ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES

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1 Georgian Mathematical Journal Volume 15 (2008), Number 1, ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES MOHAMED AZIZ TAOUDI Abstract. This paper is concerned with the definition and study of semipartial isometries on Banach spaces. This class of operators, which is a natural generalization of partial isometries from Hilbert to general Banach spaces, contains in particular the class of partial isometries recently introduced by M. Mbekhta [12]. First of all, we establish some basic properties of semi-partial isometries. Next, we introduce the notion of pseudo Moore Penrose inverse as a natural generalization of the Moore Penrose inverse from Hilbert spaces to arbitrary Banach spaces. This concept is used to carry out a classification for semi-partial isometries in Banach spaces and to provide a characterization for Hilbert spaces Mathematics subject classification: 47A53, 46B04. Key words and phrases: Isometries, partial isometries, semi partial isometries, generalized inverse, Moore Penrose inverse. 1. Introduction and Preliminaries Let X and Y be complex Banach spaces and let L(X, Y ) denote the set of bounded linear operators from X to Y, and abbreviate L(X, X) to L(X). For T L(X, Y ) we write T for its adjoint, N(T ) for its kernel and R(T ) for its range. Recall that T L(X) has a generalized inverse if there exists an operator S L(X) for which T ST = T and ST S = S. The operator S is called a generalized inverse of T ; see, for instance, [14, 15] and the references therein. In general, the generalized inverse is not unique and T L(X) has a generalized inverse if and only if N(T ) and R(T ) are closed complemented subspaces of X. An operator T L(X, Y ) is said to be an isometry if T x = x for all x X. The literature concerning isometries is vast. The book [6] contains a detailed description of isometries on function spaces and a good overview of the topic. Also, we say that T is an co-isometry if T L(Y, X ) is an isometry and T is unitary if both T and T are isometries. If H is a Hilbert space, we say that T L(H) is a partial isometry if T x = x for all x N(T ), where N(T ) is the orthogonal complement of N(T ). It is clear that, in the Hilbert space setting, every isometry is a partial isometry. Moreover, we have the following characterization of partial isometries (see [12]). Theorem 1.1. If T L(H) is a contraction (i.e., T 1), then the following statements are equivalent: (i) T is a partial isometry; ISSN X / $8.00 / c Heldermann Verlag

2 178 M. A. TAOUDI (ii) T possesses a contractive generalized inverse; (iii) γ(t ) 1, where γ(t ) stands for the reduced minimum modulus of T (see [9]). The problem of defining a partial isometry in Banach spaces was introduced in [12]. Observing that the assertion (ii) in Theorem 1.1 does not depend on the structure of Hilbert spaces, M. Mbekhta [12] suggests the following definition. Definition 1.1 ([12]). A bounded operator T on X is called a partial isometry if T is a contraction and admits a generalized inverse which is a contraction. Although such a requirement is completely natural in Hilbert spaces, it is sometimes restrictive when dealing with operators in Banach spaces. One of the disadvantages of this definition is that, in contrast with the Hilbert space setting, an isometry on a Banach space X need not to be a partial isometry. This and other like facts entail the importance of introducing a more general extension. In this paper, we introduce the class of semi-partial isometries (see Definition 2.1), which is a natural generalization of partial isometries from Hilbert spaces to general Banach spaces. This class of operators contains, among others, isometries, co-isometries, unitary operators and partial isometries (in the sense of M. Mbekhta). Our main goal here is to investigate and to classify this class of operators. The outline of the paper is as follows: Preliminary results are described in Section 1. In Section 2 we introduce the class of semi-partial isometries in Banach spaces. Some basic properties of this class are developed in this section. More precisely, after preparing some elementary properties we give a characterization of a semi-partial isometry in terms of its reduced minimum modulus. The problem of the classification of semi-partial isometries is also discussed. We first give a special attention to the class of semi-partial isometries with complemented kernels. This class is of particular interest since it inherits a desirable property from the Hilbert space setting. Namely, we show that an operator belongs to this class if and only if it is an isometry on a complement of its kernel. Further, we introduce the concept of pseudo Moore Penrose inverse of an operator acting on a Banach space, which is one of the most natural generalizations of the notion of Moore Penrose inverse from Hilbert spaces to general Banach spaces. This concept is used to investigate the class of semi-partial isometries with complemented kernels and ranges and to provide a characterization for Hilbert spaces. More precisely, we show that a Banach space is isometrically isomorphic to a Hilbert space if and only if every operator having a generalized inverse admits a pseudo Moore Penrose inverse. Further, we consider the class of semi-partial isometries with index. In particular, we improve some results from [16]. The last part of this section is devoted to the reduced minimum modulus which plays a fundamental role in the study of semi-partial isometries. Specifically, we provide a formula for the reduced minimum modulus involving norm-one projections and improving several ones obtained earlier by other authors.

