International Journal of Analysis and Applications ISSN 2291-8639 Volume 8, Number 2 (2015), 110-122 http://www.etamaths.com STRONG METRIZABILITY FOR CLOSED OPERATORS AND THE SEMI-FREDHOLM OPERATORS BETWEEN TWO HILBERT SPACES MOHAMMED BENHARRAT1 AND BEKKAI MESSIRDI2,∗ Abstract. To be able to refine the completion of C(H1, H2), the set of all closed densely defined linear operators between two Hilbert spaces H1 and H2, we define in this paper some new strictly stronger metrics than the gap metric g and we characterize the closure with respect to theses metrics of the subset L(H1, H2) of bounded elements of C(H1, H2). In addition, several operator norm inequalities concerning the equivalence of some metrics on L(H1, H2) are presented. We also establish the semi-Fredholmness and Fredholmness of unbounded operators in terms of bounded pure contractions. 1. Introduction Let H, H1, H2 be a complex Hilbert spaces endowed with the appropriate scalar product and the associated norm. The inner product in H1 × H2 is defined by < (x,y); (x′,y′) >=< x; x′ > + < y; y′ >. For T linear operator from H1 to H2, the symbols D(T) ⊂ H1, N(T) ⊂ H1 and R(T) ⊂ H2 will denote the domain, null space and the range space of T , respectively. The set G(T) = {(x,Tx) : x ∈ D(T)} ⊂ H1 × H2 is called the graph of T . The operator T is closed if and only if G(T) is a closed subset of H1 ×H2, and is densely defined if D(T) = H1, where D(T) denote the closure of D(T) in H1. The set of all closed and densely defined linear operators from H1 to H2 will be denoted by C(H1,H2). Denote by L(H1,H2) the Banach space of all bounded linear operators from H1 to H2. If H1 = H2, write C(H1,H2) = C(H1) and L(H1,H2) = L(H1). If T ∈C(H1,H2), the adjoint T∗ of T exists, is unique and T∗ ∈C(H2,H1). An operator A ∈L(H1,H2) is a pure contraction if ‖Ax‖2 < ‖x‖1 for all nonzero x in H1. We denote by L0(H1,H2) the set of all pure contractions. In [9] W. E. Kaufman showed that if T ∈ C(H) then T is represented as T = Γ(A) = A(I −A∗A)−1/2 using a unique pure contraction A defined in H, where I denote the identity in H. Since the publication of Kaufman [9] in 1978 and its follows papers, this Kaufman’s rep- resentation is used to reformulate questions about unbounded operators in terms of bounded ones: • In [9] [11], Kaufman proved that the map Γ preserves many properties of operators: self-adjontness, nonnegative conditions, normality and quasinormality. In [12] he also defined by the use of Γ−1 a metric in the space of closed densely defined Hilbert space which is stronger than the gap metric and sharing many of its properties. On the the bounded operators it is equivalent to the metric generated by the usual operator-norm. • In [5] Hirasawa showed that a pure contraction A is hyponormal if and only if Γ(A) is formally hyponormal, and if A is quasinormal then Tn = An(I − A∗A)−n/2 is quasi- normal for all integers n ≥ 2. • In [2], Cordes and Labrousse proved that if a closed and densely defined operator T is semi Fredholm then so is the bounded operator Γ−1(T) = T(I + T∗T)−1/2. 2010 Mathematics Subject Classification. 47A10, 47A30 and 47A53. Key words and phrases. Pure contractions, Closed densely defined linear operators, The gap metric, The gap topology, Semi-Fredholm operators. c©2015 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 110 STRONG METRIZABILITY FOR CLOSED OPERATORS 111 • In [1], Benharrat and Messirdi prove that if a pure contraction A on a Hlibert space H is semi Fredholm, then the closed densely defined linear operator λI −T = λI −A(I − A∗A)−1/2 is semi Fredholm operator for all λ ∈ C such that |λ| < γ(A) 1+γ(A) . Recently, J. J. Koliha in [14] extend Kaufman’s results to operators between two Hilbert spaces and showed