and Hilbert spaces. Then, it is interesting to consider generalized inverses of elements in Banach and -algebras, or more general, in rings with or without involution.

Let be a complex unital Banach algebra. An element is generalized (or inner) invertible, if there exists some such that holds. In this case is a generalized (or inner) inverse of . If , then take to obtain the following: and . Such is called a reflexive (or normalized) generalized inverse of . Finally, if , then and are idempotents. In the case of the -algebra, we can require that and are Hermitian. We arrive at the definition of the Moore-Penrose inverse in -algebras.

Definition 1.1. Let be a unital -algebra. An element is Moore-Penrose invertible if there exists some such that hold. In this case is the Moore-Penrose inverse of , usually denoted by .

If is Moore-Penrose invertible in a -algebra, then is unique, and the notation is justified.

More general, if is a unital Banach algebra, we have the following definition of Hermitian elements.

Definition 1.2. An element is said to be Hermitian if for all .

The set of all Hermitian elements of will be denoted by . Now, it is natural to consider the following definition of the Moore-Penrose inverse in Banach algebras ([1, 2]).

Definition 1.3. Let be a complex unital Banach algebra and . If there exists such that then the element is the Moore-Penrose inverse of , and it will be denoted by .

The Moore-Penrose inverse of is unique in the case when it exists.

Although the Moore-Penrose inverse has many nice approximation properties, the equality does not hold in general. Hence, it is interesting to distinguish such elements.

Definition 1.4. An element of a unital Banach algebra is said to be EP if there exists and .

The name EP will be explained latter. There is another kind of a generalized inverse that commutes with the starting element.

Definition 1.5. Let be a unital Banach algebra and . An element is the group inverse of , if the following conditions are satisfied:

The group inverse of will be denoted by which is uniquely determined (in the case when it exists).

Let be a Banach space and the Banach algebra of all linear bounded operators on . In addition, if , then and stand for the null space and the range of , respectively. The ascent of is defined as , and the descent of is defined as . In both cases the infimum of the empty set is equal to . If and , then .

Necessary and sufficient for to exist is the fact that . If is a closed range operator, then exists if and only if (see [3]). Obviously, and , for every nonnegative integer . Now the name follows: EP means “equal projections” on parallel to for all positive integers .

Finally, if is an EP element, then clearly exists. In fact, . On the other hand, if exists, then necessary and sufficient for to be EP is that is a Hermitian element of . Furthermore, in this case .

The left multiplication by is the mapping , which is defined as for all . Observe that, for , and that implies . If is both Moore-Penrose and group invertible, then and in the Banach algebra . According to [4, Remark 12], a necessary and sufficient condition for to be EP is that is EP.

A similar statement can be proved if we consider instead of , where the mapping is the right multiplication by , and defined as for all .

Let . Recall that according to [5, Hilfssatz 2(c)], for each there exist necessary unique Hermitian elements such that . As a result, the operation is well defined. Note that is not an involution, in particular does not in general coincide with . However, if and for every , with , , then is a -algebra whose involution is the just considered operation, see [5].

An element satisfying is called normal. If (), it is easy to see that is normal if and only if . An element satisfying is called a partial isometry [6].

Note that necessary and sufficient for to belong to is that . Therefore, is normal if and only if is normal. Observe that if then and .

Theorem 1.6 (see [7]). Let be a Banach space and consider such that exists and . Then the following statements hold: (i) if and only if ,(ii) if and only if . In addition, if the conditions of statements (i) and (ii) are satisfied, then is an EP operator.

Notice that is equivalent to , by . The condition is equivalent to , because [7]. Hence, by Theorem 1.6, we deduce the following.

Corollary 1.7. Let be a Banach space and consider such that exists and . Then the following statements hold:(i) if and only if ,(ii) if and only if .

There are many papers characterizing EP elements, partial isometries, or related classes (such as normal elements). See, for example [4, 723]. Properties of the Moore-Penrose inverse in various structures can be found in [1, 2, 2428].

In [8] Baksalary et al. used an elegant representation of complex matrices to explore various classes of matrices, such as partial isometries and EP. Inspired by [8], in paper [21] we use a different approach, exploiting the structure of rings with involution to investigate partial isometries and EP elements.

In this paper we characterize elements in Banach algebras which are EP and partial isometries.

2. Partial Isometry and EP Elements

Before the main theorem, we give some characterizations of partial isometries in Banach algebras in the following theorem.