3 ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES Semi-Partial Isometries 2.1. Definitions and basic properties. Definition 2.1. A bounded linear operator T on a Banach space X is called a semi-partial isometry if T x = dist(x, N(T )) for all x X. Remarks A bounded linear operator T on X is said to be an isometry if T x = x for all x X. It follows immediately from the definition that every isometry is a semi-partial isometry. 2. Every semi-partial isometry T 0 on a Banach space satisfies γ(t ) = T = If H is a Hilbert space, it is easy to check that T L(H) is a semi-partial isometry if and only if T is a partial isometry. 4. A semi-partial isometry T is an isometry if and only if N(T ) = {0}. As has been previously noticed, every isometry is a semi-partial isometry. Here is an example of a semi-partial isometry which is not an isometry. Example 2.1. Let X be a Banach space and let T be the bounded linear operator defined on X X by T (x y) = y 0, for all x, y X. It can be easily verified that T 2 = 0, N(T ) = X {0} and for all x, y X one has T (x y) = dist(x y, N(T )) = y. Thus, T is a semi-partial isometry which is not an isometry. Next, we give the following characterization of semi-partial isometries. It is crucial for our subsequent analysis. It shows, in particular, that semi-partial isometries are closely related to isometries. Theorem 2.1. Let T L(X), T 0. The following statements are equivalent: (i) T is a semi-partial isometry; (ii) γ(t ) = T = 1; (iii) T is a contraction and γ(t ) 1; (iv) the operator T : X/N(T ) X given by T ẋ = T x for all x X, is an isometry. Proof. The implications (i) (ii) (iii) are straightforward. (iii) (iv) Let x X. Since γ(t ) 1, then T x dist(x, N(T )). On the other hand, keeping in mind that T is a contraction we get for all y N(T ), T x = T (x y) x y. Hence T x inf{ x y, y N(T )} = d(x, N(T )). Accordingly, T ẋ = T x = dist(x, N(T )) = ẋ, which is the desired result.