that the mapping Γ maps the set L0(H1,H2) one-to-one onto the set C(H1,H2). More precisely, we have the following result. Theorem 1.1. [14, Theorem 5.] Let L0(H1,H2) be the set of all pure contractions from H1 to H2, C(H1,H2) the set of all closed and densely defined linear operators from H1 to H2, and G ∈L+(H1) a positive bijection. The mapping ΓG defined by ΓG(A) = AG 1/2(G1/2(I −A∗A)G1/2)−1/2G1/2, A ∈L0(H1,H2), is a bijection of L0(H1,H2) onto C(H1,H2) with the inverse Γ−1G (T) = T(G + T ∗T)−1/2, T ∈C(H1,H2). In this paper, by the use of the generalized Kaufman’s representation, we discuss some metrics in the space C(H1,H2) endowed with the gap metric. More precisely, in Section 2, we define some metrics on C(H1,H2) equivalent to the gap metric. In Section 3, we define on C(H1,H2) a metric in term of Γ−1, strictly stronger than the gap metric and is equivalent to the metric associated to the operator-norm in L(H1,H2). We characterize essentially the closure of L(H1,H2) in C(H1,H2) for this metric. In section 4, we prove some operator norm inequalities for bounded operators between two Hilbert spaces. In the last section, we establish some characterizations of Fredholm unbounded operators in terms of bounded pure contractions by treating unbounded operators between two Hilbert spaces rather than restricting the investigation to operators on a single space. 2. A strong metric for closed operators between two Hilbert spaces Recall that, if T ∈ C(H1,H2), then the operator RT = (I + T∗T)−1 is self-adjoint positive operator defined on all H1, and has a unique positive definite self-adjoint square root, which we denoted by ST . The fundamental properties of RT and ST are, see [15]: RT ,ST ∈L(H1,H2), ‖RT‖≤ 1, ‖ST‖≤ 1, ‖TRT‖≤ 1, ‖TST‖≤ 1, and (2.1) (TRT ) ∗ = T∗RT∗, (TST ) ∗ = T∗ST∗, T ∗ST∗TST = I −RT . In the sequel L+(H1) denotes the set of all positive bijective operators in L(H1). Let G be an element of L+(H1). By Theorem 1.1, for a given pure contraction A ∈L0(H1,H2), there exists a unique operator T ∈C(H1,H2) such that the equation (2.2) XGX = I −A∗A, admits a unique solution given by (2.3) X = (G + T∗T)−1/2 = G−1/2(G1/2(I −A∗A)G1/2)1/2G−1/2, with T = ΓG(A). Further, the operator TG −1/2 has domain G1/2D(T), which is clearly dense in H1. Therefore RTG−1/2 is defined on all H1 and (2.4) (G + T∗T)−1 = G−1/2RTG−1/2G −1/2. Hence (G + T∗T)−1 ∈L(H1). Let PG(T) denote the correspondence which assigns to each T ∈ C(H1,H2) the orthogonal projection from H1 ×H2 onto the graph G(T) of T . 112 BENHARRAT AND MESSIRDI Lemma 2.1. Let T ∈C(H1,H2) and G ∈L+(H1). Then (2.5) PG(TG−1/2) = ( G1/2(G + T∗T)−1G1/2 G1/2(G + T∗T)−1T∗ T(G + T∗T)−1G1/2 I −T(G + T∗T)−1T∗ ) . Proof. Let v ∈D(G1/2T∗) and X = (u,v) ∈ H1 ×H2 such that PG(TG−1/2)X = (x,TG−1/2x), for x ∈ G1/2D(T). Since G(TG−1/2)⊥ = V (G(G−1/2T∗)) with the isomorphism V from H1×H2 to H2 ×H1 defined by V (x1,x2) = (−x2,x1), we get PG(TG−1/2)X from the decomposition X = (u,v) = (x,TG−1/2x) + (−G1/2T∗y,y), y ∈D(G1/2T∗), x ∈ G1/2D(T), where x and y are the solutions of the system:{ u = x−G1/2T∗y v = TG−1/2x + y. By solving this system we get{ x = G1/2(G + T∗T)−1G1/2u + G1/2(G + T∗T)−1T∗v y = T(G + T∗T)−1G1/2u + (I −T(G + T∗T)−1T∗)v. � By (2.4), we can rewrite (2.5) as follows (2.6) PG(TG−1/2) = ( RTG−1/2 RTG−1/2G −1/2T∗ TG−1/2RTG−1/2 I −TG−1/2RTG−1/2G−1/2T∗ ) . If H1 = H2, we also have (2.7) PG(TG−1/2) = ( RTG−1/2 G −1/2T∗RG−1/2T∗ TG−1/2RTG−1/2 I −RG−1/2T∗ ) . In the case of G = IH1 Lemma 2.1 reduces to the following well-known statement, see [15]. Corollary 2.2. Let T ∈C(H1,H2). Then the orthogonal projection PG(T) in H1 ⊕H2 onto the graph G(T) of T , is given by (2.8) PG(T) = ( RT T ∗RT∗ TRT I −RT∗ ) . Definition 2.3. Let G ∈ L+(H1) and T,S ∈ C(H1,H2). The