- Email: [email protected]
Descriptions of partial isometries on Hilbert C∗ -modules
Linear Algebra and its Applications 431 (2009) 883–887
Contents lists available at ScienceDirect
Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa
Descriptions of partial isometries on Hilbert C∗ -modules Kamran Sharifi Department of Mathematics, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran
A R T I C L E
I N F O
Article history: Received 31 July 2008 Accepted 27 March 2009 Available online 2 May 2009 Submitted by R.A. Brualdi AMS classification: 46L08 46C05
A B S T R A C T
Partial isometries and Moore–Penrose inverses of bounded adjointable operators on Hilbert C∗ -modules are investigated. It is shown that a contraction between Hilbert C∗ -modules is a partial isometry if and only if it possesses a contractive Moore–Penrose inverse. A bounded adjointable operator T is a partial isometry if and only if its bounded transform is 2−1/2 T. © 2009 Elsevier Inc. All rights reserved.
Keywords: Hilbert C∗ -module Partial isometry Moore–Penrose inverse
1. Introduction Partial isometries form an attractive and important class of operators. A partial isometry is an operator whose restriction to the orthogonal complement of its null-space is an isometry. Partial isometries play a vital role in operator theory; they enter, for instance, in the theory of the polar decomposition of arbitrary operators, and they form the cornerstone of the theory of von Neumann algebras. Orthogonal projections, unitaries, isometries and co-isometries are familiar examples of partial isometries. Partial isometries in Hilbert and Banach spaces have been studied in [11–13] and in the seminal paper by Halmos and Mclaughlin in [6]. Let T in B(H, H ) be a bounded operator between two Hilbert spaces H, H . A bounded operator S in B(H , H ) is called a generalized inverse of T if TST = T , STS = S. Recently, Mbekhta has shown that a contraction on a Hilbert space is a partial isometry if and only if it has a contractive generalized inverse [11].
E-mail addresses: sharifi[email protected], sharifi@shahroodut.ac.ir 0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2009.03.042
884
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
A Hilbert C∗ -module obeys the same axioms as an ordinary Hilbert space, except that the inner product, from which the geometry emerges, takes values in an arbitrary C∗ -algebra A rather than C. Fundamental and familiar Hilbert space properties like Pythagoras’ equality, self-duality, and even decomposition into orthogonal complements must be given up. They play an important role in the modern theory of C∗ -algebra, notably, in noncommutative geometry and in quantum groups, see [5]. In this paper we first present some definitions and basic facts about bounded and unbounded operators on Hilbert C∗ -modules. Moore–Penrose inverses of bounded and unbounded operators on Hilbert C∗ -modules have been studied in [16,4]. Their works with a recent result by Mbekhta [11] motivate us to describe bounded C∗ -linear partial isometries between Hilbert C∗ -modules, in fact, a contraction is a partial isometry if and only if it has a contractive Moore–Penrose inverse. Let T be a bounded adjointable operator between two Hilbert C∗ -modules and FT = T (1 + T ∗ T )−1/2 be its bounded transform (or Woronowicz transform), we prove T is a partial isometry if and only if FT
=
1 2
T.
