# Partial Isometries and EP Elements in Banach Algebras

Authors: Dijana Mosić; Dragan S. Djordjević
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

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


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## Body

## 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 andC*-algebras, or more general, in rings with or without involution.Let𝒜 be a complex unital Banach algebra. An element a∈𝒜 is generalized (or inner) invertible, if there exists some b∈𝒜 such that aba=a holds. In this case b is a generalized (or inner) inverse of a. If aba=a, then take c=bab to obtain the following: aca=a and cac=c. Such c is called a reflexive (or normalized) generalized inverse of a. Finally, if aba=a, then ab and ba are idempotents. In the case of the C*-algebra, we can require that ab and ba are Hermitian. We arrive at the definition of the Moore-Penrose inverse in C*-algebras.Definition 1.1.
Let𝒜 be a unital C*-algebra. An element a∈𝒜 is Moore-Penrose invertible if there exists some b∈𝒜 such that
(1.1)aba=a,bab=b,(ab)*=ab,(ba)*=ba,
hold. In this case b is the Moore-Penrose inverse of a, usually denoted by a†.Ifa is Moore-Penrose invertible in a C*-algebra, then a† is unique, and the notation is justified.More general, if𝒜 is a unital Banach algebra, we have the following definition of Hermitian elements.Definition 1.2.
An elementa∈𝒜 is said to be Hermitian if ∥exp(ita)∥=1 for all t∈R.The set of all Hermitian elements of𝒜 will be denoted by ℋ(𝒜). Now, it is natural to consider the following definition of the Moore-Penrose inverse in Banach algebras ([1, 2]).Definition 1.3.
Let𝒜 be a complex unital Banach algebra and a∈𝒜. If there exists b∈𝒜 such that
(1.2)aba=a,bab=b,ab,baareHermitian,
then the element b is the Moore-Penrose inverse of a, and it will be denoted by a†.The Moore-Penrose inverse ofa is unique in the case when it exists.Although the Moore-Penrose inverse has many nice approximation properties, the equalityaa†=a†a does not hold in general. Hence, it is interesting to distinguish such elements.Definition 1.4.
An elementa of a unital Banach algebra 𝒜 is said to be EP if there exists a† and aa†=a†a.The name EP will be explained latter. There is another kind of a generalized inverse that commutes with the starting element.Definition 1.5.
Let𝒜 be a unital Banach algebra and a∈𝒜. An element b∈𝒜 is the group inverse of a, if the following conditions are satisfied:
(1.3)aba=a,bab=b,ab=ba.The group inverse ofa will be denoted by a# which is uniquely determined (in the case when it exists).LetX be a Banach space and ℒ(X) the Banach algebra of all linear bounded operators on X. In addition, if T∈ℒ(X), then N(T) and R(T) stand for the null space and the range of T, respectively. The ascent of T is defined as asc(T)=inf{n≥0:N(Tn)=N(Tn+1)}, and the descent of T is defined as dsc(T)=inf{n≥0:R(Tn)=R(Tn+1)}. In both cases the infimum of the empty set is equal to ∞. If asc(T)<∞ and dsc(T)<∞, then asc(T)=dsc(T).Necessary and sufficient forT# to exist is the fact that asc(T)=dsc(T)≤1. If T∈ℒ(X) is a closed range operator, then T# exists if and only if X=N(T)⊕R(T) (see [3]). Obviously, R((T#)n)=R(T#)=R(T)=R(Tn) and N((T#)n)=N(T#)=N(T)=N(Tn), for every nonnegative integer n. Now the name follows: EP means “equal projections” on R(Tk) parallel to N(Tk) for all positive integers k.Finally, ifa∈𝒜 is an EP element, then clearly a# exists. In fact, a#=a†. On the other hand, if a exists, then necessary and sufficient for a to be EP is that aa# is a Hermitian element of 𝒜. Furthermore, in this case a#=a†.The left multiplication bya∈𝒜 is the mapping La:𝒜→𝒜, which is defined as La(x)=ax for all x∈𝒜. Observe that, for a,b∈𝒜, Lab=LaLb and that La=Lb implies a=b. If a∈𝒜 is both Moore-Penrose and group invertible, then