CONTINUITY AND DIFFERENTIABILITY OF THE MOORE^PENROSE INVERSE IN C

CONTINUITY AND DIFFERENTIABILITY OF THE MOORE^PENROSE INVERSE IN C

-ALGEBRAS

J. J. KOLIHA

1. Continuity of the Moore^Penrose inverse

The paper gives an elementary proof of the theorem on the continuity of the Moore^Penrose inverse in aC-algebra that does not require the concept of the conorm, but uses instead a C modification of Izumino's inequality kb^yk 4ka^ykvalid whena;bhave the Moore^Penrose inverse and satisfy the inequalities kbÿak<¹₂ka^yk^ÿ1 and kbb^yÿaa^yk<1. The paper then studies the conditions for the differentiability of the Moore^Penrose inverse in aC- algebra and gives an explicit formula for the derivative.

TheMoore^Penrose inverse of an elementa of a unitalC-algebraAwith the uniteis the unique element a^yofAsatisfying the equations

aa^yaˆa; a^yaa^yˆa^y; …a^ya†ˆa^ya; …aa^y†ˆaa^y …1:1†

(see [10, 5, 11, 13]). The set of alla2Athat possess the Moore^Penrose in- verse will be denoted byA^y. It is shown in [5, Theorem 6] thata2A^y if and only if a2aAa. The elements a^ya and aa^y are Hermitian idempotents. We also writeA^ÿ1 for the set of all invertible elements inA.

It is well known that the following two results hold for the ordinary in- verse in Banach algebras.

Theorem A. If a is an invertible element of the Banach algebra Aand if a_n!a, thena_n are invertible for all sufficiently largen, anda^ÿ1_n !a^ÿ1.

Theorem B. If an are invertible elements of the Banach algebra A such thata_n!a and that the normska^ÿ1_n k are bounded, thena is invertible and a^ÿ1_n !a^ÿ1.

We discuss the validity of Theorems A and B when the ordinary inverse is replaced by the Moore^Penrose inverse inC-algebras.

Example 1.1. Theorem A is, in general, false for the Moore^Penrose in- verse in aC-algebraAas the setA^yneed not be open inA.

Received May 11, 1998.

First we observe that ifa2Ais Hermitian, then a2A^y if and only if 0 is not an accumulation spectral point of a; this follows from [6, Theorem 7].

LetAbe the C-algebra of all complex valued functions continuous on the set ‰0;1Š [ ‰2;3Š, equipped with the supremum norm. Define a and an by a…t† ˆ0 ift2 ‰0;1Š,a…t† ˆtift2 ‰2;3Š,a_n…t† ˆt=nift2 ‰0;1Šanda_n…t† ˆtif t2 ‰2;3Š. Note that a and a_n are Hermitian elements of A and that 0 is an isolated spectral point for a. Then a2A^y, and a^y is defined by a^y…t† ˆ0 if t2 ‰0;1Š, a^y…t† ˆ1=t if t2 ‰2;3Š. We have kanÿak ˆ1=n!0, however, an2= A^y for all n since …an† ˆ ‰0;1=nŠ [ ‰2;3Š, and 0 is an accumulation spectral point ofan. (See [8, Example 2.1].)

Before we can show that Theorem B holds for the Moore^Penrose inverse in anyC-algebraA, we need to derive some auxiliary results.

Theorem1.2.Let a;b2A^y. Then b^yÿa^yˆ ÿb^y…bÿa†a^y …1:2†

‡ …eÿb^yb†…bÿa†…a^y†a^y‡b^y…b^y†…bÿa†…eÿaa^y†:

The foregoing identity was first obtained by Wedin [18] for matrices and by Harte and Mbekhta [6, Theorem 5] for C-algebras. When we observe that keÿb^ybk 1, keÿaa^yk 1 and kaÿbk ˆ kaÿbk, we obtain the following result.

Theorem1.3.Let a;b2A^y. Then

ka^yÿb^yk 3maxfka^yk²;kb^yk²gkaÿbk:

…1:3†

We can now give an elementary proof of the validity of Theorem B for the Moore^Penrose inverse.

Theorem 1.4.Let a_n2A^ybe such that a_n!a and that the normska^y_nk are bounded. Then a2A^y, and a^y_n!a^y.

Proof. Letka^y_nk Mfor alln. By Theorem 1.3, kamyÿa^y_nk 3M²kamÿank;

the sequence …a^y_n† is Cauchy, and hence convergent to some element c2A.

From the continuity of the product in A we get acaˆlimnana^y_nanˆ limnanˆa; then a2aAa, and a2A^y. Another application of Theorem 1.3 yieldska^y_nÿa^yk 3M²ka_nÿak !0, and the result follows.

The conclusion thata2A^yifan!aand the normska^y_nkare bounded was obtained by Harte and Mbekhta [6, Theorems 7 and 8] as a consequence of the upper semicontinuity of the conorm onAnf0g.