Throughout the present paper we assume A to be an arbitrary C∗ -algebra (i.e. not necessarily unital). Bounded and unbounded operators appear in our context, so we simply denote bounded operators by capital letters and unbounded operators by small letters. We use the notations Dom(.), Ker (.) and Ran(.) for domain, kernel and range of operators, respectively. 2. Preliminaries
A (right) pre-Hilbert C∗ -module over a C∗ -algebra A is a right A-module E endowed with an Avalued inner product ·, · : E × E → A, (x, y) → x, y which is linear in the second variable y (and conjugate-linear in x), satisfying the conditions
x, y = y, x∗ , x, ya = x, ya for all a ∈ A, x, x 0 with equality if and only if x = 0. A pre-Hilbert A-module E is called a Hilbert A-module if E is a complete space with respect to the norm x = x, x 1/2 . As well as this scalar-valued norm, E has an A-valued ‘norm’ given by |x| = x, x1/2 . A pre-Hilbert A-submodule E of a Hilbert A-module F is an orthogonal summand if E ⊕ E ⊥ = F, where E ⊥ :={y ∈ F : x, y = 0 for all x ∈ E } is the orthogonal complement of E in F. In this case, E and E ⊥ are necessarily closed. If E and F are Hilbert A-modules, then a map T : E → F is called adjointable if there is a map T ∗ : F → E such that the equality Tx, y = x, T ∗ y holds for any x ∈ E, y ∈ F. It then follows that T (and also T ∗ ) is a bounded, linear, A-module map in the sense of T (xa) = (Tx)a for any x ∈ E, a ∈ A. The set of all bounded adjointable maps from E to F is denoted by B(E, F ). In particular, B(E, E ), which we abbreviate to B(E ), is a C∗ -algebra whose unit is denoted by 1. The papers [1,2], some chapters in [5,14], and the book by Lance [10] are used as standard sources of reference. Let T be a bounded adjointable operator in B(E, F ), then Theorem 3.2 of [10] implies that T has closed range if and only if its adjoint operator T ∗ has closed range, if and only if F = Ker (T ∗ ) ⊕ Ran(T ), if and only if E = Ker (T ) ⊕ Ran(T ∗ ). Let E and F be Hilbert A-modules and T in B(E, F ) be a bounded adjointable operator. T is called a partial isometry (from E0 to F0 ) if F0 = Ran(T ) is complemented in F and there exists a complemented submodule E0 of E such that T is isometric from E0 onto F0 and T (E0⊥ ) = 0. The operator T in B(E, F ) is a partial isometry if and only if T ∗ T is a projection in B(E ), if and only if TT ∗ is a projection in B(F ), if and only if TT ∗ T = T , if and only if T ∗ TT ∗ = T ∗ , cf. [10]. Let T ∈ B(E, F ), then a bounded adjointable operator S ∈ B(F, E ) is called the Moore–Penrose inverse of T if TST
= T , STS = S, (TS)∗ = TS and (ST )∗ = ST .
These properties imply that S is unique and ST is an orthogonal projection, moreover, Ran(S ) and Ker (T ) = Ker (ST ). Therefore E = Ker (ST ) ⊕ Ran(ST ) = Ker (T ) ⊕ Ran(S ).
(1)
= Ran(ST )
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
885
Let E, F be Hilbert A-modules, a regular operator from E to F is a densely defined closed A-linear map t : Dom(t ) ⊆ E → F such that t ∗ is densely defined and 1 + t ∗ t has dense range in E. Note that as we set A = C i.e. if we take E, F to be Hilbert spaces, then this is exactly the definition of a densely defined closed operator. We denote the set of all regular operators from E to F by R(E, F ). Clearly every bounded adjointable operator is a regular operator. Define Qt = (1 + t ∗ t )−1/2 and Ft = tQt , then Ran(Qt ) = Dom(t ), 0 Qt 1 in B(E ) and Ft ∈ B(E, F ). The bounded operator Ft is called the bounded transform of the regular operator t. The map t → Ft defines a bijection R(E, F )
→ {T ∈ B(E, F ) : T 1 and Ran(1 − T ∗ T ) is dense in F },
this map is adjoint-preserving, i.e. Ft∗ = Ft ∗ , cf. [10, Theorem 10.4]. Basic definitions and a few simple facts about regular operators on Hilbert C∗ -modules can be found in chapters 9 and 10 of [10], and the papers [3,8,15] with details. Very often there are interesting relationships between regular operators and their bounded transforms. Indeed, for a regular operator t, some properties transfer to its bounded transform Ft , and vice versa. Remarkably, a regular operator t is normal (resp., selfadjoint, positive) if and only if its bounded transform Ft is normal (resp., selfadjoint, positive), see [15, 1.15] and [8,√ Result 1.14]. We will prove that T is a partial isometry if and only if T is invariant under the transform 2F(.) . 3. Moore–Penrose inverses and partial isometries Xu and Sheng in [16] have shown that a bounded adjointable C∗ -linear operator between two Hilbert C∗ -modules admits a bounded Moore–Penrose inverse if and only if the operator has closed range. The reader should be aware of the fact that a bounded adjointable operator may admit a regular operator as its Moore–Penrose, see Theorem 3.1, Corollary 3.4 and Proposition 3.5 of [4] for more detailed information. Koliha in [7] has given an explicit representation of the Moore–Penrose inverse in an arbitrary unital C∗ -algebra. Let T be a bounded adjointable operator on a Hilbert C∗ -module and S be its bounded Moore–Penrose inverse. According to Example 3.4 of [7] we obtain S
= lim (α 1 + T ∗ T )−1 T ∗ = lim T ∗ (α 1 + TT ∗ )−1 . α→0
α→0
Now it is natural to ask for the equality S
= T ∗ . To find an answer we focus on the norms of T and S.