La†=(La)† and La#=(La)# in the Banach algebra ℒ(𝒜). According to [4, Remark 12], a necessary and sufficient condition for a∈𝒜 to be EP is that La∈ℒ(𝒜) is EP.A similar statement can be proved if we considerRa∈ℒ(𝒜) instead of La∈ℒ(𝒜), where the mapping Ra:𝒜→𝒜 is the right multiplication by a, and defined as Ra(x)=xa for all x∈𝒜.Let𝒱(𝒜)=ℋ(𝒜)+iℋ(𝒜). Recall that according to [5, Hilfssatz 2(c)], for each a∈𝒱(𝒜) there exist necessary unique Hermitian elements u,v∈ℋ(𝒜) such that a=u+iv. As a result, the operation a*=u-iv is well defined. Note that *:𝒱(𝒜)→𝒱(𝒜) is not an involution, in particular (ab)* does not in general coincide with b*a*,a,b∈𝒱(𝒜). However, if 𝒜=𝒱(𝒜) and for every h∈ℋ(𝒜), h2=u+iv with uv=vu, u,v∈ℋ(𝒜), then 𝒜 is a C*-algebra whose involution is the just considered operation, see [5].An elementa∈𝒱(𝒜) satisfying aa*=a*a is called normal. If a=u+iv∈𝒱(𝒜) (u,v∈ℋ(𝒜)), it is easy to see that a is normal if and only if uv=vu. An element a∈𝒱(𝒜) satisfying a=aa*a is called a partial isometry [6].Note that necessary and sufficient fora∈𝒜 to belong to ℋ(𝒜) is that La∈ℋ(ℒ(𝒜)). Therefore, a∈𝒱(𝒜) is normal if and only if La∈𝒱(ℒ(𝒜)) is normal. Observe that if a∈𝒱(𝒜) then La∈𝒱(ℒ(𝒜)) and La*=(La)*.Theorem 1.6 (see [7]).
LetX be a Banach space and consider T∈ℒ(X) such that T† exists and T∈𝒱(ℒ(X)). Then the following statements hold:  (i)
R(T*)⊆R(T) if and only if T=TTT†,(ii)
N(T)⊆N(T*) if and only if T=T†TT. In addition, if the conditions of statements (i)  and (ii)  are satisfied, then T is an EP operator.Notice thatR(T*)⊆R(T) is equivalent to T*=TT†T*, by R(T)=R(TT†)=N(I-TT†). The condition N(T)⊆N(T*) is equivalent to T*=T*T†T, because N(T)=N(T†T)=R(I-T†T) [7]. Hence, by Theorem 1.6, we deduce the following.Corollary 1.7.
LetX be a Banach space and consider T∈ℒ(X) such that T† exists and T∈𝒱(ℒ(X)). Then the following statements hold:(i)
T*=TT†T* if and only if T=TTT†,(ii)
T*=T*T†T if and only if T=T†TT.There are many papers characterizing EP elements, partial isometries, or related classes (such as normal elements). See, for example [4, 7–23]. Properties of the Moore-Penrose inverse in various structures can be found in [1, 2, 24–28].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.
Let𝒜 be a unital Banach algebra and consider a∈𝒱(𝒜) such that a† and a# exist. Then the following statements are equivalent:(i)
a is a partial isometry,(ii)
a#a*a=a#,(iii)
aa*a#=a#.Proof.
(i)⇒ (ii): If aa*a=a, then
(2.1)a#a*a=(a#)2(aa*a)=(a#)2a=a#.
(ii)⇒ (i): From a#a*a=a#, it follows that
(2.2)aa*a=a2(a#a*a)=a2a#=a.
(i)⇔ (iii): This part can be proved similarly.In the following result we present equivalent conditions for an bounded linear operatorT on Banach space X to be a partial isometry and EP. Compare with [21, Theorem 2.3] where we studied necessary and sufficient conditions for an element a of a ring with involution to be a partial isometry and EP.Theorem 2.2.
LetX be a Banach space and consider T∈ℒ(X) such that T† and T# exist and T∈𝒱(ℒ(X)). Then the following statements are equivalent: (i)
T is a partial isometry and EP,(ii)
T is a partial isometry and normal,(iii)
T*=T#,(iv)
TT*=T†T and T=TTT†,(v)
T*T=TT† and T=T†TT,(vi)
TT*=TT# and T=TTT†,(vii)
T*T=TT# and T=T†TT,(viii)
T*T†=T†T#,(ix)
T†T*=T#T†,(x)
T†T*=T†T# and T=TTT†,(xi)
T*T†=T#T† and T=T†TT,(xii)
T*T#=T#T† and T=T†TT,(xiii)
T*T†=T#T# and T=T†TT,(xiv)
T*T#=T#T# and T=T†TT,(xv)
TT*T#=T† and T=T†TT,(xvi)
T*T2=T and T=T†TT,(xvii)
T2T*=T and T=TTT†,(xviii)
TT†T*=T# and T=TTT†,(xix)
T*T†T=T# and T=T†TT.Proof.
(i)⇒ (ii): If T is EP, then T=TTT† and, by Corollary 1.7, T*=TT†T*. Since T is a partial isometry, we have
(2.3)TT*T#=(TT*T)(T#)2=T(T#)2=T#,T*T#T=TT†T*T#T=T†(TT*T)T#=T†TT#=T#TT#=T#.
Thus, TT*T#=T*T#T and T=T†TT imply T is normal, by [7, Theorem 3.4(i)].
