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Multiplication Operators with Closed Range in Operator Algebras

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Multiplication Operators with Closed Range in Operator Algebras J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) 1 Journal of Analysis & Number Theory An International Journal °c 2013 NSP Natural Sciences Publishing Cor. Multiplication Operators with Closed Range in Operator Algebras P. Sam Johnson Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka, Mangalore, India Received: 16 Jul. 2012; Revised 22 Oct. 2012; Accepted 23 Oct. 2012 Published online: 1 Jan. 2013 Abstract: Let B(H) denote the C¤-algebra of all bounded linear operators from a Hilbert space H into itself. Let T 2 B(H). Define LT : B(H) !B(H) by LT (S) = TS and define RT : B(H) !B(H) by RT (S) = ST . Consider the following three statements: (i) T has closed range in H, (ii) LT has closed range in B(H), (iii) RT has closed range in B(H). It is proved that all these three statements are equivalent. Some possibilities of extending this result to Banach spaces have been discussed. Keywords: Closed range operator ; Hahn-Banach extension property. 1. Introduction Many of the concrete applications of mathematics in science and engineering, eventually result in a problem involving operator equations. This problem can be usually represented as an operator equation T x = y (1) where T : X ! Y is a linear or nonlinear operator (between certain function spaces or Euclidean spaces) such as differential operator or integral operator or a matrix. The spaces X and Y are linear spaces endowed with certain norms on them. Solving linear equations with infinitely many variables is a problem of functional analysis, while solving equations with finitely many variables is one of the main themes of linear algebra. The problem of solving the equation (1) is well-posed if it asserts existence and uniqueness of a solution of (1) and the continuous dependence of the solution on the data y. It is well-known that the problem of solving the operator equation (1) is essentially well-posed if the range of T is closed. The study of operators with closed range plays a vital role in perturbation theory. The composition of closed range continuous linear operators between Frechet´ spaces does not have closed range, in general. We have given necessary and sufficient conditions for the composition operator TS to have closed range in [3, 8]. Bouldin [1,2] has given a geometric characterization in terms of the angle between two linear subspaces for operators on Hilbert spaces and for the operators on Banach spaces. The closedness of range of a linear operator T is helpful in establishing the Hyers-Ulam stability of T between Frechet´ spaces [7]. The range and null spaces of a linear operator T are denoted by R(T ) and N(T ) respectively. In this paper, we first discuss the closedness of multiplication operators on Hilbert spaces. Kulkarni and Nair [5] proved that for a bounded linear operator T from a Hilbert space H into a Hilbert space K, T has closed range in K if and only if σ(T ¤T ) ⊆ f0g [ [γ; kT k2], for some γ > 0. In this paper it is proved that ¤ 2 σ(x x) ⊆ f0g [ [γ; kxk ], for some γ > 0 if and only if Lx : A ! A defined by Lx(y) = xy has closed range in A where x is any element of a commutative C¤-algebra A with identity. Moreover, the multiplication operators with closed range are found on some classical Banach algebras. Throughout the paper, the following characterization for closed range operators on Banach spaces is used. Theorem 1. [3] Let T : X ! Y be a bounded linear operator from a Banach space X into a Banach space Y . Then T has closed range in Y if and only if there is a constant c > 0 such that for given x 2 X, there is an element y 2 X such that T x = T y and kyk · ckT xk. ¤ Corresponding author: e-mail: [email protected] °c 2013 NSP Natural Sciences Publishing Cor. 2 P. Sam Johnson: Multiplication Operators with Closed Range in Operator ... We shall apply this theorem to multiplication operators on Banach algebras. For a given Banach algebra A and a given element x in A, Lx and Rx will denote the left multiplication operator and the right