Lemma 1 (cf. [9]). Suppose that a, b are positive elements of the C∗ -algebra A, and that ac all c in A. Then a = b.
= bc for
Theorem 2. Suppose a bounded adjointable operator T in B(E, F ) possesses a Moore–Penrose inverse S in B(F, E ). Then the operator T ∗ T acts on Ran(S ) as the identity operator if and only if T = S = 1. Proof. Without loss of generality we can suppose that T is not the zero operator. Let T , S satisfy (1) and T ∗ T be the identity operator on Ran(S ), then Tx, Tx = x, x for every x in Ran(S ). Consequently, for every element y in F and x = Sy we have Sy = TSy TS y y , and so S 1. Any element x of E = Ker (T ) ⊕ Ran(S ) can be written as the sum x = x0 + Sy, for an element x0 ∈ Ker (T ) and some y ∈ F. Then the equality Sy = TSy implies
Tx = 0 + TSy = Sy = Sy, Sy 1/2
x0 , x0 + Sy, Sy 1/2
= x0 + Sy, x0 + Sy 1/2 = x, x 1/2 = x . Therefore T 1. Then T Similarly T = 1.
= TST = T S T T S , that implies S 1 and so S = 1.
886
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
Conversely, suppose T , S satisfy (1) and T = S = 1. Then for every element x in E we have Tx = TSTx T STx S Tx = Tx , and hence STx = Tx . For every x in E and a in A we have
|STx|a = a∗ STx, STx a 1/2 = (STx)a, (STx)a 1/2 = ST (xa), ST (xa) 1/2 = ST (xa) = T (xa) = a∗ Tx, Tx a 1/2 = |Tx|a . It follows from Lemma 1 that |STx| = |Tx| for all x in E. We can therefore square both side to get STx, STx = Tx, Tx. Utilizing the polarization identity, we would obtain
STx, STy = Tx, Ty, for all x, y in E . It follows ST
= T ∗ T , so S = STS = T ∗ TS, i.e. T ∗ T acts on Ran(S) as the identity operator.
Corollary 3. Let T in B(E, F ) be a contraction, i.e. T 1. Then T possesses a contractive Moore–Penrose inverse if and only if T is a partial isometry. Proof. Without loss of generality we can assume that T is not the zero operator. Let S in B(F, E ) be the Moore–Penrose inverse of T such that S 1. Then T = TST = T S T T S , that is, S 1 and so S = 1. The same argument shows that T = 1. Now Theorem 2 implies that S = T ∗ TS, which yields T = TST = TT ∗ TST = TT ∗ T . Conversely, if T is a partial isometry then S = T ∗ is the contractive Moore–Penrose inverse of T. Let T in B(E, F ) be a partial isometry and S = T ∗ be its contractive Moore–Penrose inverse. Then E = Ker (T ) ⊕ Ran(S ) and so Ran(S ) = Ker (T )⊥ ⊂ {x ∈ E : Tx = x }. In view of this inclusion, the following problem, due to M. Mbekhta [11], arises in the framework of Hilbert C∗ -modules. Problem 4. Let E, F be Hilbert A-modules and T ∈ B(E, F ) be a partial isometry. If S = T ∗ is the contractive Moore–Penrose inverse of T, characterize those C∗ -algebras A for which the following equality holds: Ran(S )
= {x ∈ E : Tx = x }.
Mbekhta in [11] and Schmoeger in [13] proved that the equality holds in Hilbert spaces and strictly convex Banach spaces. Note that from the topological point of view, a Hilbert A-module is a Banach space with the norm which is not strictly convex in general. Now we are going to give another characterization of partial isometries in terms of their bounded 1×3×5×···×(2n−1) transforms. Let n be a natural number and an = (−1)n be the binomial coeffi2n n! ∞ − 1/2 n = 1 + n=1 an x , for every x in [0, 1]. In particular, for x = 1 we have 1 + cient, then (1 + x) ∞ √1 . These elementary facts are used in the following proposition. n =1 a n = 2
Proposition 5. Let T in B(E, F ) be a√bounded adjointable operator and FT be its bounded transform. T is a partial isometry if and only if T = 2 FT .
