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

ABSTRACT: This paper deals with the recent work and developments of Cayely transforms which had been rapidly developed. ا :

ﺘﺘﻌﻠﻕﻫﺫﻩﺍﻝﻭﺭﻗﺔﺍﻝﻌﻠﻤﻴﺔﺒﺘﻘﺩﻴﻡﺍﻝﻌﻤلﺍﻝﺤﺩﻴﺙﻭ ﺍﻝﺘﻁﻭﺭﺍﺕﺍﻝﺴﺭﻴﻌﺔﻝﺘﺤﻭﻴﻼﺕﻜﻤ 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 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 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, 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 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 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).

= , Since we now in the coming deal only with the semi – axis λ ≥ 0, it is not necessary to use imaginary numbers in which maps the circumference of the unit order to transform it into a bounded curve. circle in the plane of onto the entire λ–axis–the transformation Hence we form the transformation which led to the idea of the “Cayley transformation (3,4). B = ( A – I ) ( A + I ) -1, instead of the Cayley transform (4a) or which is (7).

Since (( A + I ) f , f ) ≥ ( f,f) , the One other important analogue or feature transformation of the Cayley transform is that it can be viewed as an extension to matrices of the C = ( A + I ) -1 exists and (g , ( Cg ) ≥ ( C (3) conformal mapping . g, C g ) for all g in D C. It follows that:

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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

(C g , g ) ≥ 0 and Cg ≤ g Now due to equations (4a) and (7), we give two important lemmas, theorems and Since C is inverse of a self - adjoint some examples on positivity matrices. transformation, it is also self – adjoint (4), Lemma l : Let A Є M n (¢) s.t – 1 ∉ σ (A) and since it is bounded in its domain D C, this domain necessarily coincides with the and B = C (A) , then , entire space H, thus the transformation A = C (F) = ( I + F ) -1( I – F ) I – 2C = ( A + I ) C – 2 C = ( A – I ) C = B Proof [2] : As F = C ( A ) , we have is also self – adjoint and bounded , and ( I + A ) F = I – A or A ( I + F) = I – F . since Now notice that if Fx = -x, then x = 0, that is -1 ∉C ( F ). O≤ C ≤ I , we have B ≤ 1. Thus by (12) and since ( I + F ) -1 and 1 µ ( I + F ) commute it follows that A = C Let B = ∫ dF µ −1− o (F). be spectral composition of B. Since the Lemma 2: Let A Є M (¢) s.t -1 ∉ σ (A) transformation I – B = 2C possesses an n and let F = C (A) , then , inverse ( namely , ½ (A+I), the value 1 is not a characteristic value or eigen value of I + F = 2 (I + A) -1 (13) B , hence F is a continuous function of at the point = 1 , that is , F 1-0 = F 1 = I. If, in addition, A is invertible, then consequently we have , I – F = 2 (I + A)-1 (14)

+ µ ∞ 1 1 = λ λ Proof [3]: As F = C ( A ) , we have ∫dF µ ∫ dE ,.....( 11 ) − 1 − o 1 − µ − o I + F = I + ( I + A ) -1 ( I – A ) = ( I + A) -1 A = (I +B) (I-B) -1 = ( I + A + I – A ) = 2 I (A + I) -1 Similarly, I – F = 2 ( I + A ) -1 A . So if A where E λ = F f or = (11a) is invertible, { E λ} is obviously a spectral family over semi-axis ≥ 0. For a rigorous proof of I – F = 2 ( A -1 ( I + A )) -1 = 2 ( I + A -1)-1 (11), we can use a decomposition of the segement – 1 ≤ < 1 by means of an as claimed . infinite number of points which tend to 1. Finally notice that if F = C ( A ) , then Since E = F is the limit of polynomials λ 1− µ in B, it is obviously commutant with λ Є σ (A) ⇔ λ = ; for some σ Є (F) , A and with all the bounded 1+ µ transformations which permute (= commute) with A. which is (10b ) and (11a). 24

Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu

Now for a matrix A in each of the The converse of theorem 2 is not true. Let a foremention positivity classes, we us examine this and the above by some examine properties of its Cayley examples as follows: transform F = C( A ) , specially since A can be factored into A=(I + F) -1(I-F), we Example 2: investigate whether the factors ( I + F ) -1 and ( I – F ) belong to same positivity classes A and , conversely under what conditions does A belong to one of these positivity classes. Indeed Fallat and Then Tsatsomeros interested in factorization of the form A = X -1Y , where X and Y have certain properties such as diagonal dominance and stability (3) . They obtained result of this type by using the fact that the And Cayley transform is an and by employing the factorization of A interms of its Cayley transform then the following theorem gives us the relation of P – matrices and Cayley transforms. is not P – matrix.

Theorem 2 : Let A Є M n (C) be a P – To obtain a characterization of a P – matrix. Then F = C ( A ) is well – defined matrix interms of Cayley transform , we and both I – F and I + F are P – matrices. need the following lemma. In particular , A = G + F -1 (I – F) is a factionization of a P - matrix into Lemma 3: Let B Є M n (¢) so that σ ( B [ ( commuting ) P – matrix. α ] )= σ( B [ α ]) for all α Є { 1,2,…,n}. Proof [4]: first , since A is a P – matrix, Then B is a P – matrix if and only if every A is totally non negative matrix and has real eigenvalue of every principal no negative real eigenvalues . Hence F = submatrix of B is positive (3). C (A) is well – defined by lemma 1 and 2 and as addition of positive diangonal Remark 1 : Based on lemma3 , theorem 2 can be written as follows : If A is a P – matrices and inversion are operation -1 precerves positive matrices , it follows matrix , then ( I + F ) is a P – matrix and its every real eigenvalue of every principal that I – F and I + F are commuting P – -1 matrices. submatrix of ( I + F ) is positive. One consequence of the above result is Theorem 3: that if A is a P – matrix , then the main Let A Є M (¢) s.t σ ( A [ α ] ) = σ(A [ α]) diagonal entries of the matrix F = C ( A) n for all α Є { 1,2,..n} and -1 ∉ σ (A). Let F all have absolute value less than one (3). = C(A) then A is a P – matrix if and only

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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

if every real eigenvalue of principal nor I + A is totally non negative. submatrix of ( I + F) -1 is greater than ½ . We have seen in above how the Cayley Proof [5] : In view of lemma 1 and by transform developed with P matrix in reversing the roles of C ( A) and F in recent work of Fallat and Tsatsomers. lemma 2 , we obtain A = 2(I+F) –I. Now we shall study the extension of symmetric transformations. Also from the results of lemma 3 and -1 Remark 1 we obtain 2(I+F) – I. Cayley transforms and Deficiency Notice that in example 2 all 1-by-1 and 2- Indices by -2 prime positive matrices of Since , we have just seen , it is the self (I + F)-1 fail to satisfy the condition in (3) adjoint transformations which have theorem 3 . spectral decomposition , it is important to know wether or not a given symmetric Theorem 4 : Let B , G Є M n (R). The set matrices { BT + G (I-T) : T = diag transformation possesses a self – adjoint extension. More generally, the ( t 1 , t 2,….., t n) , t i Є [0,1 ] ( 1 ≤ i ≤ n ) } problem arises of characterizing all the contains only non singular matrices if and symmetric extension of a given symmetric only if G -1B are non singular (3) . transformations.

Now we shall give an example shows that Cayley transformations have been used in the natural question arising here , wether the study of this problem since their -1 the factorization = (I – A) (I + A) of a introduction in (1929) (4). The Cayley (3) totally nonnegative matrix . , F has transform of a symmetric transformations -1 factors (I - A ) and (I+A) are totally is defined just as for a self – adjoint nonnegative or not ? transformation previously , namely by : (3) Example 3 V = ( S – i I ) ( S + iI ) -1 ,

consider the totally nonnegative matrix just as these , we show that V isometric and that we can recover S from V by means of the formula :

S = i ( I + V ) ( I – V ) -1. and consider A = - C ( Fˆ ). By using relations (3) , (5) , (6) , written Then Fˆ = C ( -A ) = ( I – A ) -1 ( I + A ) for S instead of A , it is easy to see that if neither S is closed then V is also closed , and conversely, since every symmetric transformation S has the closed extension S** ( its cloure ) , we shall consider only closed symmetric transformations. 26

Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu

′ We know that if S is self – adjoint , its V is isometric and closed , D v and DV are Cayley transform V is unitary ; we shall closed sets , that is , subspaces of H, one show that the converse is also true. or the other of which may coincide with Suppose that V is unitary ; let g be an H. the orthogonal complements element of D s* and set g*=S*g . then ( Sf , g) = ( f , g*), for all elements f of D ′ s H–Dv and H– D are called the deficiency , and since these elements f are of the V subspaces, and their dimensions the form f = ( I – V ) h , where h runs through deficiency indices of the symmetric D =H, we have : V transformation S (or also of the isometric ( i ( I + V ) h , g ) = (( I – V ) h , g* ), or transformation V).

i (h,g) + i ( Vh , g ) = (h,g*) –( Vh,g*), for Let us recall that D v is the set of values of ′ all elements h of H. Since V is unitary S + i I and DV is the set of values of ( hence defined every where and isometric S – i I. , we can replace ( h, g ) by (Vh , Vg) and (h,g) by (Vh,Vg*) and obtain : It follows from what we have just proved that a closed symmetric transformation is (Vh, - i Vg – Vg* + g* ) = 0 , self –adjoint if and only if its deficiencies are (0,0). The values Vh of the unitary transformations V exhaust the space H; We pass to the problem of extension. It is this implies that : clear that if S′ is an extension of S (we ′ g − ig * g − ig * suppose that both S and S are symmetric g = (1-V) g* = i(1+V) 2 2 and closed), now the Cayley transform of V ′ of S′ will be an extension of the Consequently g also belongs to the Cayley transform V of S. D will be a domain of S and Sg = g* . This prove that v subspace of D′ it follows that when we S* = S ; S is therefore a self e- adjoint V transformation (4). pass from S to S′ , the deficiency indices diminish by the same (finite or infinite ) Now we are in a position to introduce the number. notion deficiency subspaces and their dimensions, and relations to Cayley We now show that conversely, every transforms , which had been studied by F (4) isometric extension of the Cayley Riez and Bela Sz Nagy in 1955 . transform V of S determines a symmetric ′ Then since in the case of an arbitrary extension S of S whose Cayley transform closed asymmetric transformation S , the V ′ equals U. domain of definition D V and the set of ′ = -1 values D V VD v do not in general Firstly observe that ( I – U ) exist , that coincide with the entire space H; but since is, ( I – U ) h = 0 , implies h = 0 . In fact, 27

Journal of Science and Technology Vol. 13 June /2012 ISSN -1605 – 427X Natural and Medical Sciences (NMS No. 1) www.sustech.edu

if (I – U ) h = 0 then for every element of Cayley transform V′consists of elements the form f = ( I – U ) g : of the form 2i φ, where φ runs through ′ DV, and that V (2i φ) = 2iU φ = U (2i φ). ( h , f ) = ( h , g ) – ( h , Ug) = (Uh, Ug) – (h, Ug) This proves that V′ = U which has to be shown. = - (( I – U ) h , Ug ) = 0 ; We note that if U is an arbitrary isometric hence is orthogonal to the set values of I transformation for which the set of values – U , and h is orthogonal to the set of of I- U is dense in H, the same reasoning values of I–U , and therefore the domain proves that : of definitions of S . Since this domain is dense in H , we necessarily have h = 0 S′ = i (I + U) (I – U) -1 is asymmetric Now let us form the transformation transformation whose Cayley transforms equals U. S′ = i ( I + U ) ( I – U ) -1 Now, with this, the problem of finding all the (closed) symmetric extensions of the which obviously is an extension of S, closed symmetric transformations reduces is symmetric ; in fact , if f and g are ′ to the problem of finding all the isometric elements of D S they are of the form extension of its Cayley transform V; this f = ( I – U ) φ , problem is obviously much simpler than the original problem. This had been done g = g ( ( I – U ) ψ, and we have by F. Riez and B. Sz – Nagy (4).

