Moore-Penrose inverse and doubly commuting elements in C*-algebras

Enrico Boasso

This work is dedicated to the memory of Professor Angel R. Larotonda, Pucho for all who knew him.

This research was supported by UBACYT and CONICET

Abstract: In this work it is proved that the Moore-Penrose inverse of the product of -doubly commuting regular -algebra elements obeys the so-called reverse order law. Conversely, conditions regarding the reverse order law of the Moore-Penrose inverse are given in order to characterize when -regular elements doubly commute. Furthermore, applications of the main results of this article to normal -algebra elements, to Hilbert space operators and to Calkin algebras will be considered.

Key words and phrases. Generalized inverse, Moore-Penrose inverse, and doubly commuting elements in a -algebra.
2000 Mathematics Subject Classification. 46L05, 47A

1. Introduction

Consider an unitary ring . An element will be said regular if it has a generalized inverse in A, that is if there exists such that

A generalized inverse is also termed a pseudo inverse.

Note that if is a regular element of and is a generalized inverse of , then and are idempotents of A, that is and .

Given a regular element, a generalized inverse of will be called normalized, if is regular and is a pseudo inverse of , equivalently if

Recall that if is a generalized inverse of , then is a normalized pseudo inverse of .

Next suppose that is a regular element and is a normalized generalized inverse of . In the presence of an involution , it is possible to enquire if the idempotents and are self-adjoint, that is whether and . In this case is called the Moore-Penrose inverse of , see [16] where this concept was introduced.

In [10] it was proved that each regular element in a -algebra has a uniquely determined Moore-Penrose inverse. The Moore-Penrose inverse of will be denote by . Therefore, the Moore-Penrose inverse of a regular element is the unique element that satisfy the following equations:

According to the uniqueness of the notion under consideration, if has a Moore-Penrose inverse, then also has a Moore-Penrose inverse and

Moreover, according to the above equations, if is a regular element, then also is and

For other properties regarding Moore-Penrose inverses in -algebras, see the works [10], [11], [14] and [16].

On the other hand, an -tuple of elements in a -algebra A will be called doubly commuting, if and , for all , , . For instance, necessary and sufficient for to be doubly commuting is that is a normal element in .

Doubly commuting operators have been studied in very different contexts, to mention only some of the most relevant works, see for example [1]-[4], [6]-[8] and [12]. In this article, doubly commuting tuples of regular -algebra elements will be consider. The main objective of this work consists in the study of the relationship between such tuples and the Moore-Penrose inverse of the product of the elements in the tuple.

In fact, given an -tuple of regular elements in a -algebra A, if the -tuple is doubly commuting, then is regular and

in particular, complies with the so-called reverse order law for the Moore-Penrose inverse. Moreover, if is such a tuple, then and are doubly commuting pairs, where , , . Consequently, and are regular and

Conversely, if is an -tuple of regular elements in a -algebra A such that the above identities are verified by and , for all , , , then is a doubly commuting -tuple of elements in the -algebra .

It is worth noticing that this characterization consists in an extension to the objects under consideration of the sufficient conditon given by J.J. Koliha in [13; 2.13] for the product of two regular elements to be Moore-Penrose invertible.

In section 2 it will be proved the aforementioned characterization. In section three, on the other hand, some applications will be developed. In fact, three sorts of objects will be considered, namely, tuples of commuting regular normal -algebra elements, tuples of doubly commuting regular Hilbert space operators, and tuples of almost doubly commuting regular Hilbert space operators (that is doubly commuting tuples of regular elements in Calkin algebras).

This article is dedicated to the memory of Professor Angel R. Larotonda, who unfortunately and unexpectedly died on January 2th 2005. Although it is not necessary to comment Professor Larotonda's work as mathematician, for his scientific publications consist in a set of achievements which speak for themselves, a few words about the man deserve to be said. The author knew Professor Larotonda for more than twenty years. During uncountable conversations shared with Professor Larotonda, this researcher always showed his condition of sensible, civilized and cultivated human being, three characteristics that seem to be far from being widespread in this time and in any time.

Acknowledgements. The author wishes to express his indebtedness to the organizers of this volume, especially to Professor G. Corach, for have invited the author to contribute with this homage.