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
887
Proof. Suppose T is a partial isometry, then T ∗ T is a projection and so ∞ FT = T (1 + T ∗ T )−1/2 = T 1 + an T ∗ T n =1
=T+
∞
= 1+ √
an TT ∗ T
n =1
∞
n= 1
an
√
T
T
=√
2
√
Conversely, let T = 2 FT , then T ∗ = 2 FT ∗ = 2 FT∗ 1 √ on Ran(T ∗ ). Therefore, for every x ∈ Ran(T ∗ ), we have 2
=
√
2 (TQT )∗
=
√
2 QT T ∗ , that is, QT
=
√ √ Tx, Tx − x, x = Tx, Tx + x, x − 2x, 2x √ √ = (1 + T ∗ T )x, x − 2x, 2x
√ √ = (1 + T ∗ T )1/2 x, (1 + T ∗ T )1/2 x − 2x, 2x √ √ = QT−1 x, QT−1 x − 2x, 2x = 0.
Consequently, Tx, Tx
= x, x for all x ∈ Ran(T ∗ ). By the polarization identity,
Tx, Ty = x, y, for all x, y in Ran(T ∗ ). It follows the operator T ∗ T acts on Ran(T ∗ ) as the identity operator, i.e. T ∗ TT ∗
= T ∗ as we required.
Acknowledgments The author would like to thank professor M. Mbekhta who sent the author some copies of his recent papers. The author is also grateful to the referee for his/her useful comments.
References [1] M. Frank, Geometrical aspects of Hilbert C∗ -modules, Positivity, 3 (1999) 215–243. [2] M. Frank, Self-duality and C∗ -reflexivity of Hilbert C∗ -moduli, Z. Anal. Anwendungen 9 (1990) 165–176. [3] M. Frank, K. Sharifi, Adjointability of densely defined closed operators and the Magajna–Schweizer Theorem, J. Operator Theory, in press. Available at: arXiv:math.OA/0705.2576 v2, 29 August 2007. [4] M. Frank, K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C∗ -modules, J. Operator Theory, in press. Available at: arXiv:math.OA/0806.0162 v1, 1 June 2008. [5] J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Non-commutative Geometry, Birkhäuser, 2000. [6] P.R. Halmos, J.E. Mclaughlin, Partial isometries, Pacific J. Math. 13 (1963) 585–596. [7] J.J. Koliha, The Drazin and Moore–Penrose inverse in C∗ -algebras, Proc. Roy. Irish Acad. Sect. A 99 (1999) 17–27. [8] J. Kustermans, The functional calculus of regular operators on Hilbert C∗ -modules revisited. Available at: arXiv:functan/9706007 v1, 20 June 1997. [9] E.C. Lance, Unitary operators on Hilbert C∗ -modules, Bull. London Math. Soc. 26 (1994) 363–366. [10] E.C. Lance, Hilbert C∗ -Modules, LMS Lecture Note Series, vol. 210, Cambridge University Press, 1995. [11] M. Mbekhta, Partial isometries and generalized inverses, Acta Sci. Math. (Szeged) 70 (2004) 767–781. [12] M. Mbekhta, Operator similar to partial isometries, Acta Sci. Math. (Szeged) 71 (2005) 663–680. [13] C. Schmoeger, On a question of Mbekhta, Extracta Math. 20 (3) (2005) 281–290. [14] N.E. Wegge-Olsen, K-theory and C∗ -algebras: a Friendly Approach, Oxford University Press, Oxford, England, 1993. [15] S.L. Woronowicz, Unbounded elements affiliated with C∗ -algebras and noncompact quantum groups, Comm. Math. Phys. 136 (1991) 399–432. [16] Q. Xu, L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C∗ -modules, Linear Algebra Appl. 428 (2008) 992–1000.