4 180 M. A. TAOUDI (iv) (i) Straightforward. The following proposition provides some examples and properties of semipartial isometries. Proposition ) A projection P 0 on a Banach space X is a semipartial isometry if and only if P = 1. 2) If T L(X) is a contraction and there is a closed subspace M of X such that N(T ) M = X and T x = x for all x M, then T is a semi-partial isometry. 3) An operator T L(X) is a semi-partial isometry if and only if T L(X ) is a semi-partial isometry. 4) If T L(X) is a semi-partial isometry and if V L(X) is an isometry, then V T is a semi-partial isometry. Proof. The direct implication is straightforward. Conversely, Let P be a projection and let x X. Clearly, N(P ) R(P ) = X. Thus, x may be written in the form x = y + P z where y N(P ) and z X. Accordingly, dist(x, N(P )) = dist(p z, N(P )) = dist(p x, N(P )) P x. (2.1) In consequence, γ(p ) 1. The use of Theorem 2.1 (iii) together with the fact that P = 1 achieves the proof. 2) Let x X and let y N(T ) and m M such that x = y + m. Then T x = T m = m dist(m, N(T )) = dist(x, N(T )). Consequently, γ(t ) 1. We conclude with Theorem 2.1 (iii). 3) This follows from the fact that γ(t ) = γ(t ) and T = T, on the basis of Theorem ) This is an immediate consequence of the fact that N(V T ) = N(T ) and V T x = T x, for all x X. Before to resume our analysis, let us recall some relevant definitions and terminologies. By a complemented subspace we mean a range of a bounded linear projection P and by a 1-complemented subspace we mean a range of a bounded linear norm-one projection. Recall that a bounded operator T + on a Hilbert space H is the Moore Penrose inverse of T L(H) if T + is a generalized inverse of T and the projections P = T T + and Q = T + T are orthogonal (i.e., P 2 = P = P and Q 2 = Q = Q ). On the other hand, norm-one projections seem to be one of the most natural generalizations of the concept of orthogonal projections from Hilbert spaces to arbitrary Banach spaces (see, for instance, [3, 5, 10, 13] and the references therein). For this reason, we introduce the following concept. Definition 2.2. We say that a bounded operator T + on a Banach space X is a pseudo Moore Penrose inverse of T L(X) if T + is a generalized inverse of T and the projections P = T T + and Q = T + T are of norm 1.

5 ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES 181 Further, the notion of hermitian operators is also a natural generalization of self-adjoint operators from Hilbert spaces to general Banach spaces. Recall that a bounded operator T on a Banach space X is said to be hermitian if exp(itt ) = 1 for every real t or, equivalently, its numerical range W (T ) R (see [2]). Following [12], we say that T + L(X) is a Moore Penrose inverse of T L(X) if T + is a generalized inverse of T and the projections T T + and T + T are hermitian. Note that every bounded operator has at most one Moore Penrose inverse (see [15]). Remarks A Moore Penrose inverse of an operator T L(X) is also a pseudo Moore Penrose inverse of T. 2. An operator T L(X) may have several pseudo Moore Penrose inverses. For example, let X = C 2 equipped with the norm (x, y) = x + y, and consider the operator T = ( Then T 2 = T and T = 1. Thus ( T is a pseudo ) Moore Penrose inverse of itself. 1 0 Consider now the operator S =. It is easy to check that S = 1, 0 0 T S = S and ST = T, therefore S is also a pseudo Moore Penrose inverse of T. 3. It has been shown [12] that if S L(X) is a generalized inverse of T L(X), then 1 T S ST γ(t ). S S In particular, if S is a pseudo Moore Penrose inverse of T then γ(t ) = 1 S. Also, if a contraction T has a contractive generalized inverse then γ(t ) 1 and therefore T is a semi-partial isometry. Now, we shall distinguish the following classes of semi-partial isometries: Definition 2.3. A semi-partial isometry T L(X) is called (a) a pseudo-isometry if there is a norm one projection P such that N(T ) = N(P ); (b) a pseudo-coisometry if R(T ) is a 1-complemented subspace in X; (c) a pseudo-unitary if T is a pseudo-isometry with complemented range. Remarks It has been shown [12] that T L(X) is a partial isometry if and only if T is at once a pseudo-isometry and pseudo-coisometry. 2. Every isometry (resp. coisometry) is a pseudo-isometry (resp. a pseudocoisometry). 3. If T is a pseudo-isometry and V is an isometry, then V T is a pseudoisometry. ).