gap metric between T and S associated to G is defined by (2.9) gG(T,S) = ∥∥PG(TG−1/2) −PG(SG−1/2)∥∥ . Note that if G = IH1 , we have the usual gap metric (cf. [8, p. 201]) for T,S ∈C(H1,H2), (2.10) g(T,S) = ∥∥PG(T) −PG(S)∥∥ for all T,S ∈C(H1,H2). Thus, for an infinite sequence (Tn) of C(H1,H2), g(Tn,T) → 0 if and only if each the following conditions hold (i) ‖RTn −RT‖→ 0, (ii) ‖TnRTn −TRT‖→ 0, (iii) ∥∥RT∗n −RT∗∥∥ → 0, (iv) ∥∥T∗nRT∗n −T∗RT∗∥∥ → 0. Similarly, we can express the convergence with respect to the metric gG as follows : Proposition 2.4. For an infinite sequence (Tn) of C(H1,H2), g(Tn,T) → 0 in the sens of Definition 2.3 if and only if each the following conditions hold (i) ∥∥RTnG−1/2 −RTG−1/2∥∥ → 0, (ii) ∥∥TnG−1/2RTnG−1/2 −TG−1/2RTG−1/2∥∥ → 0, (iii) ∥∥TnG−1/2RTnG−1/2G−1/2T∗n −TG−1/2RTG−1/2G−1/2T∗∥∥ → 0, (iv) ∥∥RTnG−1/2G−1/2T∗n −RTG−1/2G−1/2T∗∥∥ → 0. STRONG METRIZABILITY FOR CLOSED OPERATORS 113 with G ∈L+(H1). By (2.9) and (2.10) we can deduce that gG(T,S) = g(TG −1/2,SG−1/2). Let M,N be two closed linear subspaces of the Hilbert space H. Denote by PM and PN the orthogonal projection onto M and N respectively. Set δ(M,N) = ‖(I −PN )PM‖ , δ is a pseudo-distance, for its properties we can see also [2]. We define another metric on C(H1,H2) as follows, dG(T,S) = ∥∥(I −PG(TG−1/2))PG(SG−1/2)∥∥ + ∥∥∥(I −PG(S)G−1/2)PG(TG−1/2)∥∥∥ , for all T,S ∈C(H1,H2) with G ∈L+(H1). We notice that gG(T,S) = max{δ(G(TG −1/2),G(SG−1/2)),δ(G(SG−1/2),G(TG−1/2))} and dG(T,S) = δ(G(TG−1/2),G(SG−1/2)) + δ(G(SG−1/2),G(TG−1/2)). The following result is immediately obtained. Corollary 2.5. If G ∈ L+(H1); then dG and gG are equivalent metrics on C(H1,H2), in particular we have gG(T,S) ≤ dG(T,S) ≤ 2gG(T,S). Put P = (I −PG(TG−1/2))PG(SG−1/2), let us remark that (2.11) P = UT [ 0 0 TG−1/2STG−1/2SSG−1/2 −S∗TG−1/2SG −1/2SSG−1/2 0 ] US. with UT = [ STG−1/2 TTG−1/2G −1/2T∗ TG−1/2STG−1/2 S ∗ TG−1/2 ] . Then we can deduce that Corollary 2.6. If G ∈L+(H1), then for T,S ∈C(H1,H2) we have dG(T,S) = ∥∥∥TG−1/2STG−1/2SSG−1/2 −S∗TG−1/2SG−1/2SSG−1/2∥∥∥ + ∥∥∥SG−1/2SSG−1/2STG−1/2 −S∗SG−1/2TG−1/2STG−1/2∥∥∥ . Furthermore, if T,S are bounded, then dG(T,S) = ∥∥∥G−1/2S∗TG−1/2 (T −S)SSG−1/2∥∥∥ + ∥∥∥G−1/2S∗SG−1/2 (S −T)STG−1/2∥∥∥ . For an operator G ∈L+(H1), we define a third metric on C(H1,H2) by pG(T,S) = [ ‖RTG−1/2 −RSG−1/2‖ 2 + ∥∥∥TG−1/2RTG−1/2 −SG−1/2RSG−1/2∥∥∥2]1/2 . It easy to see that pG(T,S) ≤ gG(T,S). Hence Theorem 2.7. The topology induced from the gap metric gG on C(H1,H2) is strictly stronger than that induced from pG. The following example exclude the possibility that the metrics pG and gG generate the same topology even in the case of G = IH1 . 114 BENHARRAT AND MESSIRDI Example 2.8. Let H1 and H2 two separable Hilbert spaces and {φn}, {ψn} an orthonormal basis in H1, H2 respectively. Put for n ∈ N∗, Tnφk = { kψk, if k < n −kψk+1 if k ≥ n. Then, T∗nψk =   kφk, if k < n 0 if k = n −kφk if k > n and thus, RTnφk =   1 1+k2 φk, if k < n φn if k = n 1 1+(k−1)2 φk if k > n Define the operator T by Tφk = kψk, k ∈ N∗. Then, RT = RT∗ = RT∗n , ∥∥RT∗ −RT∗n∥∥ = 0, and T∗RT∗ −T∗nRT∗n ψk =   0 if k < n n 1+n2 φn if k = n k 1+k2 φk if k > n Thus ‖TRT −TnRTn‖ = ∥∥T∗RT∗ −T∗nRT∗n∥∥ ≤ 2√ 1 + n2 → 0. On the other hand, (RT −RTn )φk =   0 if k < n ( n 1+n2 − 1)φn if k = n 1−2k (1+k2)(1+(k−1)2 φk if k > n Then ‖RT −RTn‖≥ n2 √ 1 + n2 → 1. Finally, if we put Sn = T ∗ n and S = T ∗, we get pG(S,Sn) → 0 and gG(S,Sn) → 1. 