Contents lists available at ScienceDirect
Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa
Descriptions of partial isometries on Hilbert C∗ -modules Kamran Sharifi Department of Mathematics, Shahrood University of Technology, P.O. Box 3619995161-316, Shahrood, Iran
A R T I C L E
I N F O
Article history: Received 31 July 2008 Accepted 27 March 2009 Available online 2 May 2009 Submitted by R.A. Brualdi AMS classification: 46L08 46C05
A B S T R A C T
Partial isometries and Moore–Penrose inverses of bounded adjointable operators on Hilbert C∗ -modules are investigated. It is shown that a contraction between Hilbert C∗ -modules is a partial isometry if and only if it possesses a contractive Moore–Penrose inverse. A bounded adjointable operator T is a partial isometry if and only if its bounded transform is 2−1/2 T. © 2009 Elsevier Inc. All rights reserved.
Keywords: Hilbert C∗ -module Partial isometry Moore–Penrose inverse
1. Introduction Partial isometries form an attractive and important class of operators. A partial isometry is an operator whose restriction to the orthogonal complement of its null-space is an isometry. Partial isometries play a vital role in operator theory; they enter, for instance, in the theory of the polar decomposition of arbitrary operators, and they form the cornerstone of the theory of von Neumann algebras. Orthogonal projections, unitaries, isometries and co-isometries are familiar examples of partial isometries. Partial isometries in Hilbert and Banach spaces have been studied in [11–13] and in the seminal paper by Halmos and Mclaughlin in [6]. Let T in B(H, H ) be a bounded operator between two Hilbert spaces H, H . A bounded operator S in B(H , H ) is called a generalized inverse of T if TST = T , STS = S. Recently, Mbekhta has shown that a contraction on a Hilbert space is a partial isometry if and only if it has a contractive generalized inverse [11].
E-mail addresses: sharifi[email protected], sharifi@shahroodut.ac.ir 0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2009.03.042
884
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
A Hilbert C∗ -module obeys the same axioms as an ordinary Hilbert space, except that the inner product, from which the geometry emerges, takes values in an arbitrary C∗ -algebra A rather than C. Fundamental and familiar Hilbert space properties like Pythagoras’ equality, self-duality, and even decomposition into orthogonal complements must be given up. They play an important role in the modern theory of C∗ -algebra, notably, in noncommutative geometry and in quantum groups, see [5]. In this paper we first present some definitions and basic facts about bounded and unbounded operators on Hilbert C∗ -modules. Moore–Penrose inverses of bounded and unbounded operators on Hilbert C∗ -modules have been studied in [16,4]. Their works with a recent result by Mbekhta [11] motivate us to describe bounded C∗ -linear partial isometries between Hilbert C∗ -modules, in fact, a contraction is a partial isometry if and only if it has a contractive Moore–Penrose inverse. Let T be a bounded adjointable operator between two Hilbert C∗ -modules and FT = T (1 + T ∗ T )−1/2 be its bounded transform (or Woronowicz transform), we prove T is a partial isometry if and only if FT
=
1 2
T.
Throughout the present paper we assume A to be an arbitrary C∗ -algebra (i.e. not necessarily unital). Bounded and unbounded operators appear in our context, so we simply denote bounded operators by capital letters and unbounded operators by small letters. We use the notations Dom(.), Ker (.) and Ran(.) for domain, kernel and range of operators, respectively. 2. Preliminaries
A (right) pre-Hilbert C∗ -module over a C∗ -algebra A is a right A-module E endowed with an Avalued inner product ·, · : E × E → A, (x, y) → x, y which is linear in the second variable y (and conjugate-linear in x), satisfying the conditions
