CONVEXITY OF INVERSION FOR POSITIVE OPERATORS ON A HILBERT SPACE. Sangadji *
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1 CONVEXITY OF INVERSION FOR POSITIVE OPERTORS ON HILBERT SPCE Sangadji * BSTRK KONVEKSITS DRI INVERSI UNTUK OPERTOR-OPERTOR POSITIF PD RUNG HILBERT Makalah ini membahas dan membuktikan tiga teorema tentang operator-operator invertibel positif pada ruang Hilbert Teorema pertama memberikan suatu perbandingan dari nilai rerata aritmetika tergeneralisir, nilai rerata geometri tergeneralisir, dan nilai rerata harmonik tergeneralisir untuk operator-operator invertibel positif pada ruang Hilbert Untuk teorema kedua dan ketiga masing-masing memberikan tiga pertaksamaan untuk operator-operator invertibel positif pada ruang Hilbert yang ketiganya ekivalen BSTRCT CONVEXITY OF INVERSION FOR POSITIVE OPERTORS ON HILBERT SPCE This paper discusses and proves three theorems for positive invertible operators on a Hilbert space The first theorem gives a comparison of the generalized arithmetic mean, generalized geometric mean, and generalized harmonic mean for positive invertible operators on a Hilbert space For the second and third theorems each gives three inequalities for positive invertible operators on a Hilbert space that are mutually equivalent PRELIMINRIES Let X be a real or complex vector space norm on X is a non-negative real valued function L on X such that for all scalar and all x, y X the following hold: ( i x 0 x = 0 x = 0 ( ii x + y x + y ( iii x = x * Pusat Pengembangan Teknologi Informasi dan Komputasi - TN
2 real (complex normed linear space is a real (complex vector space X together with specified norm on X On such a space we have a metric ρ defined by ρ ( x, y = x y If X is complete in this metric we call X a Banach space Completeness means that if x is a sequence of elements of X such that { } n there exists an element x in X such that lim x m xn = m, n lim x x = 0 n (, Let H be a real or complex vector space n inner product on H is a function which assigns to each ordered pair of vectors in H a scalar, in such a way that for all x, y, z H and all scalar the following hold: ( i ( x + y, z = ( x, z + ( y, z ( ii ( iii ( x, y = ( x, y ( x, y = ( y, x ( iv ( x, x 0 ( x, x = 0 x = 0 Such a space H, together with a specified inner product on H, is called an inner product space In any inner product space H we have the Cauchy-Schwarz inequality: From the Cauchy-Schwarz inequality it follows that n ( x, y ( x, x( y, y, for all x, y H 0 1 / x = ( x, x constitutes a norm on H If H is complete in this norm, we call that H is a Hilbert space function f defined on a set S is called to be convex if for each x, y S and each, 0 1 we have f ( x + (1 y f ( x + (1 f ( y n operator on a Hilbert space H is called to be positive if ( x > 0 for every x H If T is a positive operator on H then for real numbers, β we define the operator T + β on H by ( T + β ( x = T( x + β, x H MIN RESULT The main result of this paper are the following three theorems The first one gives a comparison of the generalized arithmetic mean, generalized geometric mean, and generalized harmonic mean for positive invertible operators on a Hilbert space
3 For the second and third theorems each gives three inequalities for positive invertible operators on a Hilbert space that are mutually equivalent Theorem 1 Let and B be positive invertible operators on a Hilbert space H Then for 0 1 the following inequalities hold: (1 ( B 1 [(1 ] 1 / / / 1 / (1 Proof Consider the function F( x = x + 1 x for positive real numbers x and 0 1 First we need to show that F (x is a nonnegative function on the interval [ 0, Recall that F (x is continuous on [ 0, By routine calculation we get F (0 = 1 0, F(1 = 0, F'( x = (1 x and F'( x = (1 x 0, 0 x 1, F'( x = (1 x 0, 1 x < Thus we have that F (x is monotonically decreasing on [ 0,1] with 0 F( x 1 and monotonically increasing on [ 1, with F ( x 0 Since F (x is a nonnegative function on the interval, by the standard operational calculus we have for a positive operator T on H and 0 1: T + 1 T ( By replacing T with T -1 in ( we get T + 1 T, (3 and by taking inverses of both sides (3 we get T ( T + 1 (4 Combining ( and (4 we conclude T + 1 T ( T + 1 (5
4 / / Putting T = B in (5 we have + 1 ( [ ( + 1 ] (6 Finally, multiplying by 1 / on both sides in (6 we achieve 1/ 1/ 1/ B + ( [ ( + 1 ] / 1/ B + ( ( B + (1 ( 1/ 1 B 1 [(1 ] 1 / / / 1 /,, Theorem Let T be a positive invertible operator on a Hilbert space H Then the following hold and are mutually equivalent: ( i If 1 0 then T + 1 T ( ii if ( iii if > 1 then T + 1 T < 0 then T + 1 T (7 Proof Obviously assertion (i was already obtained in ( The rest is to show the equivalence of (i, (ii and (iii ( i ( ii : ssume > 1 ssertion (i is equivalent to 1 / 1 / S (1/ S + 1 (1/, ie, S S + Putting 1 / T = S we have T S + 1 Thus (i implies (ii Similarly we can get (ii implies (i ( ii ( iii : ssertion (ii is equivalent to S + 1 S Multiply both / sides of this inequality by S to obtain the equivalent inequality 1 (1 + S S b for any > 1 Then set λ = 1 < 0 and T = S We λ get λt + 1 λ T Thus (ii implies (iii Similarly we can get (iii implies (ii
5 Theorem 3 Let and B be positive invertible operators on a Hilbert space H Then the following hold and are mutually equivalent: 1 / / / 1 / ( i If 1 0 then (1 ( ( ii ( iii if > 1 then (1 1 / if < 0 then (1 ( 1 / / ( / / / 1 / 1 / (8 Proof / / Use Theorem by putting T = we have ( i If 1 0 then + 1 ( ( ii if > 1then ( iii if < 0 then + 1 ( + 1 ( (9 1 / and then multiplying (9 by on both sides we get 1/ ( i If 1 0 then (1 ( ( ii if > 1then (1 1/ ( iii if < 0 then (1 as desired ( 1/ ( 1/ 1/ 1/ CONCLUSION Inequalities in (5 are very important and fundamental Inequalities in (1 in / / Theorem 1 can be obtained from (5 by putting T = B Statements in Theorem can be derived from the first inequality in (5 Statements in Theorem 3 / / obviously can be derived from statements in Theorem by putting T = 1 / and then multiplying by on both sides
6 REFERENCES 1 HOFFMN, K, Banach Spaces of nalytic Functions, Dover Publications, Inc, New York (196 WNG, D Convex Operator Function, Internat J Math nd Sci 11 ( FURUT, T and YNGID, M, Generalized Means and Convexity of Inversion for Positive Operators, The merican Mathematical Monthly, Volume 105, Numbert 3 The Mathematical ssociation of merica (1998
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