Partial Isometries and EP Elements in Banach Algebras

and Applied Analysis 3 The left multiplication by a ∈ 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 algebraL 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 consider Ra ∈ 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. Let V 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 element a ∈ 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 for a ∈ 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 . Let X 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 that R 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. Let X 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 theMoore-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. 4 Abstract and Applied Analysis 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 A 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 a#a∗a ( a# )2 aa∗a ( a# )2 a a#. 2.1 ii ⇒ i : From a#a∗a a#, it follows that aa∗a a2 ( a#a∗a ) a2a# a. 2.2 i ⇔ iii : This part can be proved similarly. In the following result we present equivalent conditions for an bounded linear operator T 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. Let X 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 , Abstract and Applied Analysis 5 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 haveand Applied Analysis 5 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 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#. 2.3 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 T∗ TT∗T# TT∗T ( T# )2 T ( T# )2 T#. 2.4 iii ⇒ i : Using the equality T∗ T#, we get: TT∗ TT# T#T T∗T, TT∗T TT#T T. 2.5 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 TT∗ T ( T#TT∗ ) T TT∗ T# TT∗T T# TT#. 2.6 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 T# TT∗ T#T†T ( T# )2 TT†T T#, 2.7