SOME INEQUALITIES IN B(H)

Let H denote a separable Hilbert space and let B(H) be the space of bounded and linear operators from H to H. We define a subspace ∆(A,B) of B(H), and prove two inequalities between the distance to ∆(A,B) of each operator T in B(H), and the value sup{‖AnTBn−T‖ :n= 1,2, . . .}. 2000 Mathematics Subject Classification. Primary 43-XX. 1. Notations. Throughout this paper H denotes a separable Hilbert space and {en}n=1 an orthonormal basis. Let LA and RB be left and right translation operators on B(H) for A,B ∈ B(H), satisfying ‖A‖ ≤ 1 and ‖B‖ ≤ 1. Then the set ∆(A,B) is defined by ∆(A,B)= {T ∈ B(H) :ATB = T} = {T ∈ B(H) : ST = T}, (1.1) where S = LARB . An operator C ∈ B(H) is called positive, if 〈Cx,x〉 ≥ 0 for all x ∈ H. Then for any positive operator C ∈ B(H) we define trC = ∑∞n=1〈en,Cen〉. The number trC is called the trace of C and is independent of the orthonormal basis chosen. An operator C ∈ B(H) is called trace class if and only if tr|C| <∞ for |C| = (C∗C)1/2, where C∗ is adjoint of C . The family of all trace class operators is denoted by L1(H). The basic properties of L1(H) and the functional tr(·) are the following: (i) Let ‖·‖1 be defined in L1(H) by ‖C‖1 = tr|C|. Then L1(H) is a Banach space with the norm ‖·‖1 and ‖C‖ ≤ ‖C‖1. (ii) L1(H) is ∗ideal, that is, (a) L1(H) is a linear space, (b) if C ∈ L1(H) and D ∈ B(H), then CD ∈ L1(H) and DC ∈ L1(H), (c) if C ∈ L1(H), then C∗ ∈ L1(H). (iii) tr(·) is linear. (iv) tr(CD)= tr(DC) if C ∈ L1(H) and D ∈ B(H). (v) B(H)= L1(H)∗, that is, the map T → tr(T) is an isometric isomorphism of B(H) onto L1(H)∗, (see [3]). Let X be a Banach space. If M ⊂X, then M⊥ = {x∗ ∈X∗ : 〈x,x∗〉= 0, x ∈M} (1.2) is called the annihilator of M . If N ⊂X∗, then ⊥N = {x ∈X : 〈x,x∗〉= 0, x∗ ∈N} (1.3) 130 C. DUYAR AND H. SEFEROGLU is called the preannihilator of N . Rudin [4] proved for these subspaces: (i) ⊥(M⊥) is the norm closure of M in X. (ii) (⊥N)⊥ is the weak-∗ closure of N in X∗. 2. Main results Lemma 2.1. Let X be a Banach space. If P is a continuous operator in the weak-∗ topology on the dual space X∗, then there exists an operator T on X such that P = T∗. Proof. If P : X∗ → X∗, then P∗ : X∗∗ → X∗∗. We know that the continuous functionals in the weak-∗ topology on X∗ are simply elements of X, (see [4]). Then we must show that P∗x is continuous in the weak-∗ topology on X∗ for all x ∈ X. Let (x′ n) be a sequence in X∗ such that x′ n→ x′, x′ ∈X∗. Then we have 〈 P∗x,x′ n 〉= 〈x,Px′ n〉 → 〈x,Px′〉= 〈P∗x,x′〉. (2.1) Hence P∗x is continuous in the weak-∗ topology on X∗ for all x ∈ X, so P∗x ∈ X. If T is the restriction to X of P∗, then we have 〈 x,T∗x′ 〉= 〈Tx,x′〉= 〈P∗x,x′〉= 〈x,Px′〉 (2.2) for all x ∈X and x′ ∈X∗. Hence P = T∗. Definition 2.2. If P∗ is the operator T in Lemma 2.1, then P∗ is called the preadjoint operator of P . The operator x⊗y ∈ B(H) for each x,y ∈ H is defined by (x⊗y)z = 〈z,y〉x for all z ∈H. It is easy to see that this operator has the following properties: (i) T(x⊗y)= Tx⊗y . (ii) (x⊗y)T = x⊗T∗y . (iii) tr(x⊗y)= 〈y,x〉. The following lemma is an easy application of some properties of the operator x⊗y (x,y ∈H) and the functional tr(·). Lemma 2.3. (i) Suppose K is a closed subset in the weak-∗ topology of B(H). Then K is closed in the weak-∗ topology of B(H). (ii) S = LARB is continuous in the weak-∗ topology of B(H) for all A,B ∈ B(H), satisfying ‖A‖ ≤ 1 and ‖B‖ ≤ 1. Lemma 2.4. There exists a linear subspace M of L1(H) such that ∆(H)=M⊥ and M is closed linear span of {S∗X−X :X ∈ L1(H)}, where S∗ is the preadjoint operator of S. Proof. Note that ⊥∆(A,B)= {U ∈ L1(H) : 〈U,U∗〉= 0, U∗ ∈∆(A,B)}. (2.3) It is known that (⊥∆(A,B)⊥) is the weak-∗ closure of ∆(A,B) (see [4]). Then we can write (⊥∆(A,B))⊥ = ∆(A,B), since ∆(A,B) is a closed set in the weak-∗ topology of B(H). We say ⊥∆(A,B)=M . Nowwe show thatM is the closed linear span of {S∗U−U : U ∈ L1(H)}. For this, it is sufficient to prove that 〈S∗U−U,T〉 = 0 for all T ∈∆(A,B). SOME INEQUALITIES IN B(H) 131 Indeed since ST = T , we have 〈 S∗X−X,T 〉= 〈(S∗−I)X,T〉= 〈X,(S∗−I)∗T〉= 〈X,(S−I)T〉= 0. (2.4) Lemma 2.5. Let K(T) be the closed convex hull of {SnT : n = 1,2, . . .} in the weak operator topology, for a fixed T ∈ B(H). Then we have K(T)∩∆(A,B) = 0. (2.5) Proof. Assume K(T)∩∆(A,B) = 0. By Lemma 2.3, K(T) is closed in the weak-∗ topology. It is easy to see that K(T) is bounded. Then K(T) is compact in the weak-∗ topology by Alaoglu, [1]. Since S is continuous in the weak-∗ topology, if Uα → U for (Uα)α∈I ⊂ ∆(A,B), then SUα = Uα → SU . Hence ∆(A,B) is closed in the weak-∗ topology. This shows that U ∈∆(A,B). Since K(T) is compact and convex in the weak-∗ topology, and ∆(A,B) is closed in the weak-∗ topology, and K(T)∩∆(A,B) = 0, there exist some U0 ∈M and σ > 0 such that ∣∣tr(TU0)∣∣≥ σ (2.6) for all T ∈∆(A,B), (see [2]). Now we define the operators Tn ∑n k=1ST for all positive integer n. These operators are clearly in K(T). It is easy to show that the operators Tn is bounded. Also by Lemma 2.4, there is a U ∈ L1(H) such that U0 = S∗U−U . Then we have ∣∣〈Tn,U0〉∣∣= ∣∣〈Tn,S∗U−U〉∣∣= ∣∣〈STn,U〉−〈Tn,U〉∣∣ = ∣∣∣∣ 〈 S   1 n n ∑ k=1 AkTBk   ,U 〉 − 〈 1 n n ∑ k=1 AkTBk,U ∣∣∣∣ = ∣∣∣∣ 〈 1 n n ∑ k=1 Ak+1TBk+1,U 〉 − 〈 1 n n ∑ k=1 AkTBk,U ∣∣∣∣ = 1 n ∣∣〈An+1TBn+1−ATB,U〉∣∣ ≤ 1 n 2‖T‖·‖U‖. (2.7) This implies that |〈Tn,X0〉| → 0, which is a glaring contradiction to (2.6). Theorem 2.6. Let H be separable Hilbert space and T ∈ B(H). Then we have (i) d(T ,∆(A,B))≥ (1/2)supn‖ST −T‖, (ii) d(T ,∆(A,B))≤ supn‖ST −T‖. Proof. (i) We can write SnT −T = Sn(T −T0)−(T −T0)+SnT0−T0 (2.8) for each T0 ∈∆(A,B). Hence we have ∥∥SnT −T∥∥≤ ∥∥Sn∥∥∥∥T −T0∥∥+∥∥T −T0∥∥≤ 2∥∥T −T0∥∥. (2.9) 132 C. DUYAR AND H. SEFEROGLU This shows that 1 2 sup n ∥∥SnT −T∥∥≤ inf T0∈∆(A,B) ∥∥T −T0∥∥. (2.10) The inequality (2.10) gives that d ( T ,∆(A,B) )≥ 1 2 sup n ∥∥SnT −T∥∥. (2.11) (ii) Let K(T) be as Lemma 2.5. Then we can write K(T)= co{SnT :n= 1,2, . . .}. (2.12) Now take any element U = ∑nk=1λkSkT in the set co{SnT : n = 1,2, . . .}, where ∑n k=1λk = 1, λk ≥ 0. Then ‖U−T‖ = ∥∥∥∥ n ∑ k=1 λkST −T ∥∥∥∥≤ ∥∥∥∥ n ∑ k=1 λkST − n ∑ k=1 λkT ∥∥∥∥ ≤ n ∑ k=1 λk ∥∥SkT −T∥∥≤ n ∑ k=1 λkσ(T)= σ(T), (2.13) where σ(T)= supn‖ST −T‖. That is, for all U ∈ co{SnT :n= 1,2, . . .} is ‖U−T‖ ≤ sup n ∥∥SnT −T∥∥. (2.14) Since there is a sequence (Un) in co{SnT : n = 1,2, . . .} such that Un → V for all V ∈ K(T), then we write ‖V −T‖ ≤ ∥∥V −Tn∥∥+∥∥Tn−T∥∥. (2.15) If we use the inequalities (2.14) and (2.15), we easily see that ‖V −T‖ ≤ sup n ∥∥SnT −T∥∥. (2.16) Also since K(T)∩∆(A,B) = 0 by Lemma 2.5, then we obtain ∥∥T −T0∥∥≤ sup n ∥∥SnT −T∥∥ (2.17) for a T0 ∈K(T)∩∆(A,B). Hence we can write d ( T ,∆(A,B) )= inf U∈∆(A,B) ‖T −U‖ ≤ ∥∥T −T0∥∥≤ sup n ∥∥SnT −T∥∥. (2.18) This completes the proof.