The equivalence of the four conditions given in Theorem 1.6 below forC- algebras has attracted considerable attention in recent literature, and many different proofs have been given [5, 6, 9, 11, 12, 13, 14]. The arguments used in these proofs involve the concept of the conorm in a Banach algebra or, equivalently, the reduced minimum modulus of the induced regular re- presentation of an element a2A, and the concept of the gap between two subspaces of A. Our aim is to find more elementary arguments rooted in Banach algebra techniques, rather than relying on the concepts of the co- norm and of the gap between subspaces which are motivated by operator theory. To this end we adapt an inequality originally obtained by Izumino [7, Lemma 2.2] for Hilbert space operators.

Theorem1.5.If a;b2A^yare such thatkbÿak<¹₂ka^yk^ÿ1andkbb^yÿaa^yk<

1, then

kb^yk 4ka^yk:

…1:4†

Proof. Letpˆaa^y andqˆbb^y;p and qare Hermitian idempotents. We have

k…eÿp†qk²ˆ kq…eÿp†…eÿp†qk ˆ kq…eÿp†qk

ˆ kq…qÿp†qk kqkkpÿqkkqk ˆ kpÿqk<1;

hence eÿ …eÿp†q2A^ÿ1. Further, e‡a^y…bÿa† 2A^ÿ1 as ka^y…bÿa†k ka^ykkbÿak<1.

Since …eÿ …eÿp†q†bˆaa^ybˆa…e‡a^y…bÿa††, we can express b as the product

bˆ …eÿ …eÿp†q†^ÿ1a…e‡a^y…bÿa†† ˆuav;

where u;v2A^ÿ1. Then bˆbv^ÿ1a^yu^ÿ1b, b^yˆb^ybb^yˆ …b^yb†…v^ÿ1a^yu^ÿ1†…bb^y†, and

kb^yk kb^ybkkv^ÿ1kka^ykku^ÿ1kkbb^yk ku^ÿ1kkv^ÿ1kka^yk keÿ …eÿp†qkk…e‡a^y…bÿa††^ÿ1kka^yk 2…1ÿ ka^ykkbÿak†^ÿ1ka^yk 4ka^yk:

We now give an elementary proof of the main result on the continuity of the Moore^Penrose inverse, which subsumes the results of Izumino [7, Pro- position 2.3] for the case of bounded linear operators in Hilbert spaces. Si- multaneously we recover Mbekhta [9, The¨ore©me 2.2], Rakocevic[13, Theo- rem 2.2] and Harte and Mbekhta [6, Theorem 6]. For the continuity of the

Moore^Penrose inverse in Banach algebras or C-algebras see also [11, 12, 14].

Theorem 1.6.Let a_n, a be nonzero elements of A^y such that a_n!a in A.

Then the following conditions are equivalent.

a^y_n !a^y; …1:5†

ana^y_n!aa^y; …1:6†

a^y_nan!a^ya;

…1:7†

supn ka^y_nk<1:

…1:8†

Proof. The implications (1.5) ) (1.6) ) (1.8) ) (1.5) follow from the continuity of the algebra multiplication inA, Theorem 1.5, and Theorem 1.3, respectively. The preceding arguments applied to a_n and a yield (1.5) ) (1.7))(1.8))(1.5) since…c^y†ˆ …c†^y forc2A^y.

Note 1.7. The argument used in the proof of Theorem 1.5 is essentially due to Izumino [7, Lemma 1.2 and Lemma 2.2] (for Hilbert space operators).

We note that the hypotheses of the general case of [7, Lemma 1.2] should be supplemented by the assumption that the operatorABhas closed range; this does not follow from the other assumptions. For the special case when Bis invertible and A has closed range this is not needed as the equation …AB†…B^ÿ1A^y†…AB† ˆABimplies thatABhas closed range.

There are many publications dealing with the continuity of the Moore^

Penrose inverse for complex matrices, both square and rectangular, such as [1, 16, 17, 18]. We recover the following fundamental result of Penrose [10, p. 408].

Corollary1.8.Let a_n, a be nonzero pp matrices such that a_n!a. Then a^y_n!a^yif and only if there is n0such thatrank…an† ˆrank…a†for all nn0.

Proof. Leta^y_n!a^y. By the preceding theorem,a_na^y_n!aa^y. Then rank…an† ˆrank…ana^y_n† ˆtr…ana^y_n† !tr…aa^y† ˆrank…aa^y† ˆrank…a†

by the continuity of the trace.