x, y = y, x∗ , x, ya = x, ya for all a ∈ A, x, x 0 with equality if and only if x = 0. A pre-Hilbert A-module E is called a Hilbert A-module if E is a complete space with respect to the norm x = x, x 1/2 . As well as this scalar-valued norm, E has an A-valued ‘norm’ given by |x| = x, x1/2 . A pre-Hilbert A-submodule E of a Hilbert A-module F is an orthogonal summand if E ⊕ E ⊥ = F, where E ⊥ :={y ∈ F : x, y = 0 for all x ∈ E } is the orthogonal complement of E in F. In this case, E and E ⊥ are necessarily closed. If E and F are Hilbert A-modules, then a map T : E → F is called adjointable if there is a map T ∗ : F → E such that the equality Tx, y = x, T ∗ y holds for any x ∈ E, y ∈ F. It then follows that T (and also T ∗ ) is a bounded, linear, A-module map in the sense of T (xa) = (Tx)a for any x ∈ E, a ∈ A. The set of all bounded adjointable maps from E to F is denoted by B(E, F ). In particular, B(E, E ), which we abbreviate to B(E ), is a C∗ -algebra whose unit is denoted by 1. The papers [1,2], some chapters in [5,14], and the book by Lance [10] are used as standard sources of reference. Let T be a bounded adjointable operator in B(E, F ), then Theorem 3.2 of [10] implies that T has closed range if and only if its adjoint operator T ∗ has closed range, if and only if F = Ker (T ∗ ) ⊕ Ran(T ), if and only if E = Ker (T ) ⊕ Ran(T ∗ ). Let E and F be Hilbert A-modules and T in B(E, F ) be a bounded adjointable operator. T is called a partial isometry (from E0 to F0 ) if F0 = Ran(T ) is complemented in F and there exists a complemented submodule E0 of E such that T is isometric from E0 onto F0 and T (E0⊥ ) = 0. The operator T in B(E, F ) is a partial isometry if and only if T ∗ T is a projection in B(E ), if and only if TT ∗ is a projection in B(F ), if and only if TT ∗ T = T , if and only if T ∗ TT ∗ = T ∗ , cf. [10]. Let T ∈ B(E, F ), then a bounded adjointable operator S ∈ B(F, E ) is called the Moore–Penrose inverse of T if TST
= T , STS = S, (TS)∗ = TS and (ST )∗ = ST .
These properties imply that S is unique and ST is an orthogonal projection, moreover, Ran(S ) and Ker (T ) = Ker (ST ). Therefore E = Ker (ST ) ⊕ Ran(ST ) = Ker (T ) ⊕ Ran(S ).
(1)
= Ran(ST )
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
885
Let E, F be Hilbert A-modules, a regular operator from E to F is a densely defined closed A-linear map t : Dom(t ) ⊆ E → F such that t ∗ is densely defined and 1 + t ∗ t has dense range in E. Note that as we set A = C i.e. if we take E, F to be Hilbert spaces, then this is exactly the definition of a densely defined closed operator. We denote the set of all regular operators from E to F by R(E, F ). Clearly every bounded adjointable operator is a regular operator. Define Qt = (1 + t ∗ t )−1/2 and Ft = tQt , then Ran(Qt ) = Dom(t ), 0 Qt 1 in B(E ) and Ft ∈ B(E, F ). The bounded operator Ft is called the bounded transform of the regular operator t. The map t → Ft defines a bijection R(E, F )
→ {T ∈ B(E, F ) : T 1 and Ran(1 − T ∗ T ) is dense in F },
this map is adjoint-preserving, i.e. Ft∗ = Ft ∗ , cf. [10, Theorem 10.4]. Basic definitions and a few simple facts about regular operators on Hilbert C∗ -modules can be found in chapters 9 and 10 of [10], and the papers [3,8,15] with details. Very often there are interesting relationships between regular operators and their bounded transforms. Indeed, for a regular operator t, some properties transfer to its bounded transform Ft , and vice versa. Remarkably, a regular operator t is normal (resp., selfadjoint, positive) if and only if its bounded transform Ft is normal (resp., selfadjoint, positive), see [15, 1.15] and [8,√ Result 1.14]. We will prove that T is a partial isometry if and only if T is invariant under the transform 2F(.) . 3. Moore–Penrose inverses and partial isometries Xu and Sheng in [16] have shown that a bounded adjointable C∗ -linear operator between two Hilbert C∗ -modules admits a bounded Moore–Penrose inverse if and only if the operator has closed range. The reader should be aware of the fact that a bounded adjointable operator may admit a regular operator as its Moore–Penrose, see Theorem 3.1, Corollary 3.4 and Proposition 3.5 of [4] for more detailed information. Koliha in [7] has given an explicit representation of the Moore–Penrose inverse in an arbitrary unital C∗ -algebra. Let T be a bounded adjointable operator on a Hilbert C∗ -module and S be its bounded Moore–Penrose inverse. According to Example 3.4 of [7] we obtain S
= lim (α 1 + T ∗ T )−1 T ∗ = lim T ∗ (α 1 + TT ∗ )−1 . α→0
α→0
Now it is natural to ask for the equality S
= T ∗ . To find an answer we focus on the norms of T and S.