6 182 M. A. TAOUDI 4. If T is a pseudo-coisometry and V is a co-isometry, then T V is a pseudocoisometry. 5. If T is a pseudo-unitary and V is unitary, then T V and V T are pseudounitary. 6. Every isometry on L p (µ), where µ is a positive finite measure and 1 p, is a pseudo-unitary. For a deeper discussion of the properties of isometries in L p (µ) spaces we refer to [6, 11]. 7. There are several examples of pseudo-isometries in Banach spaces which are not pseudo-unitary. For example, a range of an isometry in C[0, 1] does not have to be complemented (see [4]). Now, we are in a position to state the following result, which characterizes pseudo-isometries. Theorem 2.2. Let T L(X) be a contraction. The following assertions are equivalent: 1) T is a pseudo-isometry; 2) there is an isometry V L(R(T ), X) such that V T is a norm-one projection; 3) the kernel N(T ) possesses a complement M in X which is 1-complemented and verifying T x = x for all x M; 4) There is a norm-one projection P such that N(P ) = N(T ), T P = T and T P x = P x for all x X. Proof. 2) Assume that T is a pseudo-isometry, then there is a norm-one projection P verifying N(T ) = N(P ). Define V : R(T ) X by V (T x) = P x. The map V is well -defined and satisfies P = V T. Our next objective is to show that V is an isometry. To this end, let x X, then V (T x) = P x = dist(x, N(P )) = dist(x, N(T )) = T x. 2) 3) Set M := R(P ), where P := V T. Clearly, M is a closed subspace and N(T ) M = N(P ) R(P ) = X. Moreover, it is clear that P x = x for all x M. Using the fact that V is an isometry we infer that for every x M one has x = P x = V (T x) = T x. 3) 4) It suffices to consider a norm-one projection P such that M = R(P ). 4) 1) Take M = R(P ). The fact that T is a semi-partial isometry follows immediately from Proposition 2.1 2). Also, we state the following

7 ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES 183 Proposition 2.2. Let T L(X) be a contraction. The following assertions are equivalent: 1) T is a partial isometry; 2) T has a contractive pseudo Moore Penrose inverse. Proof. 1) = 2) Suppose that T is a partial isometry, then there is a norm-one projection P such that N(T ) = N(P ), T P = T and T P x = P x for all x X. Moreover, there is a norm-one projection Q such that R(T ) = R(Q) and QT = T (see Remarks 2.3, see also [12]). Next, let x X, then there is y X such that Qx = T y. Set, T + x = P y. We claim that T + is well defined. Indeed, if Qx = T y = T z, then y z N(T ) = N(P ) and hence P y = P z. Clearly, T T + = Q and T + T = P. Accordingly, T T + T = T. Moreover, for all x X one has T + x = P y = T P y = T y = Qx x. In consequence, T + is a contractive generalized inverse of T. In addition, 1 T T + T T + 1, and 1 T + T T + T 1. Hence, T T + = T + T = 1. In consequence, T + is a contractive pseudo Moore Penrose inverse of T. 2) = 1) Let T + be a contractive pseudo Moore Penrose inverse of T. As previously mentioned, T is a semi-partial isometry. Moreover, the projections T T + and T + T are of norm 1 and verify N(T + T ) = N(T ) and R(T T + ) = R(T ). Using Remark 2.3 1) we get the desired result. Remark 2.4. In [3], Campbell, Faulkner and Sine gave an example of an isometry T on a C(K) space such that the range of T is complemented (and thus T is pseudo unitary) but not 1-complemented in C(K) (T is not a partial isometry). This motivates us to give a characterization of pseudo unitary operators with non- 1-complemented ranges. This is the purpose of our next result. Theorem 2.3. Let T L(X) be a contraction. The following assertions are equivalent: 1) T is pseudo-unitary; 2) T has a generalized inverse S such that ST x = T x and T Sx = Sx for all x X. Proof. 