Let X be a Banach space.If M ⊂ X, then is called the preannihilator of N. Rudin [4] proved for these subspaces:

Main results
Lemma 2.1.Let X be a Banach space.If P is a continuous operator in the weak- * topology on the dual space X * , then there exists an operator T on X such that P = T * .
Proof.If P : X * → X * , then P * : X * * → X * * .We know that the continuous functionals in the weak- * topology on X * are simply elements of X, (see [4]).Then we must show that P * x is continuous in the weak- * topology on X * for all x ∈ X.Let (x n ) be a sequence in X * such that x n → x , x ∈ X * .Then we have P * x, x n = x, P x n → x, P x = P * x, x . (2.1) Hence P * x is continuous in the weak- * topology on X * for all x ∈ X, so P * x ∈ X.If T is the restriction to X of P * , then we have for all x ∈ X and x ∈ X * .Hence P = T * .
Definition 2.2.If P * is the operator T in Lemma 2.1, then P * is called the preadjoint operator of P .
The operator x ⊗ y ∈ B(H) for each x, y ∈ H is defined by (x ⊗ y)z = z, y x for all z ∈ H.It is easy to see that this operator has the following properties: The following lemma is an easy application of some properties of the operator x ⊗ y (x,y ∈ H) and the functional tr(•).

Lemma 2.3. (i) Suppose K is a closed subset in the weak- * topology of B(H). Then K is closed in the weak- * topology of B(H).
(ii [4]).Then we can write

Lemma 2.4. There exists a linear subspace
For this, it is sufficient to prove that S * U − U,T = 0 for all T ∈ ∆(A, B).
Indeed since ST = T , we have Lemma 2.5.Let K(T ) be the closed convex hull of {S n T : n = 1, 2,...} in the weak operator topology, for a fixed T ∈ B(H).Then we have (2.5) is closed in the weak- * topology.It is easy to see that K(T ) is bounded.Then K(T ) is compact in the weak- * topology by Alaoglu, [1].Since S is continuous in the weak- * topology, if Since K(T ) is compact and convex in the weak- * topology, and ∆(A, B) is closed in the weak- * topology, and K(T ) ∩ ∆(A, B) = 0, there exist some U 0 ∈ M and σ > 0 such that tr T U 0 ≥ σ (2.6) for all T ∈ ∆(A, B), (see [2]).Now we define the operators T n n k=1 S k T for all positive integer n.These operators are clearly in K(T ).It is easy to show that the operators T n is bounded.Also by Lemma 2.4, there is a U ∈ L 1 (H) such that U 0 = S * U −U .Then we have (2.7) This implies that | T n ,X 0 | → 0, which is a glaring contradiction to (2.6).

Theorem 2.6. Let H be separable Hilbert space and T ∈ B(H). Then we have
Proof.(i) We can write for each T 0 ∈ ∆(A, B).Hence we have where Since there is a sequence If we use the inequalities (2.14) and (2.15), we easily see that (2.16) Also since K(T ) ∩ ∆(A, B) = 0 by Lemma 2.5, then we obtain This completes the proof.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

1 .
Notations.Throughout this paper H denotes a separable Hilbert space and {e n } ∞ n=1 an orthonormal basis.Let L A and R B be left and right translation operators on B(H) for A, B ∈ B(H), satisfying A ≤ 1 and B ≤ 1.Then the set ∆(A, B) is defined by∆(A, B) = T ∈ B(H) : AT B = T } = {T ∈ B(H) : ST = T , (1.1)where S = L A R B .An operator C ∈ B(H) is called positive, if Cx, x ≥ 0 for all x ∈ H. Then for any positive operator C ∈ B(H) we define tr C = ∞ n=1 e n ,Ce n .The number tr C is called the trace of C and is independent of the orthonormal basis chosen.An operator C ∈ B(H) is called trace class if and only if tr |C| < ∞ for |C| = (C * C) 1/2 , where C * is adjoint of C. The family of all trace class operators is denoted by L 1 (H).The basic properties of L 1 (H) and the functional tr(•) are the following: Let K(T ) be as Lemma 2.5.Then we can writeK(T ) = co S n T : n = 1, 2,... .(2.12) Now take any element U = n k=1 λ k S k T in the set co{S n T : n = 1, 2,...}, where

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation