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Revista de la Unión Matemática Argentina
versão impressa ISSN 0041-6932versão On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.46 n.2 Bahía Blanca jul./dez. 2005
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 = aba.](/img/revistas/ruma/v46n2/2a0313x.png)
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
![a = aba, b = bab.](/img/revistas/ruma/v46n2/2a0328x.png)
![b](/img/revistas/ruma/v46n2/2a0329x.png)
![a](/img/revistas/ruma/v46n2/2a0330x.png)
![c = bab](/img/revistas/ruma/v46n2/2a0331x.png)
![a](/img/revistas/ruma/v46n2/2a0332x.png)
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:
![a = aa†a, a† = a†aa †, (a†a)* = a†a, (aa†)* = aa†.](/img/revistas/ruma/v46n2/2a0349x.png)
![a](/img/revistas/ruma/v46n2/2a0350x.png)
![a *](/img/revistas/ruma/v46n2/2a0351x.png)
![* † † * (a ) = (a ) .](/img/revistas/ruma/v46n2/2a0352x.png)
![a](/img/revistas/ruma/v46n2/2a0353x.png)
![a†](/img/revistas/ruma/v46n2/2a0354x.png)
![(a †)† = a.](/img/revistas/ruma/v46n2/2a0355x.png)
![* C](/img/revistas/ruma/v46n2/2a0356x.png)
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
![n n-1 n ∏ † ∏ † ∏ † ( ai) = an- i = ai, i=1 i=0 i=1](/img/revistas/ruma/v46n2/2a0375x.png)
![∏n i=1 ai](/img/revistas/ruma/v46n2/2a0376x.png)
![a](/img/revistas/ruma/v46n2/2a0377x.png)
![(ai,aj)](/img/revistas/ruma/v46n2/2a0378x.png)
![(a*i,aj)](/img/revistas/ruma/v46n2/2a0379x.png)
![i](/img/revistas/ruma/v46n2/2a0380x.png)
![j ∈ [[1,n]]](/img/revistas/ruma/v46n2/2a0381x.png)
![i ⁄= j](/img/revistas/ruma/v46n2/2a0382x.png)
![aiaj](/img/revistas/ruma/v46n2/2a0383x.png)
![a*iaj](/img/revistas/ruma/v46n2/2a0384x.png)
![† † * † *† (aiaj) = (ajai), (aiaj) = (aja i) .](/img/revistas/ruma/v46n2/2a0385x.png)
![a = (a1,...,an)](/img/revistas/ruma/v46n2/2a0386x.png)
![n](/img/revistas/ruma/v46n2/2a0387x.png)
![C *](/img/revistas/ruma/v46n2/2a0388x.png)
![(ai,aj)](/img/revistas/ruma/v46n2/2a0389x.png)
![(a*,aj) i](/img/revistas/ruma/v46n2/2a0390x.png)
![i](/img/revistas/ruma/v46n2/2a0391x.png)
![j ∈ [[1, n]]](/img/revistas/ruma/v46n2/2a0392x.png)
![i ⁄= j](/img/revistas/ruma/v46n2/2a0393x.png)
![a](/img/revistas/ruma/v46n2/2a0394x.png)
![n](/img/revistas/ruma/v46n2/2a0395x.png)
![C*](/img/revistas/ruma/v46n2/2a0396x.png)
![A](/img/revistas/ruma/v46n2/2a0397x.png)
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],
![† † bc = cb .](/img/revistas/ruma/v46n2/2a03132x.png)
![(b,c*)](/img/revistas/ruma/v46n2/2a03133x.png)
![b†c* = c*b†.](/img/revistas/ruma/v46n2/2a03134x.png)
![† (b ,c)](/img/revistas/ruma/v46n2/2a03135x.png)
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
![n∏ n∏- 1 ∏n ( ai)† = a†n-i = a†i. i=1 i=0 i=1](/img/revistas/ruma/v46n2/2a03171x.png)
Proof. Consider and
two regular elements of
such that the pair
is doubly commuting. According to [13; 2.13],
is regular and
![(bc)† = c†b†.](/img/revistas/ruma/v46n2/2a03177x.png)