1) = 2) Suppose that T is pseudo unitary, and let P and Q be two projections verifying P = 1, N(P ) = N(T ), T P = T, R(Q) = R(T ) and QT = T. Now, let x X, then there is y X such that Qx = T y. Put, Sx = P y. Arguing as in Proposition 2.2 we infer that S is well defined and verify, T S = Q and ST = P. Accordingly, T ST = T and ST S = S, hence S is a generalized inverse of T. Moreover, for all x X we have ST x = P x = dist(x, N(P )) = dist(x, N(T )) = T x, and Sx = P y = ST y = T y = Qx = T Sx. 2) = 1) Suppose that T admits a generalized inverse S such that ST x = T x and T Sx = Sx for all x X. Put P = ST and Q = T S. One

8 184 M. A. TAOUDI verifies immediately that P = 1, N(T ) = N(P ), T P = T, R(Q) = R(T ) and QT = T. This achieves the proof. Remark 2.5. It is worthwhile to mention that the concepts of semi-partial isometries, pseudo-isometries, pseudo-co-isometries and pseudo-unitary operators coincide in the Hilbert space setting. Definition 2.4. A bounded linear operator T is said to be a power semipartial isometry if T n is a semi-partial isometry for every integer n 1. Proposition 2.3. Let T L(X) be a contraction and suppose that N(T k ) R(T ) for k = 1, 2,.... Then the following assertions are equivalent: 1) T is similar to a semi-partial isometry; 2) T is similar to a power semi-partial isometry. Proof. The implication 2) 1) is obvious. Conversely, let S L(X) be invertible such that S 1 T S is a semi-partial isometry, then γ(s 1 T S) 1. In view of Theorem 2.1 it suffices to show that γ(s 1 T k S) 1. To this end, let us first notice that N(S 1 T k S) R(S 1 T S). Next, thanks to [7, Lemma 1] we infer that γ(s 1 T k S) γ(s 1 T S) k 1. This achieves the proof. An immediate consequence of Proposition 2.3 is the following result. Corollary 2.1. Let T L(X) be a contraction and suppose that N(T k ) R(T ) for k = 1, 2,.... Then the following assertions are equivalent: 1) T is a semi-partial isometry; 2) T is a power semi-partial isometry Semi-partial isometries with an index. From now on, α(t ) denotes the dimension of N(T ) and β(t ) stands for the codimension of R(T ). The quantities α(t ) and β(t ) are called respectively the nullity and the deficiency of T. If min{α(t ), β(t )} <, we say that T has an index. The index of T is then defined by ind(t ) = α(t ) β(t ). Recall that T L(X) is a semi-fredholm operator, if R(T ) is closed and T has an index. We will denote by SF (X) the set of semi-fredholm operators on X. Our next task is to investigate semi-partial isometries with an index. Our results improve some earlier ones from [16]. Proposition 2.4. Let T L(X) be a non-zero semi-partial isometry and U L(X). (1) If α(t ) < α(u), then T U 1. (2) If R(U) is closed and β(t ) < β(u), then T U 1. Proof. (1) By Lemma V.1.1 in [8] there is x N(U) such that 1 = x = dist(x, N(T )). Further, the use of Theorem 2.1, (iii) leads to 1 = γ(t ) T x = T x Ux T U x = T U.