Theorem 2.1. Let be a unital Banach algebra and consider such that and exist. Then the following statements are equivalent:(i) is a partial isometry,(ii), (iii).

Proof. (i) (ii): If , then
(ii) (i): From , it follows that
(i) (iii): This part can be proved similarly.

In the following result we present equivalent conditions for an bounded linear operator on Banach space to be a partial isometry and EP. Compare with [21, Theorem 2.3] where we studied necessary and sufficient conditions for an element of a ring with involution to be a partial isometry and EP.

Theorem 2.2. Let be a Banach space and consider such that and exist and . Then the following statements are equivalent: (i) is a partial isometry and EP,(ii) is a partial isometry and normal,(iii), (iv) and ,(v) and ,(vi) and ,(vii) and ,(viii), (ix), (x) and ,(xi) and ,(xii) and ,(xiii) and ,(xiv) and ,(xv) and ,(xvi) and ,(xvii) and ,(xviii) and ,(xix) and .

Proof. (i) (ii): If is EP, then and, by Corollary 1.7, . Since is a partial isometry, we have Thus, and imply is normal, by [7, Theorem 3.4(i)].
(ii) (iii): The condition is normal and [7, Theorem 3.4(vii)] imply . Because is a partial isometry, we have
(iii) (i): Using the equality , we get: By [7, Theorem 3.3], is normal gives is EP. The condition (i) is satisfied.
(ii) (iv): By [7, Theorem 3.4(ii)], is normal gives and . Now Since is normal implies is EP, then .
(iv) (vi): Assume that and . Then implying and . By [7, Theorem 3.4(ii)], is normal and, by [7, Theorem 3.3], is EP. Therefore, .
(vi) (ii): Let and . Then which yields that is a partial isometry and normal by [7, Theorem 3.4(x)].
(ii) (v) (vii) (ii): These implications can be proved in the same way as (ii) (iv) (vi) (ii) using [7, Theorem 3.4(i)] and [7, Theorem 3.4(ix)].
(i) (viii): From (i) follows (iii) and is EP which gives (viii).
(viii) (xi): Suppose that . Then Hence, is Hermitian and is EP. Now condition (xi) is satisfied by
(xi) (xvi): The assumptions and give, by Corollary 1.7,
(xvi) (xiv): Multiplying by from the right side, we get . Hence, satisfies condition (xiv).
(xiv) (xii): If and , then we see that . Thus, by (vii) (i), we get that is EP, and
(xii) (vii): Applying and , we obtain the condition (vii):
(i) (ix) (x) (xvii): Similarly as (i) (viii) (xi) (xvi).
(xvii) (vi): Suppose that and . Then and the condition (vi) is satisfied.
(xiii) (xi): Multiplying the equality by from the right side, we obtain . So, we deduce that condition (xi) holds.
(xi) (xiii): By (xi), we have that is EP and condition (xiii) is satisfied.
(xv) (i): Let and . Now, we observe that Therefore, is a partial isometry and EP.
(i) (xv): The hypothesis is EP gives and, because (i) implies (iii),
(xviii) (iii): By the assumption and , we obtain .
(iii) (xviii): From , we get and is EP implying .
(iii) (xix): Analogy as (iii) (xviii).

Now, we return to a general case, that is, is a complex unital Banach algebra, and is both Moore-Penrose and group invertible.

Corollary 2.3. Theorem 2.2 holds if we change for an arbitrary complex Banach algebra , and one changes by an such that and exist.

Proof. If satisfies the hypothesis of this theorem, then satisfies the hypothesis of Theorem 2.2. Now, if any one of statements (i)–(xix) holds for , then the same statement holds for . Therefore, is a partial isometry and EP in . By [4, Remark 12], it follows that is EP in . It is well-known that if then and . Since is a partial isometry, , that is, . So, we deduce that and a partial isometry in .
A similar statement can be proved if we consider instead of .

The cancellation property and the identity are important when we proved the equivalent statements characterizing the condition of being a partial isometry and EP in a ring with involution in [21]. Since is not in general an involution, and it is not clear if the cancellation property holds for Moore-Penrose invertible elements of , in most statements of Theorem 2.2 an additional condition needs to be considered.

Acknowledgment

The authors are supported by the Ministry of Science and Technological Development, Serbia, Grant no. 174007.