3. A new strong metric than the gap metric Let G ∈L+(H1) and T,S ∈C(H1,H2). We define another metric in terms of Γ−1G , given in Theorem 1.1, as follows, qG(T,S) = ∥∥Γ−1G (T) − Γ−1G (S)∥∥ . Clearly C(H1,H2) is isometric to the subset L0(H1,H2) of the unit ball in L(H1,H2) under the operator-norm, so that qG(T,S) ≤ 2 for all T,S ∈ C(H1,H2). The related convergence in the space C(H1,H2), called quotient-convergence associated to G. The purpose of the following theorem is to prove that the metric qG is stronger than dG. Theorem 3.1. Let G ∈ L+(H1). The metric topology induced by qG is stronger than that induced by the gap metric gG in C(H1,H2). Proof. Let T ∈C(H1,H2) and (Tn) an infinite sequence of C(H1,H2), such that qG(Tn,T) → 0. By Theorem 1.1 then we can write T = ΓG(A) (resp Tn = ΓG(An)) with a unique positive contraction A ∈L0(H1,H2) (resp. An ∈L0(H1,H2) for all n). Thus, (G + T∗T)−1/2G(G + T∗T)−1/2 = I −A∗A, and (G + T∗nTn) −1/2G(G + T∗nTn) −1/2 = I −A∗nAn. Therefore, by (2.3) the orthogonal projections PG(TG−1/2) and PG(TnG −1/2 n ) are easily computed from (2.3), and we obtain respectively, (3.1) PG(TG−1/2) = ( UG−1U UG−1UG−1/2(ΓG(A)) ∗ AG−1/2U I −AG−1/2UG−1/2(ΓG(A))∗ ) , STRONG METRIZABILITY FOR CLOSED OPERATORS 115 and (3.2) PG(TnG−1/2) = ( UnG −1Un UnG −1UnG −1/2(ΓG(An)) ∗ AnG −1/2Un I −AnG−1/2UnG−1/2(ΓG(An))∗ ) , where U = (G1/2(I −A∗A)G1/2)1/2 and Un = (G1/2(I −A∗nAn)G1/2)1/2 for all n ∈ N. Consequently, if An converges to A in L0(H1,H2), then Un converges to U and this assures the convergence PG(TnG−1/2) −→ PG(TG−1/2) as n −→∞, hence gG(Tn,T) → 0. � In the following example, we show that is not possible that the metrics qG and gG generate the same topology even for G = IH1 . Example 3.2. Let H1 and H2 be two separable Hilbert spaces and {φn}, {ψn} an orthonormal basis in H1, H2 respectively. Put for n ∈ N∗, Tnφk = { kψk, if k < n −kψk if k ≥ n. Then, T∗nψk = { kφk, if k < n −kφk if k ≥ n and thus, RTnφk = 1 1 + k2 φk, RT∗n ψk = 1 1 + k2 ψk. If we define the operator T by Tφk = kψk, k ∈ N∗. Then, T = Γ(A) where Aφk = k√1+k2 ψk, A ∈L0(H1,H2), we see that the conditions (i)-(iv) of Proposition 2.4 holds. Thus gG(Tn,T) → 0. On the other hand, as n −→∞, qG(Tn,T) = ‖An −A‖ = 2n √ 1 + n2 → 2. We have also the following result: Corollary 3.3. The topology induced on C(H1,H2) by the metric qG is strictly stronger than the topology induced by the metric dG. Lemma 3.4. Let G ∈L+(H1). An operator T ∈C(H1,H2) is bounded if and only if ∥∥Γ−1G (T)∥∥ < 1. In this case, (3.3) ‖T‖ = ∥∥Γ−1G (T)∥∥∥∥G1/2∥∥2√ 1 − ∥∥Γ−1G (T)∥∥2 . Proof. Let T ∈C(H1,H2) a bounded operator, then for all x ∈ H1, we have ‖x‖2 = ∥∥∥G1/2(G + T∗T)−1/2x∥∥∥2 + ∥∥∥T(G + T∗T)−1/2x∥∥∥2 ≤ [ 1 + ∥∥∥G−1/2∥∥∥2 ‖T‖2]∥∥∥G1/2(G + T∗T)−1/2x∥∥∥2 . Thus, ∥∥∥G1/2(G + T∗T)−1/2x∥∥∥2 ≥ 1 1 + ∥∥G−1/2∥∥2 ‖T‖2 ‖x‖2 . Consequently∥∥∥T(G + T∗T)−1/2x∥∥∥2 = ‖x‖2 −∥∥∥G1/2(G + T∗T)−1/2x∥∥∥2 ≤ ‖T‖2∥∥G1/2∥∥2 + ‖T‖2 ‖x‖2 . Hence (3.4) ∥∥∥T(G + T∗T)−1/2∥∥∥ ≤ ‖T‖√∥∥G1/2∥∥2 + ‖T‖2 < 1. 116 BENHARRAT AND MESSIRDI Conversely, assume that ∥∥Γ−1G (T)∥∥ < 1. Then I − (Γ−1G (T))∗Γ−1G (T) is invertible and for all x ∈ H1, we have〈 G1/2(I − (Γ−1G (T)) ∗Γ−1G (T))G 1/2x,x 〉 = ∥∥∥G1/2x∥∥∥2 −∥∥∥Γ−1G (T)G1/2x∥∥∥2 ≥ [ 1 − ∥∥Γ−1G (T)∥∥2]∥∥∥G1/2x∥∥∥2 . Hence ∥∥∥∥[G1/2(I − (Γ−1G (T))∗Γ−1G (T))G1/2]−1/2 ∥∥∥∥ ≤ 1√ 1 − ∥∥Γ−1G (T)∥∥2 . Since T = Γ−1G (T)G 1/2 [ G1/2(I − (Γ−1G (T)) ∗Γ−1G (T))G 1/2 ]−1/2 G1/2, we obtain (3.5) ‖T‖≤ ∥∥Γ−1G (T)∥∥∥∥G1/2∥∥2√ 1 − ∥∥Γ−1G (T)∥∥2 . This implies the boundedness of T . Furthermore, by (3.4) and (3.5) we obtain (3.3). � Theorem 3.5. L(H1,H2) is dense open subset of C(H1,H2) endowed with the metric qG. Proof. L(H1,H2) is an open subset of C(H1,H2) with respect to the metric qG follows im- mediately from Lemma 3.4. Now suppose that T ∈ C(H1,H2), then Γ−1G (T) is in the unit closed ball of L(H1,H2), relative to the operator-norm. Hence { nn+1 Γ −1 G (T)} is a sequence {An} of operators such that for each n, ‖An‖ < 1 and ∥∥An − Γ−1G (T)∥∥ −→ 0. For each n, we put Tn = ΓG(An), by Theorem 1.1 each Tn is in L(H1,H2) and, clearly, qG(Tn,T) =∥∥Γ−1G (Tn) − Γ−1G (T)∥∥ −→ 0. This complete the proof. � Definition 3.6. Let T1,T2 ∈C(H1,H2). We put ΣG(T1,T2) = [ 2qG(T1,T2) 2 + ∥∥ST1G−1/2 −ST2G−1/2∥∥2 + ∥∥∥SG−1/2T∗1 −SG−1/2T∗2 ∥∥∥2 ]1/2 . ΣG is a metric on C(H1,H2) and note that the sequence defined in the Example 2.8 converges on C(H1,H2) for the metric qG but is not convergent for the metric ΣG. Theorem 3.7. The topology induced on C(H1,H2) by the metric ΣG is strictly stronger than the topology induced from the metric qG. By Theorem 3.7 and Theorem 3.3 we obtain the following results. Corollary 3.8. The topology induced on C(H1,H2) by the metric ΣG is strictly stronger than the topology induced from the gap metric gG. Theorem 3.9. L(H1,H2) is dense open subset of C(H1,H2) endowed with the metric ΣG. For the proof of this theorem we need the following lemma Lemma 3.10. If T ∈C(H1,H2) and B ∈L(H1,H2) such that ΣG(T,B) < 1√ 1 + ∥∥BG−1/2∥∥2 , then T ∈L(H1,H2). Proof. Let x ∈ G1/2D(T), then for all y ∈ H2,〈 TG−1/2x,y 〉 − 〈 x,G−1/2B∗y 〉 = 〈 (x,TG−1/2x), (−G−1/2B∗y,y) 〉 = 〈 PG(TG−1/2)(x,TG −1/2x), (I −PG(BG−1/2))(−G −1/2B∗y,y) 〉 , STRONG METRIZABILITY FOR CLOSED OPERATORS 117 by using Schwarz inequality, (3.6) ∣∣∣〈(T −B)G−1/2x,y〉∣∣∣ ≤ gG(T,B) ∥∥∥(x,TG−1/2x)∥∥∥∥∥∥(−G−1/2B∗y,y)∥∥∥ . Setting y = (T −B)G−1/2x in (3.6) it follows that ∥∥∥(T −B)G−1/2x∥∥∥ ≤ ΣG(T,B)√‖x‖2 + ∥∥TG−1/2x∥∥2√1 + ∥∥BG−1/2∥∥2. Let us put ΣG(T,B) √ 1 + ∥∥BG−1/2∥∥2 = 1 − �, � > 0. Thus, ∥∥∥TG−1/2x∥∥∥ ≤ ∥∥∥BG−1/2x∥∥∥ + (1 − �)[‖x‖ + ∥∥∥TG−1/2x∥∥∥], finally ∥∥∥TG−1/2x∥∥∥ ≤ 1 � [ ∥∥∥G1/2∥∥∥ + ∥∥∥BG−1/2∥∥∥] ∥∥∥G−1/2x∥∥∥ , what shows that T is bounded from H1 to H2. � Proof of Theorem 3.9. From Lemma 3.10 L(H1,H2) is an open subset of C(H1,H2). Now, we show the density. Let T ∈C(H1,H2), then there exists an unique pure contraction A such that A = Γ−1G (T). We put Tn = n n+1 A as in the proof of Theorem 3.5. Then Tn ∈ L(H1,H2) and qG(Tn,T) −→ 0. On the other hand, let RGT = (G + T ∗T)−1 and SGT = (G + T ∗T)−1/2, then∥∥RGTn −RGT ∥∥ = ∥∥SGTnT∗nTnSGTn −SGT T∗TSGT ∥∥ = ∥∥SGTnT∗nTnSGTn + SGTnT∗nTSGT −SGTnT∗nTSGT −SGT T∗TSGT ∥∥ ≤ ∥∥SGTnT∗n∥∥∥∥TnSGTn −TSGT ∥∥ + ∥∥SGTnT∗n −SGT T∗∥∥∥∥TSGT ∥∥ ≤ [∥∥SGTnT∗n∥∥ + ∥∥TSGT ∥∥]qG(Tn,T). Thus, limn→+∞ ∥∥RGTn −RGT ∥∥ = 0, hence limn→+∞∥∥SGTn −SGT ∥∥ = 0. By (2.4), we observe that STG−1/2 = (G 1/2RGT G 1/2)1/2, then we conclude that lim n→+∞ ∥∥STnG−1/2 −STG−1/2∥∥ = ∥∥∥SG−1/2T∗n −SG−1/2T∗∥∥∥ = 0. Thus ΣG(Tn,T) −→ 0, this shows the density of L(H1,H2) in C(H1,H2). � 4. Some equivalent metrics for bounded operators between two Hilbert spaces In this section we present several operator norm inequalities to compare the metric qG, the gap metric gG, and the usual operator norm metric. More presicily, we show that these three metrics are equivalent in L(H1,H2). Our results extend those obtained in [12] and [13] to the bounded operators between two Hilbert spaces. Lemma 4.1. If T1,T2 ∈L(H1,H2), then ‖T1 −T2‖≤ 1 2 ∥∥(SGT1 )−1 + (SGT2 )−1∥∥qG(T1,T2) + 1 2 ∥∥(RGT1 )−1∥∥∥∥(RGT2 )−1∥∥∥∥Γ−1G (T1) + Γ−1G (T2)∥∥2 qG(T1,T2). 118 BENHARRAT AND MESSIRDI Proof. Let T1,T2 ∈L(H1,H2), we have ‖T1 −T2‖ = ∥∥Γ−1G (T1)(SGT1 )−1 − Γ−1G (T2)(SGT2 )−1∥∥ ≤ 1 2 qG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 12 ∥∥T1SGT1 + T2SGT2∥∥∥∥(SGT1 )−1 − (SGT2 )−1∥∥ ≤ 1 2 qG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 14 ∥∥T1SGT1 + T2SGT2∥∥∥∥(RGT1 )−1 − (RGT2 )−1∥∥ ≤ 1 2 qG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 1 4 ∥∥(RGT1 )−1∥∥∥∥(RGT2 )−1∥∥∥∥T1SGT1 + T2SGT2∥∥∥∥RGT1 −RGT2∥∥ . Since ∥∥RGT1 −RGT2∥∥ = ∥∥(Γ−1G (T1))∗Γ−1G (T1) − (Γ−1G (T2))∗Γ−1G (T2)∥∥ ≤ qG(T1,T2) ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥ ,(4.1) it follows the desired inequality. � Lemma 4.2. If T1,T2 ∈L(H1,H2), then qG(T1,T2) ≤ (1 + 1 4 ‖T1 + T2‖ 2 )‖T1 −T2‖ . Proof. Let T1,T2 ∈L(H1,H2), we have