9 ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES 185 (2) Arguing by duality and using the formulas β(t ) = α(t ), β(u) = α(u ) together with the fact that T remains a semi-partial isometry, the result becomes an immediate consequence of (1). Corollary 2.2. If T and S are semi-partial isometries on X and if S < 1, then (1) α(t ) = α(s) and β(t ) = β(s). (2) T SF (X) if and only if S SF (X). (3) If T SF (X) and ind(t ) 0, then T λi SF (X), S λi SF (X) and ind(t λi) = ind(s λi) 0 for all λ D where D is the open disc centered at 0 with radius 1. T Proof. (1) Without loss of generality we may assume that T 0. By virtue of Proposition 2.4 we infer that α(t ) α(s) and β(t ) β(s). By symmetry, we also get α(s) α(t ) and β(s) β(t ). This achieves the proof. (2) This is an immediate consequence of (1). (3) Suppose that T SF (X). In view of (2) we derive that S SF (X). By using [8, Theorem V.1.6] together with γ(t ) = γ(s) = 1, we obtain that T λi SF (X), S λi SF (X), ind(t λi) = ind(t ) and ind(s λi) = ind(s) for all λ C with λ < 1. On the other hand, from the assertion (1) it follows that ind(t ) = ind(s). Hence ind(t λi) = ind(s λi) for all λ C with λ < Hilbert spaces and pseudo Moore Penrose inverses. In this section we shall give a characterization of Banach spaces which are isometrically isomorphic to Hilbert spaces by the concept of pseudo Moore Penrose inverse. Following [12], we introduce the following notation: ING(X) = {T L(X) : T has a generalized inverse}; P IMP (X) = {T L(X) : T has a pseudo Moore Penrose inverse}; IMP (X) = {T L(X) : T has a Moore Penrose inverse}. It is clear that IMP (X) P IMP (X) ING(X) and the equality holds in the Hilbert spaces setting. Moreover, we have Theorem 2.4. For a Banach space X with dim(x) 3, the following statements are equivalent: (i) X is isometrically isomorphic to a Hilbert space; (ii) P IMP (X) = ING(X). Proof. The direct implication being obvious, let us prove the converse. It suffices to show that every 2-dimensional subspace of X can be obtained as the range of a norm-one projection (see [1]). Let Y be a 2-dimensional subspace of X and let P be the projection onto Y. It is clear that P ING(X). In consequence, P has a pseudo Moore Penrose inverse, say P +. Hence, taking into account

10 186 M. A. TAOUDI the obvious equality Y = R(P ) = R(P P + ) we deduce that Y is a range of a norm-one projection, which achieves the proof. Remarks Consider the operator T = ( As has been previously noticed, we have T 2 = T and T = 1. Thus T is a pseudo Moore Penrose inverse of itself. Hence T P IMP (X). Let us show that T / ( IMP ) (X). To do so, we first claim that T is not hermitian. Indeed, 1 for f = and X = 1 i 2 (1, i) we have f(x) = 1 and f(t (X)) = 1 (1 + i) ( 2 ) 1 0 W (T ). This proves our claim. Next, consider the operator S =. It is 0 0 easily seen that S is hermitian, S = 1, T S = S and ST = T. If T admits a Moore Penrose inverse T +, then T T + is hermitian. On the other hand, (T T + T S) 2 = T T + + T S T T + T S T ST T + = T T + + T S T T + T S = 0. ). Keeping in mind that T T + T S is hermitian, we derive T T + T S = r(t T + T S) = 0. In consequence, T T + = T S = S, Easy computations show that T + T must be of the form ( ) 1 + t 1 t T + T =. t t which is not hermitian. Hence, T IMP (X). 2. It is worthwhile to notice that the isometry considered on X = C(K) (see [3]) belongs to ING(X) but not to P IMP (X). Indeed, if T admits a pseudo Moore Penrose inverse T +, then R(T ) = R(T T + ). Thus the range of T is 1-complemented, which is absurd The reduced minimum modulus. The reduced minimum modulus turns out to be of fundamental importance in the study of semi-partial isometries. Thus we provide the following useful formula for the reduced minimum modulus. This result improves the previous one obtained in [12]. Theorem 2.5. For T L(X) we have γ(t ) = inf{ T Q : Q 0; Q 2 = Q and N(T ) N(Q)} = inf{ QT : Q 0; Q 2 = Q and R(Q) R(T )}. Proof. Let Q 0 be a projection satisfying N(T ) N(Q). It is easy to see that there is x 0 X such that Qx 0 N(T ). Moreover, dist(qx 0, N(T ))γ(t ) T Qx 0 = T Q(Qx 0 ) T Q dist(qx 0, N(T Q)) T Q dist(qx 0, N(T )).