![a = (a1,...,an, an+1)](/img/revistas/ruma/v46n2/2a03178x.png)
![A](/img/revistas/ruma/v46n2/2a03179x.png)
![∏n i=1 ai](/img/revistas/ruma/v46n2/2a03180x.png)
![n n- 1 (∏ a )† = ∏ a† . i n-i i=1 i=0](/img/revistas/ruma/v46n2/2a03181x.png)
![∏n+1 ∏n i=1 ai = ( i=1ai)an+1](/img/revistas/ruma/v46n2/2a03182x.png)
![n+1 n-1 ∏ † † ∏ † ( ai) = an+1( an-i). i=1 i=0](/img/revistas/ruma/v46n2/2a03183x.png)
![∏n d = i=1 ai](/img/revistas/ruma/v46n2/2a03184x.png)
![(an+1,aj)](/img/revistas/ruma/v46n2/2a03185x.png)
![j = 1,...,n](/img/revistas/ruma/v46n2/2a03186x.png)
![(d,an+1)](/img/revistas/ruma/v46n2/2a03187x.png)
![da = ∏n+1 a n+1 i=1 i](/img/revistas/ruma/v46n2/2a03188x.png)
![n∏+1 ∏n ( ai)† = (dan+1) † = a †n+1d† = a†n+1-i. i=1 i=0](/img/revistas/ruma/v46n2/2a03189x.png)
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
![n n- 1 n ∏ † ∏ † ∏ † ( bi) = bn-i = bi. i=1 i=0 i=1](/img/revistas/ruma/v46n2/2a03207x.png)
![π,σ : [[1,n]] → [[1,n]]](/img/revistas/ruma/v46n2/2a03208x.png)
![τ : [[1,n]] → [[1,n]]](/img/revistas/ruma/v46n2/2a03209x.png)
![τ = π -1σ](/img/revistas/ruma/v46n2/2a03210x.png)
![πτ = σ](/img/revistas/ruma/v46n2/2a03211x.png)
![π](/img/revistas/ruma/v46n2/2a03212x.png)
![n](/img/revistas/ruma/v46n2/2a03213x.png)
![b](/img/revistas/ruma/v46n2/2a03214x.png)
![σ](/img/revistas/ruma/v46n2/2a03215x.png)
![n](/img/revistas/ruma/v46n2/2a03216x.png)
![c](/img/revistas/ruma/v46n2/2a03217x.png)
![bτ(j) = cj](/img/revistas/ruma/v46n2/2a03218x.png)
![j = 1,...,n](/img/revistas/ruma/v46n2/2a03219x.png)
![∏n ∏n ∏n ∏n ( bi)† = b†= c†= ( ci)†. i=1 i=1 i i=1 i i=1](/img/revistas/ruma/v46n2/2a03220x.png)
![(a,b)](/img/revistas/ruma/v46n2/2a03221x.png)
![* C](/img/revistas/ruma/v46n2/2a03222x.png)
![A](/img/revistas/ruma/v46n2/2a03223x.png)
![pict](/img/revistas/ruma/v46n2/2a03224x.png)
Consequently,
![pict](/img/revistas/ruma/v46n2/2a03225x.png)
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:
![† † * † *† (aiaj) = (ajai), (aiaj) = (aja i) .](/img/revistas/ruma/v46n2/2a03245x.png)
![a](/img/revistas/ruma/v46n2/2a03246x.png)
![n](/img/revistas/ruma/v46n2/2a03247x.png)
![A](/img/revistas/ruma/v46n2/2a03248x.png)
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
![(ab)† = (ba)†, (a*b)† = (ba*)†.](/img/revistas/ruma/v46n2/2a03270x.png)
![a](/img/revistas/ruma/v46n2/2a03271x.png)
![b](/img/revistas/ruma/v46n2/2a03272x.png)
![ab](/img/revistas/ruma/v46n2/2a03273x.png)
![ba](/img/revistas/ruma/v46n2/2a03274x.png)
![a*b](/img/revistas/ruma/v46n2/2a03275x.png)
![ab *](/img/revistas/ruma/v46n2/2a03276x.png)
![(a,b)](/img/revistas/ruma/v46n2/2a03277x.png)
![ab](/img/revistas/ruma/v46n2/2a03278x.png)
![* a b](/img/revistas/ruma/v46n2/2a03279x.png)
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
![m1 mn aα = (a 1 ,...,an ).](/img/revistas/ruma/v46n2/2a03288x.png)
![aα](/img/revistas/ruma/v46n2/2a03289x.png)
![n](/img/revistas/ruma/v46n2/2a03290x.png)