qG(T1,T2) = ∥∥Γ−1G (T1) − Γ−1G (T2)∥∥ = ∥∥∥∥12 (T1 −T2)(SGT1 + SGT2 ) + 12 (T1 + T2)(SGT1 −SGT2 ) ∥∥∥∥ ≤‖T1 −T2‖ + 1 2 ‖T1 + T2‖ ∥∥SGT1 −SGT2∥∥ ≤‖T1 −T2‖ + 1 2 ‖T1 + T2‖ ∥∥(SGT1 )−1 − (SGT2 )−1∥∥ ≤‖T1 −T2‖ + 1 4 ‖T1 + T2‖ ∥∥(RGT1 )−1 − (RGT2 )−1∥∥ = ‖T1 −T2‖ + 1 4 ‖T1 + T2‖‖T∗1 T1 −T ∗ 2 T2‖ ≤‖T1 −T2‖(1 + 1 4 ‖T1 + T2‖ 2 ). � Combining Lemma 4.1 and Lemma 4.2, we obtain the following result: Theorem 4.3. Let G ∈ L+(H1). The restriction of the metric qG to L(H1,H2) is equivalent to the operator-norm. Note that this theorem is extended to the unbounded operators between two Hilbert spaces, the result was shown by W. E. Kaufman in [12, Theorem 2] in the case of unbounded operators defined on a single Hilbert space and when G = IH1 . Lemma 4.4. If T1,T2 ∈L(H1,H2), then qG(T1,T2) ≤ 1 2 [∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 12 ‖G + T∗1 T1‖‖G + T∗2 T2‖ ] gG(T1,T2). STRONG METRIZABILITY FOR CLOSED OPERATORS 119 Proof. Let T1,T2 ∈L(H1,H2), we have qG(T1,T2) = ∥∥T1RGT1 (SGT1 )−1 −T2RGT2 (SGT2 )−1∥∥ ≤ 1 2 ∥∥T1RGT1 −T2RGT2∥∥∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 1 2 ∥∥T1RGT1 + T2RGT2∥∥∥∥(SGT1 )−1 − (SGT2 )−1∥∥ ≤ 1 2 gG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 12 ∥∥(SGT1 )−1 − (SGT2 )−1∥∥ ≤ 1 2 gG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 14 ∥∥(RGT1 )−1 − (RGT2 )−1∥∥ ≤ 1 2 gG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 14 ∥∥(RGT1 )−1∥∥∥∥(RGT2 )−1∥∥∥∥(RGT1 ) − (RGT2∥∥ ≤ 1 2 gG(T1,T2) ∥∥(SGT1 )−1 + (SGT2 )−1∥∥ + 14 ∥∥(RGT1 )−1∥∥∥∥(RGT2 )−1∥∥gG(T1,T2). � Lemma 4.5. If T1,T2 ∈L(H1,H2), then g2G(T1,T2) ≤ [ ( ∥∥∥G1/2∥∥∥4 + 1) ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥2 + ∥∥∥G1/2∥∥∥2 ] q2G(T1,T2) + 2 ∥∥∥G1/2∥∥∥2 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥3/2 q3/2G (T1,T2) + 1 2 ∥∥∥G1/2∥∥∥2 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥3 qG(T1,T2). Proof. By using the representation (2.5), we get g2G(T1,T2) ≤ ∥∥∥G1/2∥∥∥4 ∥∥RGT1 −RGT2∥∥2 + 2 ∥∥∥G1/2∥∥∥2 ∥∥TG1 RGT1 −T2RGT2∥∥2 + ∥∥T1RGT1T∗1 −T2RGT2T∗2 ∥∥2 . We have∥∥T1RGT1 −T2RGT2∥∥ ≤ 12 qG(T1,T2) ∥∥SGT1 + SGT2∥∥ + 12 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥∥∥SGT1 −SGT2∥∥ ≤ qG(T1,T2) + 1 2 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥∥∥SGT1 −SGT2∥∥ ≤ qG(T1,T2) + 1 2 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥∥∥RGT1 −RGT2∥∥1/2 ≤ qG(T1,T2) + 1 2 ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥3/2 q1/2G (T1,T2). In view of these estimations and the fact that, by (4.1), both ∥∥T1RGT1T∗1 −T2RGT2T∗2 ∥∥ and∥∥RGT1 −RGT2∥∥ are majorized by qG(T1,T2) ∥∥Γ−1G (T1) + Γ−1G (T2)∥∥ we get the required inequality. � Combining Lemma 4.4 and Lemma 4.5 we obtain: Theorem 4.6. In L(H1,H2) the metric qG is equivalent to the gap metric gG. Combining Theorem 4.3, Theorem 4.6 and Corollary 2.5 we deduce: Corollary 4.7. The metrics qG, gG, pG and the operator-norm metric are equivalent on L(H1,H2). 120 BENHARRAT AND MESSIRDI 5. Pure contractions and semi-Fredholm operators In this section, by the use of the generalized Kaufman’s representation, we present some results concerning the characterization of unbounded semi Fredholm operators in terms of bounded ones. We begin by introduce now some important classes of operators in Fredholm theory. In the sequel, for every T ∈ C(H1,H2), let α(T) and β(T) be the nullity and the deficiency of T defined as α(T) := dim N(T), and β(T) := codimR(T). If the range R(T) of T is closed and α(T) < ∞ (resp. β(T) < ∞), then T is called an upper (resp. a lower) semi-Fredholm operator. If T is either upper or lower semi-Fredholm, then T is called a semi-Fredholm operator, and the index of T is defined by ind(T) := α(T) − β(T). If both α(T) and β(T) are finite, then T is a called a Fredholm operator. In the following, A