11 ON A GENERALIZATION OF PARTIAL ISOMETRIES IN BANACH SPACES 187 Accordingly, γ(t ) T Q. In consequence, γ(t ) inf{ T Q : Q 0; Q 2 = Q and N(T ) N(Q)}. To show the other inequality, let (x n ) n 1 be a sequence of X satisfying dist(x n, N(T )) = 1 and T x n γ(t ). According to the Hahn Banach theorem it follows that for each n there is a continuous linear form ϕ n X such that ϕ n = ϕ n (x n ) = 1 and N(T ) N(ϕ n ). Set Q n := ϕ n x n given by Q n (x) = ϕ n (x)x n. One can easily verify that Q 2 n = Q n and N(T ) N(Q n ). Accordingly, inf{ T Q : Q 0; Q 2 = Q and N(T ) N(Q)} T Q n = T x n. Letting n, we obtain the desired inequality. assertion follows easily. By duality, the second Corollary 2.3. If T is a partial isometry and Q is a norm-one projection such that N(T ) N(Q) (resp. R(T ) R(Q)), then T Q = 1 (resp. QT = 1). Acknowledgment The author would like to thank Professor M. Barraa for helpful discussions and the referee for drawing his attention to the reference [6]. References 1. D. Amir, Characterizations of inner product spaces. Operator Theory: Advances and Applications, 20. Birkhäuser Verlag, Basel, F. F. Bonsall and J. Duncan, Numerical ranges of operators on normed spaces and of elements of normed algebras. London Mathematical Society Lecture Note Series, 2. Cambridge University Press, London New York, S. Campbell, G. Faulkner and R. Sine, Isometries, projections and Wold decompositions. Operator theory and functional analysis (Papers, Summer Meeting, Amer. Math. Soc., Providence, R.I., 1978), pp , Res. Notes in Math., 38, Pitman, Boston, Mass. London, S. Z. Ditor, Averaging operators in C(S) and lower semicontinuous sections of continuous maps. Trans. Amer. Math. Soc. 175(1973), G. D. Faulkner and J. E. Huneycutt, Jr., Orthogonal decomposition of isometries in a Banach space. Proc. Amer. Math. Soc. 69(1978), No. 1, R. J. Fleming and J. E. Jamison, Isometries on Banach spaces: function spaces. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 129. Chapman & Hall/CRC, Boca Raton, FL, K. H. Forster and M. A. Kaashoek, The asymptotic behaviour of the reduced minimum modulus of a Fredholm operator. Proc. Amer. Math. Soc. 49(1975), S. Goldberg, Unbounded linear operators: Theory and applications. McGraw-Hill Book Co., New York Toronto, Ont. London, T. Kato, Perturbation theory for linear operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. Springer-Verlag New York, Inc., New York, S. Kinnunen, On projections and Birkhoff-James orthogonality in Banach spaces. Nieuw Arch. Wisk. (4) 2(1984), No. 2,

12 188 M. A. TAOUDI 11. H. E. Lacey, The isometric theory of classical Banach spaces. Die Grundlehren der mathematischen Wissenschaften, Band 208. Springer-Verlag, New York Heidelberg, M. Mbekhta, Partial isometries and generalized inverses. Acta Sci. Math. (Szeged) 70(2004), No. 3-4, P. L. Papini, Some questions related to the concept of orthogonality in Banach spaces. Orthogonal projections. Boll. Un. Mat. Ital. (4) 9(1974), R. Penrose, A generalized inverse for matrices. Proc. Cambridge Philos. Soc. 51(1955), V. Rakočević, Moore-Penrose inverse in Banach algebras. Proc. Roy. Irish Acad. Sect. A 88(1988), No. 1, Ch. Schmoeger, Partial isometries on Banach spaces. Preprint, semlv. Author s address: (Received ) Département de Mathématiques et Informatique Faculté des Sciences de Gabès Cité Erriadh 6072, Zrig, Gabès Tunisie mataoudi@gmail.com