Research Article on Local Minimum and Orthogonality of Normal

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International Journal of Modern Science and Technology Vol. 2, No. 2, 2017. Page 70-73. http://www.ijmst.co/ ISSN: 2456-0235.

Research Article

On Local Minimum and Orthogonality of Normal Derivations in Cp-Classes B. O. Owino, N. B. Okelo*, O. Ongati School of Mathematics and Actuarial Science, Jaramogi Oginga Odinga University of Science and Technology, P.O Box 210-40601, Bondo-Kenya *Corresponding author’s e-mail: [email protected] Abstract

The present paper gives some results on local minimum and orthogonality of normal derivations in Cp- Classes. We employ some techniques for normal derivations due to Mecheri, Hacene, Bounkhel and Anderson. Let Cp Cp be normal, then the linear map = attains a local minimum at x Cp if and only if z Cp such that ) Also let Cp, and let have the polar decomposition , then the map attains local minimum on Cp at T if and only if . Regarding orthogonality, let S Cp and let N(S) have the polar decomposition N(S)=U|N(S)|, then for X Cp if . Moreover, the map has a local minimum at x if and only if for y . Keywords: Banach space; Hilbert space; Gateaux derivative; Orthogonality; Schatten-p Class. Introduction understood if we have two pair of projections P Normal derivation has been an area of and Q in terms of a generating self adjoint interest for many mathematicians and operator A which are implementing normal researchers particularly their properties for derivations. example [1,2]. In their study they gave results on In [8] it is noted that the trace class the global minimum of linear maps. In this operators, denoted by C(H), is the set of all paper, we give results on the conditions for the compact operators A ∈ L(H), for which the linear map to attain local minimum in Cp- eigenvalues according to multiplicity, are Classes. We have used polar decomposition [3], summable. The I(H) of L(H) admits a trace tensor product [4] and inner product [5] to arrive function tr(T), for any complete orthonormal at the main results. system in H. As a Banach spaces K(H) can be identiﬁed with the dual of the ideal K of compact In [6], the knowledge of structural properties operators by means of the linear isometry of of the underlying C*-algebra is undoubtedly one normal derivations. It was shown in [9] that, if A of the cornerstones in solving the problem of is a unital C* Orc(A). It allows that if A is any norms of elementary operators. Equivalently the C*-algebra then K is sef-adjoint.We now recall spectral theorem helps a great deal in some properties of the complete regularization of understanding Hilbert space theory. The spectral Prim(A) for a C*-algebra (see [110] for further theory gives the unitary invariants of self-adjoint operator A on a Hilbert space H in terms of its details). For P, Q ∈ Prim(A) (Prim(A)). Then ˜ multiplicity measureclasses {m} and provide is an equivalence relation on Prim(A) and the unitarily equivalent model operators. Problems equivalence classes are closed subsets of concerning pairs of projections play a Prim(A). It follows that there is a one-to-one fundamental role in the theory of operator correspondence tween Prim(A)/ ˜ and a set of algebras of normal derivations. [7] shows that if closed two-sided ideals of A given by the P and Q are projections then the C*-algebra intersection of the ideals in the equivalence class generated by P, Q has a concrete realisation as [P] of P. The set of ideals obtained in this way is an algebra of 2 × 2-matrix valued functions on denoted by Glimm(A) and we identify this set [0, 1], so its representation theory is well with Prim(A)/ ˜ by the correspondence above. If A is unital then Glimm(A) consists of the ideals