denotes a pure contraction from H1 to H2, and T the closed and densely-defined operator ΓG(A) = AG 1/2B−1G1/2 from H1 to H2, with B = (G 1/2(I − A∗A)G1/2)1/2 such that G−1/2BG−1/2 is the unique solution of the equation (2.2) with G ∈GL+(H1). Note that since A is a pure contraction, B is a positive and injective element of L(H1). Recall that the reduced minimum modulus of a non-zero operator T is defined by γ(T) = inf x∈N(T)⊥ ‖Tx‖ ‖x‖ If T = 0 then we take γ(T) = ∞. Note that (see [8]): γ(T) > 0 ⇔ R(T) is closed. Lemma 5.1 ([8]). (1) If δ(M,N) < 1 then dim M ≤ dim N. (2) δ(M,N) = δ(N⊥,M⊥). The main results of this section is: Theorem 5.2. Let A ∈ L0(H1,H2). If A is upper semi-Fredholm operator then λC − ΓG(A) is upper semi-Fredholm operator for all C ∈L(H1,H2) and |λ| < γ(A) 1+γ(A) ‖G1/2‖ ‖C‖ . Proof. Let A ∈ L0(H1,H2), C ∈ L(H1,H2) and B denote the positive member (G1/2(I − A∗A)G1/2)1/2 of L0(H). Since A is a pure contraction, B is one-to-one with dense range in H1, and the fact that λC − ΓG(A) = (λCG−1/2BG−1/2 −A)G1/2B−1G1/2, it follows that to prove λC − ΓG(A) is upper semi-Fredholm operator it suffices to prove that (λCG−1/2BG−1/2 −A) is upper semi-Fredholm one. For each nonzero x in H1, ‖x‖ 2 −‖Ax‖2 = ∥∥BG−1/2x∥∥2; thus∥∥∥BG−1/2x∥∥∥ ≤‖x‖ + ‖Ax‖ . Hence ∥∥∥CG−1/2BG−1/2x∥∥∥ ≤ ∥∥∥G−1/2∥∥∥∥∥∥BG−1/2x∥∥∥ ≤‖C‖ ∥∥∥G−1/2∥∥∥ (‖x‖ + ‖Ax‖).(5.1) Let λ in C. We prove that if |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) then 0 < γ(λCG −1/2BG−1/2 − A) < ∞ and hence R(λCG−1/2BG−1/2 − A) is closed. First if we use (5.1) with λx instead of x and by [7, Theorem 1a], we obtain that γ(λCG−1/2BG−1/2 − A) > 0 for |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) . Now to prove that γ(λCG−1/2BG−1/2 − A) < ∞, we proceed by contraposition. In fact γ(λCG−1/2BG−1/2 −A) = ∞ implies that (λCG−1/2BG−1/2 −A)x = 0 for all x ∈ H1. Hence ‖Ax‖ = |λ| ∥∥∥CG−1/2BG−1/2x∥∥∥ ≤‖C‖|λ|∥∥∥G−1/2∥∥∥ (‖x‖ + ‖Ax‖), and so (5.2) γ(A)‖x‖≤‖Ax‖≤ |λ|‖C‖ ∥∥G−1/2∥∥ 1 −|λ|‖C‖ ∥∥G−1/2∥∥ ‖x‖ STRONG METRIZABILITY FOR CLOSED OPERATORS 121 for x ∈ N(A)⊥ with x 6= 0. It follows that |λ| ≥ ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) . We next prove that (5.3) δ(N(λCG−1/2BG−1/2 −A),N(A)) ≤ |λ|‖C‖ ∥∥G−1/2∥∥ (1 −|λ|‖C‖ ∥∥G−1/2∥∥)γ(A). Let x ∈ H1, γ(A) ∥∥(I −PN(A))PN(λCG−1/2BG−1/2−A)x∥∥ ≤ ∥∥APN(λCG−1/2BG−1/2−A)x∥∥ . Since PN(λCG−1/2BG−1/2−A)x ∈ N(λCG−1/2BG−1/2 −A) by the same calculation given before we have γ(A) ∥∥(I −PN(A))PN(λCG−1/2BG−1/2−A)x∥∥ ≤ |λ|‖C‖ ∥∥G−1/2∥∥ (1 −|λ|‖C‖ ∥∥G−1/2∥∥) ‖x‖ . Recalling the definition of δ(N,M), this proves (5.3). The right side of (5.3) is smaller than one if |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) , thus Lemma 5.1 shows that (5.4) α(λCG−1/2BG−1/2 −A) ≤ α(A) for |λ| < ∥∥G1/2∥∥γ(A) ‖C‖(1 + γ(A)) . We then conclude that λCG−1/2BG−1/2−A is upper semi-Fredholm operator for |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) . This complete the proof of the theorem. � Theorem 5.3. Let A ∈ L0(H1,H2) is a lower semi-Fredholm operator. Then λC − ΓG(A) is a lower semi-Fredholm operator for all C ∈L(H1,H2) and λ such that |λ| < ‖G1/2‖γ(A) ‖C‖(1+γ(A)) . Proof. Since R(A) is closed, by the first part of the proof of Theorem 5.2, R(λCG−1/2BG−1/2− A) is closed and R(λCG−1/2BG−1/2 − A) = N(λG−1/2B∗G−1/2C∗ − A∗)⊥ for all |λ| < ‖G1/2‖γ(A) ‖C‖(1+γ(A)) . From (5.3) we deduce that δ(R(λCG−1/2BG−1/2 −A)⊥,R(A)⊥) = δ(N(λG−1/2B∗G−1/2C∗ −A∗),N(A∗)) ≤ |λ|‖C‖ ∥∥G−1/2∥∥ (1 −|λ|‖C‖ ∥∥G−1/2∥∥)γ(A), because γ(A) = γ(A∗). Now by Lemma 5.1 we have β(λCG−1/2BG−1/2 −A) ≤ β(A) for |λ| < ∥∥G1/2∥∥γ(A) ‖C‖(1 + γ(A) . Consequently, λCG−1/2BG−1/2 −A is lower semi-Fredholm one for all |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) and hence λC − ΓG(A) is lower semi-Fredholm operator for all λ such