Lemma 1 (cf. [9]). Suppose that a, b are positive elements of the C∗ -algebra A, and that ac all c in A. Then a = b.
= bc for
Theorem 2. Suppose a bounded adjointable operator T in B(E, F ) possesses a Moore–Penrose inverse S in B(F, E ). Then the operator T ∗ T acts on Ran(S ) as the identity operator if and only if T = S = 1. Proof. Without loss of generality we can suppose that T is not the zero operator. Let T , S satisfy (1) and T ∗ T be the identity operator on Ran(S ), then Tx, Tx = x, x for every x in Ran(S ). Consequently, for every element y in F and x = Sy we have Sy = TSy TS y y , and so S 1. Any element x of E = Ker (T ) ⊕ Ran(S ) can be written as the sum x = x0 + Sy, for an element x0 ∈ Ker (T ) and some y ∈ F. Then the equality Sy = TSy implies
Tx = 0 + TSy = Sy = Sy, Sy 1/2
x0 , x0 + Sy, Sy 1/2
= x0 + Sy, x0 + Sy 1/2 = x, x 1/2 = x . Therefore T 1. Then T Similarly T = 1.
= TST = T S T T S , that implies S 1 and so S = 1.
886
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
Conversely, suppose T , S satisfy (1) and T = S = 1. Then for every element x in E we have Tx = TSTx T STx S Tx = Tx , and hence STx = Tx . For every x in E and a in A we have
|STx|a = a∗ STx, STx a 1/2 = (STx)a, (STx)a 1/2 = ST (xa), ST (xa) 1/2 = ST (xa) = T (xa) = a∗ Tx, Tx a 1/2 = |Tx|a . It follows from Lemma 1 that |STx| = |Tx| for all x in E. We can therefore square both side to get STx, STx = Tx, Tx. Utilizing the polarization identity, we would obtain
STx, STy = Tx, Ty, for all x, y in E . It follows ST
= T ∗ T , so S = STS = T ∗ TS, i.e. T ∗ T acts on Ran(S) as the identity operator.
Corollary 3. Let T in B(E, F ) be a contraction, i.e. T 1. Then T possesses a contractive Moore–Penrose inverse if and only if T is a partial isometry. Proof. Without loss of generality we can assume that T is not the zero operator. Let S in B(F, E ) be the Moore–Penrose inverse of T such that S 1. Then T = TST = T S T T S , that is, S 1 and so S = 1. The same argument shows that T = 1. Now Theorem 2 implies that S = T ∗ TS, which yields T = TST = TT ∗ TST = TT ∗ T . Conversely, if T is a partial isometry then S = T ∗ is the contractive Moore–Penrose inverse of T. Let T in B(E, F ) be a partial isometry and S = T ∗ be its contractive Moore–Penrose inverse. Then E = Ker (T ) ⊕ Ran(S ) and so Ran(S ) = Ker (T )⊥ ⊂ {x ∈ E : Tx = x }. In view of this inclusion, the following problem, due to M. Mbekhta [11], arises in the framework of Hilbert C∗ -modules. Problem 4. Let E, F be Hilbert A-modules and T ∈ B(E, F ) be a partial isometry. If S = T ∗ is the contractive Moore–Penrose inverse of T, characterize those C∗ -algebras A for which the following equality holds: Ran(S )
= {x ∈ E : Tx = x }.
Mbekhta in [11] and Schmoeger in [13] proved that the equality holds in Hilbert spaces and strictly convex Banach spaces. Note that from the topological point of view, a Hilbert A-module is a Banach space with the norm which is not strictly convex in general. Now we are going to give another characterization of partial isometries in terms of their bounded 1×3×5×···×(2n−1) transforms. Let n be a natural number and an = (−1)n be the binomial coeffi2n n! ∞ − 1/2 n = 1 + n=1 an x , for every x in [0, 1]. In particular, for x = 1 we have 1 + cient, then (1 + x) ∞ √1 . These elementary facts are used in the following proposition. n =1 a n = 2
Proposition 5. Let T in B(E, F ) be a√bounded adjointable operator and FT be its bounded transform. T is a partial isometry if and only if T = 2 FT .