![A](/img/revistas/ruma/v46n2/2a03291x.png)
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
![∏n ∏n ( amii)† = (a †i)mi. i=1 i=1](/img/revistas/ruma/v46n2/2a03311x.png)
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
![aα = (am11 ,...,amnn ).](/img/revistas/ruma/v46n2/2a03314x.png)
![a α](/img/revistas/ruma/v46n2/2a03315x.png)
![A](/img/revistas/ruma/v46n2/2a03316x.png)
![QED](/img/revistas/ruma/v46n2/2blackbox1.png)
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
![n n ∏ ni † ∏ † ni ( bi ) = (bi) . i=1 i=1](/img/revistas/ruma/v46n2/2a03338x.png)
![(a,b)](/img/revistas/ruma/v46n2/2a03339x.png)
![C *](/img/revistas/ruma/v46n2/2a03340x.png)
![A](/img/revistas/ruma/v46n2/2a03341x.png)
![pict](/img/revistas/ruma/v46n2/2a03342x.png)
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:
![pict](/img/revistas/ruma/v46n2/2a03360x.png)
where , that is the restriction to
of
. Then, it is not diffuclt to prove that
![T = TST, S = ST S, (T S)* = T S, (ST )* = ST,](/img/revistas/ruma/v46n2/2a03364x.png)
![S](/img/revistas/ruma/v46n2/2a03365x.png)
![T](/img/revistas/ruma/v46n2/2a03366x.png)
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
![∏n † n∏-1 † ∏n † ( Ti) = Tn-i = Ti . i=1 i=0 i=1](/img/revistas/ruma/v46n2/2a03376x.png)
![i](/img/revistas/ruma/v46n2/2a03377x.png)
![j = 1,...,n](/img/revistas/ruma/v46n2/2a03378x.png)
![i ⁄= j](/img/revistas/ruma/v46n2/2a03379x.png)
![TiTj](/img/revistas/ruma/v46n2/2a03380x.png)
![TjTi](/img/revistas/ruma/v46n2/2a03381x.png)
![T*i Tj](/img/revistas/ruma/v46n2/2a03382x.png)
![TjT*i](/img/revistas/ruma/v46n2/2a03383x.png)
![(TiTj)† = (TjTi)†, (T *i Tj)† = (TjT*i )†,](/img/revistas/ruma/v46n2/2a03384x.png)
![T](/img/revistas/ruma/v46n2/2a03385x.png)
![n](/img/revistas/ruma/v46n2/2a03386x.png)
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
![n n-1 n ∏ ni † ∏ † ni ∏ † ni ( Ti ) = (Tn-i) = (T i ) . i=1 i=0 i=1](/img/revistas/ruma/v46n2/2a03395x.png)
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,
![∏n ∏n ( Ti)† - T †i ∈ K(H). i=1 i=1](/img/revistas/ruma/v46n2/2a03428x.png)
![i](/img/revistas/ruma/v46n2/2a03429x.png)
![j = 1,...,n](/img/revistas/ruma/v46n2/2a03430x.png)
![i ⁄= j](/img/revistas/ruma/v46n2/2a03431x.png)
![TiTj](/img/revistas/ruma/v46n2/2a03432x.png)
![TjTi](/img/revistas/ruma/v46n2/2a03433x.png)
![T*i Tj](/img/revistas/ruma/v46n2/2a03434x.png)
![TjT*i](/img/revistas/ruma/v46n2/2a03435x.png)
![(TiTj)† - (TjTi)† ∈ K(H), (T *Tj)† - (TjT *)† ∈ K(H), i i](/img/revistas/ruma/v46n2/2a03436x.png)
![T](/img/revistas/ruma/v46n2/2a03437x.png)
![n](/img/revistas/ruma/v46n2/2a03438x.png)
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
![∏n ∏n ( T nii)† - (Ti† )ni ∈ K(H). i=1 i=1](/img/revistas/ruma/v46n2/2a03450x.png)
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.
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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