2. Main Results

In this section the relationship between the Moore-Penrose inverse and doubly commuting tuples of regular -algebra elements will be studied. In fact, the characterization described in the previous section will be proved. In first place, a property of doubly commuting tuples is discussed.

Remark 2.1. Let be an -tuple of elements of a -algebra . Consider a permutation, and define an -tuple in such a way that is either or . Then, it is easy to prove that is doubly commuting if and only if is.

Furthermore, note that the following facts are equivalents:

i) is a doubly commuting -tuple of elements of ,

ii) for each , , , is a pair of doubly commuting elements of .

Proposition 2.2. Let be an -tuple of regular elements in a -algebra . Consider the -tuple . Then, is doubly commuting if and only if is.

Proof. According to Remark 2.1, it is enough to prove that a pair of regular elements is doubly commuting if and only if is.

Suppose that is a doubly commuting pair of regular elements of . Then, according to Theorem 5 of [10],

Moreover, since according to Remark 2.1 is a doubly commuting pair, it is clear that

Consequently, is a doubly commuting pair.

However, according to Remark 2.1, is a doubly commuting pair, and thanks to what has been proved, is a doubly commuting pair. Therefore, according again to Remark 2.1, is a doubly commuting pair.

Conversely, if is a doubly commuting pair, since and , according to the first part of the proof, is a doubly commuting pair

Note that in [13] Theorem 5 of [10] was proved using the Drazin inverse, see [13; 2.12]. Therefore, Proposition 2.2 can also be derived using the Drazin inverse, see [13; 2.12] and [13; 2.13].

Remark 2.3. Let be an -tuple of regular elements in a -algebra . Consider a permutation, and define an -tuple in the following way. Given , is either , , or . Then, according to Remark 2.1 and to Proposition 2.2, is doubly commuting if and only if is.

Furthermore, according to Remark 2.1 and to Proposition 2.2, the following facts are equivalent:

i) is an -tuple of doubly commuting regular elements of ,

ii) for each , , , is a pair of doubly commuting regular elements of .

Next follows the first part of our characterization. In fact, in the following theorem it will be proved that the product of the elements in a doubly commuting tuple of regular elements satisfy the so-called reverse order law for the Moore-Penrose inverse.

Theorem 2.4. Let be an -tuple of doubly commuting regular elements of a -algebra . Then, is regular and

Proof. Consider and two regular elements of such that the pair is doubly commuting. According to [13; 2.13], is regular and

Next consider a doubly commuting tuple of regular elements of . Suppose that is regular and

In order to conclude the proof of the reverse order law for the Moore- Penrose inverse, it is enough to prove that is regular and

Consider . According to Remark 2.1, is a doubly commuting pair for , which implies that is a doubly commuting pair of regular elements. Therefore, according to the part of the theorem that has been proved, is regular, and

The last identity is a consequence of [13; 2.13] or Proposition 2.2.

Remark 2.5. Let be an -tuple of doubly commuting regular elements in a -algebra . Consider a permutation and consider an -tuple , where is either , , or , . Then, is an -tuple of doubly commuting regular elements of . Consequently, according to [13; 2.13] or Proposition 2.2 and to Theorem 2.4, is regular and

Next consider two permutations . Let be the permutation , that is . Next associate to an -tuple as in the previous paragraph. Then, it is possible to associate to an -tuple such that , for . Consequently, according to [13; 2.13] or Proposition 2.2 and to Theorem 2.4,

For instance, if is a pair of doubly commuting regular elements of the -algebra , the following identities hold.

Consequently,

Finally, recall that according to Remark 2.1, is a doubly commuting -tuple of regular elements of if and only if is a doubly commuting pair of regular elements of , for , , . Therefore, for each such pair the above identities hold.

Next follows the converse of Theorem 2.4.

Theorem 2.6. Consider an -tuple of regular elements in a -algebra . Suppose that for all , , , , , and are regular and comply with the following identities:

Then is a doubly commuting -tuple of regular elements of .

Proof. According to Remark 2.1, it is enough to prove that is doubly commuting for all , , . However, since and are regular and , it is clear that . Similarly, . Therefore, is a doubly commuting pair, , , .