(ii)⇒ (iii): The condition T is normal and [7, Theorem 3.4(vii)] imply T*=TT*T#. Because T is a partial isometry, we have
(2.4)T*=TT*T#=(TT*T)(T#)2=T(T#)2=T#.
(iii)⇒ (i): Using the equality T*=T#, we get:
(2.5)TT*=TT#=T#T=T*T,TT*T=TT#T=T.
By [7, Theorem 3.3], T is normal gives T is EP. The condition (i) is satisfied.
(ii)⇒ (iv): By [7, Theorem 3.4(ii)], T is normal gives TT*T#=T#TT* and T=TTT†. Now
(2.6)TT*=T(T#TT*)=T(TT*)T#=(TT*T)T#=TT#.
Since T is normal implies T is EP, then TT*=T#T=T†T.
(iv)⇒ (vi): Assume that TT*=T†T and T=TTT†. Then
(2.7)T#(TT*)=T#T†T=(T#)2TT†T=T#,
implying
(2.8)TT*T#=T(T#TT*)T#=TT#T#=T#,
and T#TT*=TT*T#. By [7, Theorem 3.4(ii)], T is normal and, by [7, Theorem 3.3], T is EP. Therefore, TT*=T†T=TT†=TT#.
(vi)⇒ (ii): Let TT*=TT# and T=TTT†. Then
(2.9)T(TT*)=TTT#=T=(TT#)T=TT*T,
which yields that T 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) T*=T# and T is EP which gives (viii).
(viii)⇒ (xi): Suppose that T*T†=T†T#. Then
(2.10)TT#=T2(T#)2=TTT†T(T#)2=TT(T†T#)=TTT*T†=TT(T*T†)TT†=TTT†T#TT†=TTT†TT#T†=TT†.
Hence, TT# is Hermitian and T is EP. Now condition (xi) is satisfied by
(2.11)T*T†=T†T#=T#T†,T†TT=TT†T=T.
(xi)⇒ (xvi): The assumptions T*T†=T#T† and T=T†TT give, by Corollary 1.7,
(2.12)T*T2=(T*T†)TT2=T#T†TT2=(T#)2TT†TT2=T.
(xvi)⇒ (xiv): Multiplying T*T2=T by (T#)3 from the right side, we get T*T#=T#T#. Hence, T satisfies condition (xiv).
(xiv)⇒ (xii): If T*T#=T#T# and T=T†TT, then we see that T*T=(T*T#)T2=T#T#T2=T#T. Thus, by (vii) ⇔ (i), we get that T is EP, and
(2.13)T*T#=(T*T)(T#)2=T#T(T#)2=(T#)2=T#T†.
(xii)⇒ (vii): Applying T*T#=T#T† and T=T†TT, we obtain the condition (vii):
(2.14)T*T=(T*T#)T2=T#(T†T2)=T#T.
(i)⇒ (ix) ⇒ (x) ⇒ (xvii): Similarly as (i) ⇒ (viii) ⇒ (xi) ⇒ (xvi).
(xvii)⇒ (vi): Suppose that T2T*=T and T=TTT†. Then TT*=T#T2T*=T#T and the condition (vi) is satisfied.
(xiii)⇒ (xi): Multiplying the equality T*T†=T#T# by TT† from the right side, we obtain T*T†=T#T†. So, we deduce that condition (xi) holds.
(xi)⇒ (xiii): By (xi), we have that T is EP and condition (xiii) is satisfied.
(xv)⇒ (i): Let TT*T#=T† and T=T†TT. Now, we observe that
(2.15)TT*T=(TT*T#)T2=T†TT=T,T†=TT*T#=T#T(TT*T)(T#)2=T#TT(T#)2=T#.
Therefore, T is a partial isometry and EP.
(i)⇒ (xv): The hypothesis T is EP gives T=T†TT and, because (i) implies (iii),
(2.16)TT*T#=TT#T#=T#=T†.
(xviii)⇒ (iii): By the assumption TT†T*=T# and T=TTT†, we obtain T*=TT†T*=T#.
(iii)⇒ (xviii): From T*=T#, we get
(2.17)TT†T*=TT†T#=TT†T(T#)2=T#,
and T is EP implying T=TTT†.
(iii)⇔ (xix): Analogy as (iii) ⇔ (xviii).Now, we return to a general case, that is,𝒜 is a complex unital Banach algebra, and a∈𝒜 is both Moore-Penrose and group invertible.Corollary 2.3.
Theorem2.2 holds if we change ℒ(X) for an arbitrary complex Banach algebra 𝒜, and one changes T by an a∈𝒜 such that a† and a# exist.Proof.
Ifa∈𝒜 satisfies the hypothesis of this theorem, then La∈ℒ(𝒜) satisfies the hypothesis of Theorem 2.2. Now, if any one of statements (i)–(xix) holds for a, then the same statement holds for La. Therefore, La is a partial isometry and EP in ℒ(A). By [4, Remark 12], it follows that a is EP in 𝒜. It is well-known that if a∈𝒱(𝒜) then La∈𝒱(ℒ(𝒜)) and La*=(La)*. Since La is a partial isometry, LaLa*La=La, that is, Laa*a=La. So, we deduce that aa*a=a and a a partial isometry in 𝒜.
A similar statement can be proved if we considerRa∈ℒ(𝒜) instead of La∈ℒ(𝒜).The cancellation property and the identity(ab)*=b*a* are important when we proved the equivalent statements characterizing the condition of being a partial isometry and EP in a ring with involution ℛ in [21]. Since *:𝒱(𝒜)→𝒱(𝒜) is not in general an involution, and it is not clear if the cancellation property holds for Moore-Penrose invertible elements of 𝒱(𝒜), in most statements of Theorem 2.2 an additional condition needs to be considered.



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