Received: 12.01.2017; Received after Revision: 04.02.2017; Accepted: 10.02.2017; Published: 27.02.2017 ©International Journal of Modern Science and Technology. All rights reserved. 70 Owino et al., 2017. On Local Minimum and Orthogonality of Normal Derivations in Cp-Classes of A generated the maximal ideals of the centre to x X if for all complex there of A, as studied by Glimm [11]. The quotient holds . map G: Prim(A) → Glimm(A) is called the Definition 2.5 ([11], Section 2) Let T B(H) be complete regularization map on a norm compact. Then s1 (T) s2 (T) … 0 are the topology. singular values of T i.e the eigenvalues of T = This topology is completely regular, counted according to multiplicity and Hausdorﬀ, weaker than the quotient topology arranged in descending order. For 1 p ∞, Cp= (and equal to it when A is s-unital [1]) and hence Cp (H) is the set of those compact T B(H) with makes f continuous. The ideals in Glimm(A) are called imm ideals and the equivalence classes for finite p-norm, p= = ˜ in Prim(A) will sometimes be referred to as . imm classes. For normal derivations, T is a linear operator on a finite dimensional complex Result and discussions Hilbert space H, then every commutator of the In this section we give the main results. We form AX — XA has trace 0 and consequently 0 discuss the conditions under which the map belongs to the numerical range W(AX-XA). attains local minimum in -class. However, if H is infinite dimensional, then, as Theorem 3.1 Let Cp Cp be normal, then the shown by [2], there exist bounded operators A linear map = attains a local and B on T such that W(AB-BA) is a vertical minimum at x Cp if and only if z Cp such that line segment in the open right half-plane. The ) present paper initiates a study of the class D of Proof. For necessity we have operators A on H which have the property that 0 For sufficiency: belongs to W(AX—XA)~ for every bounded operator X on H. We call such operators finite, Assume the stated condition and choose . Then the term being suggested by the facts that D Let M, then contains all normal operators, all compact by [7] operators, all operators having a direct summand of finite rank, and the entire C*-algebra generated by each of its members. Lastly, the by hypothesis main focus in this paper is to study the local by sub-additivity [8] miminum and orthogonality in Cp-classes for normal derivations. Theorem 3.2 Let Cp, and let have the Preliminaries polar decomposition . Then the map attains local minimum on Cp at T if and In this section, we start by defining some key only if . terms that are useful in the sequel. Proof: Suppose that N has a local minimum on Definition 2.1 ([4], Definition 1.2) A Banach Cp at T. Then, space is a complete normed space. ( ) (1) Definition 2.2 ([3], Definition 33.1) A Hilbert if and only if Z C i.e space H is an inner product space which is p complete under the norm induced by its inner for Z Cp. product. This implies that Definition 2.3 ([5,] Definition 3.1) Let f be a (2) for Z Cp. function on an open subset U of a Banach space Choose arbitrary vectors f and g in the Hilbert X into Banach space Y. f is Gateaux space H and let f g be rank one operator given differentiable at x U if there is bounded and by x . Now take Z= f g, then one has linear operator T: X Y such that Tx(h)= by [9]

for every h X. The operator Then inequality (1) is equivalent to T is called the Gateaux derivative of f at x. Definition 2.4 ([6], Definition 0.1) Let X be a for Z Cp, or equivalently by [10]. complex Banach space. Then y X is orthogonal

for all f and g in x,x〉 where is H. Suppose f=g such that , then one positive. However, was arbitrary and therefore, obtains x,x〉 Hence, and so We note that the set . is the Theorem 3.4 Let S Cp and let N(S) have the numerical range of on that polar decomposition N(S)=U|N(S)|. Then, is convex set and its closure is a closed convex for X Cp if set. By inequality (2), it must contain the value . of positive real part[12], under all rotation about the origin and it must contain the origin so that Proof. Let We need to we obtain some vector f in H such show that B,A. Since we are that where is considering Cp, we know positive. Given is arbitrary, we that . Given that Cp obtain . Hence, admits pseudo-unitarily invariant norm, then i.e By induction Conversely, if U admits nilpotency and hence then also admits nilpotency. Suppose is isometric Using the above and also co-isometric, admits argument, we find nilpotency. All nilpotent operators belong to the that for Cp. kernel of . Hence, By this we get inequality (2). since is the adjoint of U. Corollary 3.3 Let T Cp and let N(T) have the Theorem 3.5 The map has a local minimum polar decomposition N(T)=U|N(T)|. Then the at x if and only if map attains local minimum on Cp at T if and (5) only if . for y . Proof: Suppose that N has a local minimum on Proof. For necessity we combine the result [2, Cp at T, then we have Theorem 2.1] and the following inequality, ( (X)) (3) if and only if X Cp i.e Conversely assume inequality (5) is satisfied and (X))} for X Cp. we observe that This implies (X))} (4) for X Cp. Take any two arbitrary vectors x and y in H and let x y to be the rank one operator given by a Let X= x y. Then (X))= X). Inequality (3) is For this we have equivalent to X)} for X Cp. Equivalently [13], y,x)〉 for all x,y Let y be arbitrary and put Assume x=y and that then, Then, . By x,x)〉 The set inequality (5) we have and x,x)〉: is the hence by subadditivity of and the numerical range of on that linearity of we obtain is convex set and its closure is a closed convex set. By inequality (4), it must contain a value of positive real part, under all rotation about the origin and it must contain the origin so that we have some vector x in H so that [14]

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