Remark 2.7. In [13] sufficient conditions for a product to be Moore-Penrose inversible were given. Actually, thanks to the Theorem 2.6 it is now possible to state the following characterization. Let and be two regular elements in a -algebra A. If is a doubly commuting pair, then , , and are regular and

On the other hand, if and are two regular elements such that , , and are regular and they comply with the above identities, then is a doubly commuting pair. Furthermore, in this case and comply the reverse order law for the Moore-Penrose inverse.

3. Some applications

In this section several applications of the main results of this work will be considered. In first place, commuting tuples of normal elements in a -algebra will be studied.

Remark 3.1. Let be a doubly commuting -tuple of elements in a -algebra . Let be an -tuple of nonnegative entire numbers and define the -tuple

Then, it is not difficult to prove that is a doubly commuting -tuple of elements of .

On the other hand, if is a regular normal element of , then is regular and , for all .

In fact, since is regular, (-times) is a doubly commuting -tuple of regular elements of . Therefore, according to Theorem 2.4, is regular and .

In the following theorem the characterization of the previous section will be applied to commuting tuples of regular normal elements. This result is an extension of [13; 2.14] to the case under consideration.

Theorem 3.2. Let be a commuting -tuple of regular normal elements in a -algebra . Let be an -tuple of nonnegative entire numbers. Then is regular and

Proof. First of all recall that according to the Flugede-Putnam Theorem, [5; IX, 6.7] or [9; 9.6.7], is a doubly commuting tuple.

Next consider, as in Remark 3.1, the -tuple

Then, according to Remark 3.1, is a doubly commuting tuple of regular elements of . Consequently, according to Theorem 2.4, Remark 2.5 and Remark 3.1 again, the proof of the Theorem is concluded.

Remark 3.3. Let be a commuting -tuple of regular normal elements in a -algebra . Consider, as in Remark 2.3, a permutation, and define an -tuple , where given , is either , , or . Next consider the -tuple , where is an -tuple of nonnegative integers. Then, according to Proposition 2.2, Remark 3.1, Theorem 3.2 and Theorem 10 of [10], is an -tuple of doubly commuting regular normal elements of . Therefore, according to Theorem 3.2, is regular and

For example, if is a pair of commuting regular normal elements in a -algebra , then

are regular elements of , for and . Furthermore, their Moore-Pensrose inverses can be calculated according to the first part of the present Remark.

Next the main results of the present work will be applied to -tuples of regular Hilbert space operators.

Let be a Hilbert space and consider , the -algebra of all bounded and linear maps defined in . Recall that is regular as an operator if and only if is a regular element of . Moreover, necessary and sufficient for to be a regular operator is the fact that the range of , , is a closed subspace of , see for example [9; 3.8]. Note that in this case the Moore-Penrose inverse can be described in a direct way.

In fact, consider a bounded Hilbert space operator with closed range. Define as follows:

where , that is the restriction to of . Then, it is not diffuclt to prove that

that is, is the Moore-Penrose inverse of .

On the other hand, an -tuple of continuous linear maps defined in , , is doubly commuting as operators if and only if it is doubly commuting as elements of . In the following theorems the relationship between doubly commuting tuples of regular operators and the Moore-Penrose inverse will be studied.

Theorem 3.4. Let be an -tuple of regular Hilbert space operators. Then, if the -tuple is doubly commuting, is a regular operator and

Conversely, if for all , , , , , and are regular operators which comply with the following identities:

then, is a doubly commuting -tuple of regular Hilbert space operators.

Proof. It is a consequence of Theorems 2.4 and 2.6.

Theorem 3.5. Let be an -tuple of commuting regular normal operators defined in a Hilbert space . Next consider the -tuple , where is an -tuple of nonnegative integers. Then, is regular and

Proof. It is a consequence of Theorem 3.2.

The last application of the results of the previous section concerns Calkin algebras.

Recall that if is a Hilbert space and denotes the closed ideal of all compact operators defined in , then the Calkin algebra of , , is a -algebra. Moreover, tha natural quotient map is a -algebra morphism, see [5; 4] or [15; 4.1.16].

Furthermore, note that if is a regular operator, then is a regular element of . In addition, it is not difficult to prove that if is the Moore-Penrose inverse of a regular operator , then .

On the other hand, recall that an -tuple is said to be almost commuting (resp. almost doubly commuting), if is a commuting tuple (resp. a doubly commuting tuple) in the -algebra , that is if (resp. and ) belong to , for , , , see [6], [7] and [8]. In the following theorems the relationship between -tuples of almost doubly commuting regular operators and the Moore-Penrose inverse will be studied.

