Microsoft Word - N. Sah _115-118_doc.doc N.P. Sah / BIBECHANA 10 (2014) 115-117 : BMHSS, p.115 (Online Publication: Dec., 2013) BIBECHANA A Multidisciplinary Journal of Science, Technology and Mathematics ISSN 2091-0762 (online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Relation between saturated and normal operators Nagendra Pd. Sah Dept. of Mathematics, M.M.A.M.Campus, Biratnagar. . Article history: Received 6 December, 2013 Abstract A vector space X with algebra of all linear maps L(X) from X into itself and the ideal of all finite dimensional linear maps with dual (Conjugate) transformation T* to T from X' to itself form a relation in terms of relatively regular and linearly independent which is sufficient for mentioned title. Keywords: Relatively regular, saturated, normal operator. 1. Introduction Let X be a vector space, L(X)be the algebra of all linear map of X into itself and be the ideal of all finite dimensional linear maps L(X). Let X* be the algebraic dual space X with elements x*, y*,…… and T L(X), then we define by y*(x) = x*(Tx), x* X*,a linear form y* X*. The map is called the (algebraic) dual or conjugate transformation T* to T. It is a linear map of X* into itself and is uniquely characterized by < Tx, x*> = , x X, x* X*[1]. If X' is a linear subspace of X', therefore a vector space of linear forms, we denote by L(X') the set of all T L(X), whose dual transformation T* maps the space X' into itself. Properties of dual transformation T* (1) (T1+T2)* = T1* + T2* (2) (αT)* = αT* (3) (T1 T2)*= T2* T1* Definition (1): If for a continuous operator T, there exists a continuous operator S with T S T = T, then T is called relatively regular. Definition (2): The algebra RRRR of operators on a vector space is called normal, if every finite dimensional operator T from RRRR is relatively RRRR - regular. Definition (3): An algebra RRRR of operators on vector space X is called saturated, if corresponding to any pair of finite sets {x1,……..xn}, {y1…..yn} ⊂X, Where {x1, x2……..xn} is linearly independent and there exists T RRRR with Txγ = yγ , γ = 1,2…….n. Theorem (1.1): Every operator T L(X) is relatively L(X)-regular. Proof: We have X = N(T) + U and X = B(T) + C. If P is the projection of X onto B(T) along C, Q the projection of X onto N(T) along U and To the restriction of T to U, then To is a bijective linear N.P. Sah / BIBECHANA 10 (2014) 115-117 : BMHSS, p.116 (Online Publication: Dec., 2013) map of U onto B(T). Therefore S = To -1 P L(X). For x = n + u, n = Q x N(T), u = (I - Q) x U, we have S Tx = To -1 P Tx = To -1 T x = To -1T (n+u) = To -1Tu = To -1 Tou = u = (T-Q)x. Therefore, S T = I – Q and T S T = T – TQ = T. Hence T is relatively regular. Theorem (1.2): Every saturated operator algebraRRRR is normal. Proof :Let x1,…….xn be a basis of the image space of T yγ ,1≤ γ ≤n, be so chosen that T yγ = xγ , 1≤ γ ≤n, then on account of being saturated, there exist S with S xγ = yγ and hence also with T S xγ = xγ , 1≤ γ ≤n. Then for T x =∑ = n 1 )( γ γγα xx We obtain T S T x = xxxxx T)(TS)( n 1 n 1 ==∑∑ == γ γγ γ γγ αα Hence T is relatively RRRR -regular. Definition (4): If X' be a linear space of linear functionals on X, then we define the set of all finite dimensional maps of the form Tx =∑ γγ xxx )(' , 'X' ∈γx and X∈γx by γ (X'). γ (X') is an algebra if E is a topological vector space with dual space E' then γ (E')= F(E). Theorem (1.3): If X is a vector space and X' is a total, then γ (X') is saturated; every super algebra. i.e. γ (X') is normal. Proof: If {x1,……..,xn} is linearly independent subset and {y1,…….,yn} arbitrary from X, then by definition. (2) corresponding to the elements x1,…….,xn there exist linear forms x'1,…….x'n on X' such that ( ) MM' γγ ∂=xx , 1≤ γ , M≤n[2]. Since x' is total, we define T by T x =∑ = n 1 )(' γ γγ yxx , then T γ (X') and TxM =∑ = n 1 M )(' γ γ xx = yM, 1≤M≤ n. Hence γ (X') is saturated. Lemma(1):T*(X') ⊂X'. And L(X') is an algebra containing I. Proof: We have γ (X') ⊂ L(X')[4] If T γ (X') then Tx = ∑ = >< n 1 ', γ γγ xxx with X'' ∈γx and X∈γx Then for arbitrary X'' ∈γx , we have N.P. Sah / BIBECHANA 10 (2014) 115-117 : BMHSS, p.117 (Online Publication: Dec., 2013) = = <∑ = >< n 1 , '', γ γγ xxxx > = ∑ = ><< n 1 , ''', γ γγγ xxxx > Therefore T*x' = ∑ = ∈>< n 1 X''', γ γ xxx Theorem (1.4): If X' is a total vector space of linear forms on X, then L(X') is normal. The result follows from Lemma (1) and theorem (1.3) [3]. 2. Conclusions The super algebra of a saturated operator is again saturated [5]. Every saturated operators of algebra RRRR is normal. If E is a topological vector space with total dual space E', then L(E) is always saturated. But if L(E) is not normal for every topological vector space E, then L(E) is not always saturated [6]. References 1. A.E. Taylor: Introduction to Functional Analysis, New-York John Wiley and Sons Inc. London, 1956. 2. A.P. Robertson and W. Robertson, Topological Vector Spaces Cambridge University Press, 1964. 3. B. Conway John, A Course on Functional Analysis Springer, Verlang, 1985. 4. M.M. Day, Normed Linear Spaces, Berline, Springer, Verlag, 1958. 5. M.M. Day, Normed Linear Spaces, Berlin, Springer, Verlang, 1958. 6. R.G. Cooke, Linear operators, Macmillan, London, 1953.