that |λ| < ‖G1/2‖γ(A) ‖C‖(1+γ(A)) . � Corollary 5.4. If A ∈ L0(H1,H2) is a semi-Fredholm operator (resp. Fredholm operator), then λC−ΓG(A) is a semi-Fredholm operator (resp. Fredholm operator) for all C ∈L(H1,H2) and λ such that |λ| < ‖ G1/2‖γ(A) ‖C‖(1+γ(A)) . We proceed as in the proof of [2, Lemma 5.2, p. 708], by taking in count that the operator T is defined between two Hilbert spaces, we can easily check the following result. Proposition 5.5. If T ∈ C(H1,H2) is a semi-Fredholm operator (resp. Fredholm operator), then A = T(G + T∗T)−1/2 is a semi-Fredholm operator (resp. Fredholm operator) from H1 to H2, and N(A) = N(T),N(A ∗) = N(T∗). 122 BENHARRAT AND MESSIRDI Proof. It is easy to see that N(T) = {x ∈ H1 : (G + T∗T)−1Gx = x}. Since G ∈ GL+(H1), (G+T∗T)−1 is bounded self-adjoint and (G+T∗T)−1G leaves N(T) as well as N(T)⊥ invariant, so this two subspaces are invariant by (G + T∗T)−1 and its square root. Accordingly N(T) = N(A). It is also clear that y ∈ N(A∗) if and only if 〈 T(G + T∗T)−1/2x,y 〉 = 0 for all x ∈ H1 i.e. 〈 T(G + T∗T)−1z,y 〉 = 0 for all z ∈ H1 i.e if y ∈ N((T(G + T∗T)−1)∗) = N(T∗). Thus N((T(G + T∗T)−1/2)∗) = N(T∗). � By Proposition 5.5 and Corollary 5.4 we obtain the following results Theorem 5.6. Let A ∈ L0(H1,H2) . Then A is a semi-Fredholm operator (resp. Fredholm operator) if and only if ΓG(A) is a semi-Fredholm operator (resp. Fredholm operator). In this case ind(A) = ind(ΓG(A)). Remark 5.7. Theorems 5.2, 5.3 and 5.6 generalize [1, Theorem 1], [1, Theorem 2] and [1, Theorem 3] respectively, by taking H1 = H2 = H and G = C = I. References [1] M. Benharrat, B. Messirdi. Semi-Fredholm operators and pure contractions in Hilbert space. Rend. Circ. Mat. Palermo, 62 (2013), 267–272. [2] H. O. Cordes and J. P. Labrousse, The invariance of the index in the metric space of closed operators. J. Math. Mech. 12 (1963), 693–719. [3] G. Djellouli, S. Messirdi and B. Messirdi, Some Stronger Topologies for Closed Operators in Hilbert Space. Int. J. Contemp. Math. Sciences, 5(25) (2010), 1223–1232. [4] S. Goldberg, Unbounded Linear Operators. McGraw-Hill, New-York, (1966). [5] G. Hirasawa, Quotient of bounded operators and Kaufman’s theorem. Math. J. Toyama Univ. 18 (1995), 215-224. [6] S. Izumino, Quotients of bounded operators, Proc. Amer. Math. Soc. 106 (1989), 427–435. [7] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators. J. Anal. Math. 6 (1958), 261–322. [8] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, (1966). [9] W. E. Kaufman, Representing a closed operator as a quotient of continuous operators, Proc. Amer. Math. Soc. 72 (1978), 531–534. [10] W. E. Kaufman, Semiclosed operators in Hilbert space, Proc. Amer. Math. Soc. 76 (1979), 67–73. [11] W. E. Kaufman, Closed operators and pure contractions in Hilbert space, Proc. Amer. Math. Soc. 87 (1983), 83–87. [12] W. E. Kaufman, A strong metric for closed operators in Hilbert space, Proc. Amer. Math. Soc. 90 (1984), 83–87. [13] F. Kittaneh, On some equivalent metrics for bounded operators on Hilbert space Proc. Amer. Math. Soc. 110 (1990), 789–798. [14] J. J. Koliha, On Kaufman’s theorem J. Math. Anal. Appl. 411(2014), 688–692. [15] J.P. Labrousse, Quelques topologies sur des espaces d’opérateurs dans des espaces de Hilbert et leurs application. I, Faculté des Sciences de Nice (Math.), 1970. [16] J. Weidmann, Linear Operators in Hilbert Spaces, Springer (1980). 1Laboratory of Fundamental and Applicable Mathematics of Oran, Department of Mathematics and informatics, National Polytechnic School of Oran, BP 1523 Oran-El M’naouar, Oran, Algeria 2Laboratory of Fundamental and Applicable Mathematics of Oran, Department of Mathematics, University of Oran 1, BP 1524 Oran-El M’naouar, Oran, Algeria ∗Corresponding author