Theorem 3.6. Let be an -tuple of regular operators defined in the Hilbert space . Suppose that is regular. Then, if is an almost doubly commuting tuple of operators,

Conversely, if for all , , , , , and are regular operators such that

then is an almost doubly commuting -tuple of regular Hilbert space operators.

Proof. It is a consequence of Theorem 3.4 and the above recalled facts regarding Calkin algebras.

Theorem 3.7. Let be an -tuple of almost commuting regular normal operators defined in a Hilbert space . Next consider the -tuple , where is an -tuple of nonnegative integers, and suppose that is regular . Then, is an -tuple of almost doubly commuting regular operators defined in , and

Proof. It is a consequence of Theorems 3.2, 3.5 and the recalled facts regarding Calkin algebras.

Remark 3.8. It is worth noticing that Theorems 3.2 and 3.4-3.7 can be extended to other tuples of -algebra elements and of Hilbert space operators respectively in the same way that it was done in Remarks 2.3 and 2.5, that is considering permutations and new tuples defined by Moore-Penrose inverses and adjoints.

References

[1]    E. Albrecht and M. Ptak, Invariant subspaces for doubly commuting contractions with rich Taylor spectrum, J. Operator Theory 40 (1998), 373-384.         [ Links ]

[2]    M. Cho, R. Curto, T. Huruya and W. Zelazko, Cartesian form of Putnam's inequality for doubly commuting hyponormal -tuples, Indiana Univ. Math. J. 49 (2000), 1437-1448.         [ Links ]

[3]    M. Cho and A. T. Dash, On the joint spectra of doubly commuting -tuples of seminormal operators, Glasgow Math. J. 26 (1985), 47-50.         [ Links ]

[4]    M. Cho, B. P. Duggal and W. Y. Lee, Putnam's inequality of doubly commuting -tuples for log-hyponormal operators, Math. Proc. Royal Ir. Acad. Sect. A 100 A (2000), 163-169.         [ Links ]

[5]    J. B. Conway, A Course in Functional Analysis, Springer Verlag, New York, Berlin, Heidelberg, Tokio, 1985.         [ Links ]

[6]    R. Curto, On the connectedness of invertible -tuples, Indiana Univ. Math. J. 29 (1980), 393-406.         [ Links ]

[7]    R. Curto, Fredholm and invertible -tuples of operators. The deformation problem, Trans. Amer. Math. Soc. 266 (1981), 129-159.         [ Links ]

[8]    R. Curto, Spectral inclusion for doubly commuting subnormal -tuples, Proc. Amer. Math. Soc. 83 (1981), 730-734.         [ Links ]

[9]    R. Harte, Invertibility and Singularity for Bounded Linear Operators, Marcel Dekker, New York and Basel, 1988.         [ Links ]

[10]    R. Harte and M. Mbekhta, On generalized inverses in -algebras, Studia Math. 103 (1992), 71-77.         [ Links ]

[11]    R. Harte and M. Mbekhta, On generalized inverses in -algebras II, Studia Math. 106 (1993), 129-138.         [ Links ]

[12]    I. H. Jeon, On joint essential spectra of doubly commuting -tuples of -hyponormal operators, Glasgow Math. J. 40 (1998), 353-358.         [ Links ]

[13]    J. J. Koliha, The Drazin and Moore-Penrose inverse in -algebras, Math. Proc. Royal Ir. Acad. Sect. A 99 A (1999) no. 1, 17-27.         [ Links ]

[14]    M. Mbekhta, Conorme et inverse généralisé dans les -algèbres, Canadian Math. Bull. 35 (1992), 512-522.         [ Links ]

[15]    G. J. Murphy, -Algebras and Operator Theory, Academic Press, Boston, (1990).         [ Links ]

[16]    R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc. 51 (1955), 406-413.         [ Links ]

Enrico Boasso
Departamento de Matemática,
Facultad de Ciencias Exactas y Naturales,
Universidad de Buenos Aires,
Ciudad Universitaria - Pab. I,
(1428) Buenos Aires, Argentina.
eboasso@dm.uba.ar

Recibido: 17 de agosto de 2005
Aceptado: 7 de agosto de 2006