Conversely, suppose thatrank…an† ˆrank…a†for allnn0. By a result of Wedin [18], inequality (1.3) reduces to

ka^y_nÿa^yk 3ka^y_nkka^ykkanÿak; nn0:

Write "_n ˆ3ka^ykka_nÿak. Then ka^y_nk ka^yk ‡ ka^y_nk"_n, and ka^y_nk

…1ÿ"_n†^ÿ1ka^yk 2ka^yk whenever nn₀ and 0"_n¹₂. The result then fol- lows from the preceding theorem.

We restate the theorem on the continuity of the Moore^Penrose inverse forC-algebra-valued functions, mainly because of its application in the next section to the differentiation of the Moore^Penrose inverse. In the following, J denotes an interval, t0 an element of J, and a…t† a C-algebra valued function defined for allt2J. Bya^y…t†we denote the Moore^Penrose inverse a…t†^yofa…t†.

Theorem1.9.Let a…t†be a function with values in a C-algebra A defined on an interval J such that06ˆa…t† 2A^yfor all t2J, and that a…t†is continuous at t₀. The following conditions are equivalent.

a^y…t†is continuous att₀; …1:9†

a…t†a^y…t†is continuous att₀; …1:10†

a^y…t†a…t†is continuous att₀; …1:11†

there is >0 such that sup

jtÿt0j<ka^y…t†k<1:

…1:12†

2. Differentiability of the Moore^Penrose inverse

The differentiation of the Moore^Penrose inverse for matrices was first stu- died by Golub and Pereyra in [4] and by Decell in [2]; Wedin obtained the equation (2.2) which leads to the explicit formula for the derivative. Drazin [3] investigated the problem in the setting of associative rings, and gave a unified derivation of the differentiation formulae for the Moore^Penrose in- verse and the Drazin inverse.

In this section J again denotes an interval, t₀ an element of J, and a:J!AaC-algebra valued function. Bya⁰…t†we denote the derivative of a…t†att, and bya^y…t†the Moore^Penrose inverse a…t†^y.

Theorem2.1.Let a…t†be a C-algebra valued function defined on an interval J such that06ˆa…t† 2A^yfor all t2J and that a…t†is differentiable at t0. Then the function a^y…t† is differentiable at t₀ if and only if one of the conditions (1.9)^(1.12)is satisfied. The derivative…a^y†⁰ˆ …a^y†⁰…t₀†is given by

…a^y†⁰ˆ ÿa^ya⁰a^y‡ …eÿa^ya†…a⁰†…a^y†a^y‡a^y…a^y†…a⁰†…eÿaa^y†;

…2:1†

where a, a, a^y, a⁰stand for a…t0†, a…t0†, a^y…t0†, a⁰…t0†, respectively.

Proof. First we observe that ifa…t†is differentiable, then so isa…t†, and

…a†⁰…t† ˆ …a⁰†…t†:

From Theorem 1.2 we get a^y…t† ÿa^y…t0†

tÿt₀ ˆ ÿa^y…t†a…t† ÿa…t0† tÿt₀ a^y…t0† …2:2†

‡ …eÿa^y…t†a…t††a…t† ÿa…t₀†

tÿt0 …a^y†…t0†a^y…t0†

‡a^y…t†…a^y†…t†a…t† ÿa…t0†

tÿt₀ …eÿa…t₀†a^y…t₀††:

If one of the conditions (1.9)^(1.12) is satisfied, thena^y…t†is continuous att₀, and we can take the limit ast!t0 in (2.2). This proves (2.1).

Conversely, ifa^y…t†is differentiable at t0, then it is also continuous at t0, and all of the equivalent conditions (1.9)^(1.12) are satisfied.

Note 2.2. The arguments in the foregoing proof depend on the fact thatt is a real variable as we make use of the formula

da…t†

dt ˆ da…t†

dt

;

which is false whentis complex; this suggests that in the preceding theorem the differentiability with respect to a real variable cannot be replaced by analyticity. This is confirmed by observing that ifa…t†anda^y…t†were analy- tic,a^y…t†a…t†anda…t†a^y…t†would be constant, aska^y…t†a…t†k ˆ ka…t†a^y…t†k ˆ1.

For finite matrices, the preceding theorem together with Corollary 1.8 yields the following result due to Golub and Pereyra [4, Theorem 4.3].

Corollary2.3.Let a…t†be a function defined on the interval J whose values are nonzero pp matrices, differentiable at t₀. Then a^y…t†is differentiable at t₀ if and only ifrank…a…t††is constant in some intervaljtÿt0j< . The derivative …a^y†⁰…t0†is given by(2.1).

We mention that applications of the differentiation of the Moore^Penrose inverse include optimization with nonlinear equality constraints, generalized Newton's method and stability of perturbed least squares problems (see [4]).

Acknowledgement. I wish to thank Professor V. Rakocevicfor drawing my attention to the topic of this paper and for graciously providing me with copies of his recent works, including the yet unpublished paper [15].

I am indebted to the referee whose careful reading helped to eliminate several inaccuracies.

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