Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu On the Development and Recent Work of Cayley Transforms [I] Adam Abdalla Abbaker Hassan Nile Valley University, Faculty of Education- Department of Mathematics ABSTRACT: This paper deals with the recent work and developments of Cayely transforms which had been rapidly developed. ا : : ﺘﺘﻌﻠﻕ ﻫﺫﻩ ﺍﻝﻭﺭﻗﺔ ﺍﻝﻌﻠﻤﻴﺔ ﺒﺘﻘﺩﻴﻡ ﺍﻝﻌﻤل ﺍﻝﺤﺩﻴﺙ ﻭ ﺍﻝﺘﻁﻭﺭﺍﺕ ﺍﻝﺴﺭﻴﻌﺔ ﻝﺘﺤﻭﻴﻼﺕ ﻜﻤ ﺎﻴﻠﻲ . INTRODUCTION In this study we shall give the basic is slightly simplified variant of that of important transforms and historical Langer (1, 2) . background of the subject, then introduce -1 the accretive operator of Cayley transform Cayley Transforms V = ( A – iI)(A+iI) and a maximal accretive and dissipative First we introduce the transformation B = operator, and relate this with Cayley -1 ( I + T*T) and Transform which yield the homography in function calculus. After this we will study C = T ( I + T*T) -1 in order to study the the fractional power and maximal Cayely transforms. accretive operators; indeed all this had been developed by Sz.Nagy and If the linear transform T is bounded , it is C. Foias (1) . The method of extending and clear that the transformation B appearing accretive (or dissipative) operator to above is also bounded , symmetric , and a maximal one via Cayley transforms such that 0 ≤ B ≤ I ; (3) . modeled on Von Neumann’s theory on symmetric operators, is due to Philips (1) C = TB is then bounded too. If T is a fractional powers of operators A in Hilbert linear transformation with dense domain, space , or even in Banachh space, such as we know that T*, and consequently also T*T, exist, but we know nothing of their – A is infinitesimal generator of a conts (4) one parameter sem–group of contractions, domains of definition . The proof of have been studied or constructed by following theorem gives a rather different authors using different methods , surprising fact: we shall appear Sz. Nagy’s method here. Theorem 1: If the linear transformation T Many had been worked in the uniqueness is closed and if its domain is dense in H, theorem (in its form on dissipative the transformations B = ( I + T*T )-1 , C operators) , MaCaev and Palant [in case of = T ( I + T*T ) -1 are defined every where bounded operators] and Langer in the B ≤ 1 C ≤ ;1 general case. The proof of Belasz – Nagy and bounded, 17 Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu Moreover, B is symmetric and positive. From which we have Proof: In order to prove this theorem, we Bh 2 + Ch 2 = f 2 + g 2 ≤ h 2 , use the graph of T (4) which in this case is a closed linear set. Therefore: Let h be arbitrary element of H. since G r B ≤ ,1 C ≤ .1 V and GT* are complementary orthogonal For any element u in the domain of T*T, subspaces of N (4), we can decompose the we have ((I +T*T) u, u)=(u , u)+(Tu, Tu) ≥ element {h , 0 } of N into the sum of an (u, u) V element of G T and element of GT* , and Hence ( I + T*T) u = 0 implies u = 0 there is only one way. This assures that the inverse -1 { h , 0} = { f , Tf } + {T *g , - g } (1) transformation (I + T*T) exists. According to equation (2), it is defined This means, passing to the component, every where and equal to B ; that the system of equations : B = ( I + T*T) -1 h = f + T*g , 0 = Tf - g The transformation B is symmetric and has a unique solution f in D T and g in D T*. positive in fact writing f = Bh, g = Ch ( Bu , v ) = ( Bu , ( I + T*T) BV) = (Bu , Bv ) + ( Bu, T*TBv) = we define two transformation of H into it self which are obviously linear. = (Bu , Bv) + (T*TBu,Bv) = (( I + T*T ) Bu , Bv ) = ( u , Bv ) and ( Bu , Bv ) = The system of equations can then be ( Bu , ( I + T*T) Bu ) = ( Bu , Bu ) + written in the form : (TBu , TBu) ≥ 0. I = B + T*C , O = TB – C This completes the proof of the theorem. The transformations B and C, which play From which : an essential role in the above discussion, C = TB, I = B + T*TB = (I + T*T) B (2) are obviously the symmetric components of the normal transformations. Now since the two terms in the second 2 -1 -1 member of (1) are