K. Sharifi / Linear Algebra and its Applications 431 (2009) 883–887
887
Proof. Suppose T is a partial isometry, then T ∗ T is a projection and so ∞ FT = T (1 + T ∗ T )−1/2 = T 1 + an T ∗ T n =1
=T+
∞
= 1+ √
an TT ∗ T
n =1
∞
n= 1
an
√
T
T
=√
2
√
Conversely, let T = 2 FT , then T ∗ = 2 FT ∗ = 2 FT∗ 1 √ on Ran(T ∗ ). Therefore, for every x ∈ Ran(T ∗ ), we have 2
=
√
2 (TQT )∗
=
√
2 QT T ∗ , that is, QT
=
√ √ Tx, Tx − x, x = Tx, Tx + x, x − 2x, 2x √ √ = (1 + T ∗ T )x, x − 2x, 2x
√ √ = (1 + T ∗ T )1/2 x, (1 + T ∗ T )1/2 x − 2x, 2x √ √ = QT−1 x, QT−1 x − 2x, 2x = 0.
Consequently, Tx, Tx
= x, x for all x ∈ Ran(T ∗ ). By the polarization identity,
Tx, Ty = x, y, for all x, y in Ran(T ∗ ). It follows the operator T ∗ T acts on Ran(T ∗ ) as the identity operator, i.e. T ∗ TT ∗
= T ∗ as we required.
Acknowledgments The author would like to thank professor M. Mbekhta who sent the author some copies of his recent papers. The author is also grateful to the referee for his/her useful comments.
References [1] M. Frank, Geometrical aspects of Hilbert C∗ -modules, Positivity, 3 (1999) 215–243. [2] M. Frank, Self-duality and C∗ -reflexivity of Hilbert C∗ -moduli, Z. Anal. Anwendungen 9 (1990) 165–176. [3] M. Frank, K. Sharifi, Adjointability of densely defined closed operators and the Magajna–Schweizer Theorem, J. Operator Theory, in press. Available at: arXiv:math.OA/0705.2576 v2, 29 August 2007. [4] M. Frank, K. Sharifi, Generalized inverses and polar decomposition of unbounded regular operators on Hilbert C∗ -modules, J. Operator Theory, in press. Available at: arXiv:math.OA/0806.0162 v1, 1 June 2008. [5] J.M. Gracia-Bondía, J.C. Várilly, H. Figueroa, Elements of Non-commutative Geometry, Birkhäuser, 2000. [6] P.R. Halmos, J.E. Mclaughlin, Partial isometries, Pacific J. Math. 13 (1963) 585–596. [7] J.J. Koliha, The Drazin and Moore–Penrose inverse in C∗ -algebras, Proc. Roy. Irish Acad. Sect. A 99 (1999) 17–27. [8] J. Kustermans, The functional calculus of regular operators on Hilbert C∗ -modules revisited. Available at: arXiv:functan/9706007 v1, 20 June 1997. [9] E.C. Lance, Unitary operators on Hilbert C∗ -modules, Bull. London Math. Soc. 26 (1994) 363–366. [10] E.C. Lance, Hilbert C∗ -Modules, LMS Lecture Note Series, vol. 210, Cambridge University Press, 1995. [11] M. Mbekhta, Partial isometries and generalized inverses, Acta Sci. Math. (Szeged) 70 (2004) 767–781. [12] M. Mbekhta, Operator similar to partial isometries, Acta Sci. Math. (Szeged) 71 (2005) 663–680. [13] C. Schmoeger, On a question of Mbekhta, Extracta Math. 20 (3) (2005) 281–290. [14] N.E. Wegge-Olsen, K-theory and C∗ -algebras: a Friendly Approach, Oxford University Press, Oxford, England, 1993. [15] S.L. Woronowicz, Unbounded elements affiliated with C∗ -algebras and noncompact quantum groups, Comm. Math. Phys. 136 (1991) 399–432. [16] Q. Xu, L. Sheng, Positive semi-definite matrices of adjointable operators on Hilbert C∗ -modules, Linear Algebra Appl. 428 (2008) 992–1000.