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,
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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
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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
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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 . ∫φ ∫m−1 m−1 2 2∏ iφ = = e dF φ (F0 o, F2∏ I) ∫0 m λ or Af = dE λ f ∫m − 1 be the spectral decomposition (4) of the unitary transformation V, using relation where we have set E λ = F-2arccot λ; (8), we can deduce the spectral {E } obviously is a spectral family over decomposition of A from that of V in the λ (-∞ , ∞) for spectral family, see SZ – following manner: Nagy (1). We begin by observing that F is a Φ Let us denote the spectral family of A, continuous function of Φ not only at the considered as a transformation in L , by point Φ = 0, but also at the point Φ=2 π. n {E λ,n} ; it is a spectral family over some 20
Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu
finite segment of the λ- axis determined by the bounds of A in L n. Setting f m = (E m- Em-1)f , the domain of definition of J therefore consists of the According to lemma (4) , there exists a self element f for which the series (4). – adjoint transformation E λ of H which ∞ 2 ∞ ∞ m 2 2 2 reduces in each L n to E λ,n . It is easy to see fJ = (J f ,f )= λd λfE ∑ −∞ m m ∑−∞ m m m ∑−∞∫− m that E λ is also a projection, and that m1 moreover it possesses the following properties. Converges; for equivalently , since E λfm = a) E λ≤ E for λ< , Eλf –Em-1f b) E λ+o = E λ c) E λ → 0 for λ → -∞ and E λ → I In the interval m-1 ≤ λ ≤ m, those for for λ → ∞ which the integral It is therefore a spectral family over the ∞ 2 λ 2 entire line ( - ∞ , ∞) , ∫ d E λ f ( 10 ) − ∞ Now we establish the formula Converges for these f , ∞ == ∞ = ∞ m λ = m λ = λ λ Jf ∑ Jm fm ∑ dE λ fm ∑ dE λ f. A ∫ dE , .( )9 −∞ −∞∫m−1 m−1 −∞ But since neither the domain of It is clear that if f belongs to the domain
integration nor the function under the of f, the same is true of E f and we have integral sign is bounded, it is first necessary to make precise the meaning of ∞ ∞ ∞ an integral of this type. = = = = JEµf ∑−∞Jm( µfE )m ∑−∞Jm µfE m Eµ∑−∞ mfJ m EµJf, Now denoting the integral in right hand side of eq (9) by J. Then this definition hence E J = JE (10a) will be valid for an arbitrary spectral family (this means that in the definition Now instead of starting with the sequence we shall only make use of properties of integers: a) and c ) of the family of projections E λ.. Let us consider the projections m = 0 , ± 1 , ± 2 , …………. we start with
Em – Em-1 ( m = 0 , ± 1 , ± 2 ,) another sequence of real number which goes to infinity in both directions, we and the corresponding subspace K m into arrive at the same definition of integral J. itself. Making use of lemma (4) , we define This being the case , inorder to establish the integral J as the uniquely determined self – adjoint transformation in H which formula (9) – that is , the given self –
reduces to the transformation J m in each adjoint transformation A is equal to the subspace K m.
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Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu
Integral J formed strarting with the we have thus arrived to a new formula ∞ spectral family of A. It suffices by virture = λ (4) A ∫ dE λ of lemma , to everify that the two self – −∞ adjoint transformations A and J coincide It is in this manner that J. Von Neu Mann in each of the orthogonal subspaces L n ( n in (1929), first proved the spectral = 1,2 , …) but for an element f of L n we composition of unbounded self – adjoint (4) have , by definition , transformation . Definition 2: Eλf = E λ,nf A symmetric transformation S is said to Since { E λ,n } is spectral family over the be lower semi bounded if there exist a real finite interval [ a , b ] , E λ f is constant for λ < a and λ ≥ b , and consequently the quantity. integral (10) converges ; hence f belongs ( Sf, f) ≥ c ( f,f ) to the domain of J and we have ∞ m b = λ = λ = For all f in D S it is said to be upper semi – Jf ∑−∞ ∫ dE λ f ∫ dE λ,n f Af m−1 a−o bounded if the opposite in equality is valid. If , in particular , ( Sf,f) ≥ 0. by the definition of { E λ,n} a spectral family corresponding to A in the We shall say, following the definition set (4) subspaces L n . This completes the proof down for bounded transformation, that S (4) of the fundamental formula (9). is positive . Now since we have defined that the Since every semi – bounded symmetric integral : transformation obtained from a positive transformation T by one or other of the ∞ formula : S = T + cI, S = -T + cI , λdE λ ∫−∞ it suffices to consider positive is self – adjoint transformation , and transformations in the sequel, namely the which reduces in each of the subspaces L m positive classes of matrices in special case = an n – by – n matrices A ( A εMn (¢) is called a positive matrix if every principal (F − F )H = (E − E )H φm φm −1 m m−1 minor of A is positive). Among the positive matrices we consider : an (invertible ) M – matrix is a real non – to the bounded self – adjoint positive an inverse M – matrix and hence transformation a positive matrix it self ; a ( Hermitian ) positive definite matrix is simply a m λ = ± ± Hermitian P – matrix (3). Our interest here ∫ dE λ ( m ,0 ,1 2 ,.....) m −1 and later lies in considering the Cayley
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Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu
transform of matrices in positivity classes T (Z) = above. But also our interest is in general positive from the complex plane into itself. In this (3) transofroms which relate to Cayley regard Stein and Tans sky both transform, therefore for a positive considered the Cayley transform , for the self – adjoint transformation A, the most part indirectly, when they provided spectral decomposition can be deduced connections between matrices stable very simply from that for abounded self – matrices ( i,e matrices for which Re (A) < adjoint transformation. This is done with 0 for all eigenvalues λ ) and convergent the aid of a linear transformation of semi– matrices ( i.e. those matrices A for which axis λ ≥ 0 into a finite segement of –
axis. In both of these papers the Key connection For example, the transformation came via Lyapunov’s equation AC+GA* =-I, and the Cayley transforms (3). The use = (10b) of the Cayley transform for stable matrices was recently made explicit in the which carries the semi – axis λ ≥ 0 into paper of Haynes (3). He proved that a the segement -1 ≤ ≤ 1. This is the matrix B is convergent if and only if there analogue of the linear transformation exists a astable matrix A such that: B = C (-A).