multiplication operator, respectively; that is; Lx(y) = xy and Rx(y) = yx, for every y 2 A. (1) If A is a Banach algebra with a multiplicative identity element e, then L0 and Le have closed ranges. (2) If x is invertible, then Lx has closed range ; this result follows from the next proposition. (3) If A is a commutative Banach algebra with an identity element e and with a non trivial idempotent element e0, then 0 0 0 0 0 Le0 has closed range (because e xn ! x implies that e xn = e e xn ! e x = x). Proposition 2 Let A be a Banach algebra with multiplicative identity e. Then x 2 A is right invertible in A if and only if Lx(A) = A. ¡1 ¡1 ¡1 Proof. Suppose x is right invertible in A with right inverse x . Then for y 2 A, Lx(x y) = xx y = ey = y. Thus Lx(A) = A. Conversely, assume that Lx(A) = A. Then for e 2 A, there is a y 2 A such that e = Lx(y) = xy so that x is right invertible with the right inverse y. We now find multiplication operators with closed ranges in some classical Banach algebras. Proposition 3 For 1 · p < 1, the collection of all x 2 `p such that Lx has closed range is the space 1 c00 = fx = (xn)n=1 : xn = 0 for all except for finitely many ng : 1 Proof.If x 2 c00, then the range of Lx is finite dimensional and hence it is closed in `p. If x = (xn)n=1 2= c00, then there 1 1 1 (m) is a subsequence (xnm )m=1 of (xn)n=1 such that jxnm j < m and xnm 6= 0 for every m. For each nm, define x 2 `p by ½ (m) m if i = n x = m i 0 elsewhere. (m) (m) (m) (m) (m) (m) If, for given m, there is a z 2 `p such that Lxz = Lxx , then znm = m so that kz kp ¸ m and kLxx kp · (m) (m) 1. In this case, there is no positive constant c such that kz kp · ckLxx kp, for every m. Therefore Lx does not have closed range in `p; by theorem 1. Proposition 4 For a given element x 2 `1, Lx has closed range in `1 if and only if x = (0; 0; 0;:::), or, infn fjxnj : xn 6= 0g > 0. This proposition follows from the next proposition. The argument used in the previous proof shall be extended to the following proposition. Proposition 5 Let (X; M; ¹) be a (non-negative) positive measure space. For a given element f 2 L1(X), Lf has closed range in L1(X) if and only if f = 0 almost everywhere on X or ess. inf. fjf(x)j : x 2 X; f(x) 6= 0g > 0 on fx 2 X : f(x) 6= 0g. Proof.If f = 0, then Lf (L1(X)) = f0g so that it has closed range in L1(X). Suppose f 6= 0 almost everywhere on X and assume that 0 < ess. inf. fjf(x)j : x 2 X; f(x) 6= 0g = c; say; on fx 2 X : f(x) 6= 0g. Then for each g 2 L1(X), define h such that ½ g(x) if f(x) 6= 0 h(x) = 0 if f(x) = 0 then ckhk1 · kfgk1 = kLf (g)k1 and Lf (g) = Lf (h). Thus, by theorem 1, Lf has closed range in L1(X). Suppose f 6= 0 almost everywhere on X and ess. inf. fjf(x)j : x 2 X; f(x) 6= 0g = 0 1 on fx 2 X : f(x) 6= 0g. Then we can find a sequence (En)n=1 of disjoint measurable subsets of X with positive measures 1 such that jf(x)j · n , for all x 2 En, and f(x) 6= 0 almost everywhere on En. For each integer n, define gn 2 L1(X) by ½ n if x 2 E g (x) = n n 0 elsewhere. If, for each n, there is a hn 2 L1(X) such that Lf (hn) = Lf (gn), then hn(x) ¸ n on En almost everywhere, khnk1 ¸ n and kLf (gn)k · 1. In this case, there is no positive constant c > 0 such that ckhnk1 · kLf (gn)k1, for every n. Therefore Lf does not have closed range in L1(X), by theorem 1. °c 2013 NSP Natural Sciences Publishing Cor. J. Ana. Num. Theor. 1, No. 1, 1-5 (2013) / www.naturalspublishing.com/Journals.asp 3 It is possible to find multiplication operators with closed range to each individual Banach algebra. The following theorem gives a characterization of closed range multiplication operators on commutative C¤-algebras with multiplicative identity elements. Theorem 6. Let A be a commutative C¤-algebra over C with a multiplicative identity element. Let x 2 A. Then the following three statements are equivalent: (1) Lx has closed range in A. (2) 0 is not a limit point of σ(x). (3) σ(x¤x) ⊆ f0g [ [γ; kxk2], for some γ > 0. Proof. By the Gelfand-Naimark theorem [10], let us assume that A = C(¢), the algebra of all complex valued continuous functions on the maximal ideal space ¢ of A, where ¢ is a compact Hausdorff space under the Gelfand-topology.
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