orthogonal, we have is C + iB = ( A + iI ) (I + A ) = (A – iA) obtained. This transformation and its adjoint, C – iB -1 2 2 2 2 2 2 2 = (A –iI ) ) are more generally the = = + * − = + + * + -1 h ho},{ f,{ Tf} {T g, g} f Tf Tg g transformations R z=(A-zI) , where z is a real or complex parameter , also play an 18 Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu essential role in other proofs of the (A-iI) f. But it then follows from the theorem. equation (( A-iI)f,h) = 0 that is the domain of (A-iI)* = A + iI and that ( A + iI) h Now the existence of = 0 . -1 R±i = ( A ± iI ) Hence h = 0, which contractdicts the hypothesis that h ≠ 0. can be proved directly from the relation 2 2 2 The transformations R ±i are therefore (A±iI )f =(Af,Af)± fi ,( Af)±i(Af f), + ff ),( =Af + f defined every where and bounded. The (3) same is true for R z = R x+iy when y ≠ 0 , since ( A – (x + iy ) I ) -1 = − In fact, it show that neither of equations 1 A xI −1 ( − iI ) . Of course R z can exist y y (A-iI) f = 0 , ( A + iI) f = 0 and be bounded even for certain real is possible unless f = 0 , which suffices for values of the parameter Z. the existence of the inverse . Furthermore, Now returning to relation (3). It showed ≥ we see that (A m iI ) f f , that (A − iI ) f = A + iI ) f , − which implies that That is, A−iI )( A+iI ) 1 g = g . The ≥ g R ± ig . (4) transformation ……………….(4a). for all elements g in the domain of R i, V = (A-iI) (A+iI) -1, R-i , respectively. called the Cayley transformation of A and Now these domains coincide with the it is therefore isometric (4). It is defined for entire space, this will follow from the fact element of the form: that these domains are: g = ( A + iI ) f (5) a) closed, and b) everywhere dense in H. by Vg = (A – iI) f (6) Proposition a) follows from the fact that the transformations R i and R -i are where f runs through D A. Then g and Vg continuous (consequence of (4) and closed each run through the entire space H. (since A and A ± iI are closed). Hence V is also unitary. Proposition b) is proved , for example for We give another equivalent definition. Ri ; in the following manner. If the domain of R , which is a linear set, not i Definition 1. every where dense in H , there would be an element h ≠0, orthogonal to the domain Let A ε M n (¢) s.t , I + A is invertible. Ri , that is to all elements of the form 19 Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu The Cayley transform of A , denoted by If not, V would have the characteristic C (A) is defined to be (3): value I; hence (I-V) -1 would not exist, contradicting (8). C(A) = ( I + A ) -1 ( I – A ) (7) Let us decompose the interval (0 , 2 Π) by The Cayley transform , not surprisingly , (3) means of an infinite number of points was defined in 1846 by Cayley . He having the two end points for limit points , proved that if A is skew Hermitain, then say by menas of the points φ for which - C(A) is unitary and the conversely, m φ provided of course that C(A) is exist. This cot m = m ( m = 0 , ± 1 , ± 2 , ) feature is useful e.g, in solving matrix equations subject to the solution being unitary by transforming them into − The projections P m = F φ F φ − equation for skew –Hermitian matrices, m m 1 later we shall discuss this point deeply. Are then pair wise orthogonal (4) and Now it is easy to recover A starting with ∞ V. It follows from (5) and (6), by addition P = lim Fφ − lim Fφ = I − 0 = I. ∑ m φ → ∏ φ → −∞ 2 0 and subtraction that ( I + V ) g = 2Af, (I-V)g = 2if, The projection P m, being permutable (= commutant) with V, is also From that we see that (I – V) g = 0 implies commutable with A ; the subspace L m that f = 0 and consequently , by (5), g = 0 corresponding to p m therefore reduces the also hence ( I – V) -1 exist and 2Af=(I+V) transformations V and A. Since the (I-V) -12if, that is, function (1- eiØ )-1 is bounded in the interval Ø m-1 ≤ Ø ≤ Øm we have, for f in A = i ( I + V ) ( I – V ) 1 (8) Lm: Example 1 : -1 Af = AP m f = i (I + V) (I – V) Pmf φ φm φ Let V = m iφ iφ −1 = i 1( +e )( 1−e ) dF φ f = (−Cot )dF φ f .