S′ f = ( i ( I + U ) φ, S′ g = i ( I + U ) ψ; In fact, inorder to extend V; we have only to map the deficiency subspaces H–DV, or Hence , in as much as ( φ, ψ)=(U φ, U ψ) a subspace of the latter, isometrically into ′ = ( f , g) = (i ( I + U ) φ , ( I – U ) ψ) the deficiency subspace H– Dv ; this is a complished, for example with the aid of = i [ ( U φ , ψ)- (φ,U ψ ) ] = (( I – U ) φ , i two orthonormal systems taken in H–Dv ′ ′ ( I + U) ψ) = (f, S g). and in H- DV . It is thus possible to exhaust the deficiency subspaces with the smaller Finally the relation f = ( I – U ) φ implies domain ; the corresponding symmetric that transformation S′ will then be a maximal extension of S. If two deficiency subspace S′ S′ f = i ( I + U ) φ , ( + iI )f = 2 i φ , are of the same dimension, they can be ( S′ – iI )f = 2iU φ, exhausted simultaneously, and we obtain a unitary extension of V, and consequently from which we see that the domain of the a self -adjoint extension of S.

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∞ The two deficiency subspaces are of the S g = i C 1g1 + i ( 2C 1 + o ∑ 1 C K K same dimension, they can be exhausted C g ) + i( 2C +2C )g + ….. simultaneously, and we obtain a unitary 2 2 1 2 3 extension of V, and consequently a self-ad for all elements f = for ∑ C g K joint extension of S. K which ( I – Vo )f has a meaning , that is, F. Ries and Bela Sz–Nagy (4) summarized for which for all elements: the essential points of the above in the following theorem. 2 + + 2 + + + 2 + C1 C1 C2 C1 C2 C3 ...... Theorems 5 : In order for the closed symmetric transformation S to be maximal it is necessary and sufficient that converges ( for these f we have , in ∞ one or the other of its deficiency indices = particular , ∑ C k o ). be equal to 0 ; inorder to admit a self 1 adjoint transformation as an extension , it

is necessary and sufficient that its Therefore the transformation S o has (0, 1) deficiency indices be equal ; finally , for deficiency indices; it is called the inorder that it its self be self-adjoint , it is elementary symmetric transformation. Now necessary and sufficient that its two it can be shown that every symmetric deficiency indices be equal to 0. transformations S 0 of a H of arbitrary dimension with the deficiency Example 4: Let H be a Hilbert space indices (0,m) (where m is an arbitrary finite whose dimension is denumerably infinite, or infinte cardinal number) is composed of of an isometric non – unitary m elementary symmetric transformations transformation V o ; let { g n} be plus possibly a self –adjoint transformation a complete orthonormal sequence in H , in the following sense ; there are m and set mutually orthogonal subspaces K α with ∞ = ∞ denumerably infinite dimension in H, each V C g C g + o ∑ 1 k k ∑ 1 k k 1 of which reduces S to an elementary symmetric transformation, s.t, in the Then the domain of V is the entire space, o subspace K ′of elements orthogonal to all while the set of values V f has the o the K (a subspace which may consist of orthogonal complement of dimension1 α the single element 0), S reduces to a self – consisting of elements of the form Cg . It is 1 adjoint transformation (4). easily shown that the set of values of I – Vo is dense in H. Hence V is the Cayley o Now the problem of maximal symmetric transform of symmetric transform : transformation whose deficiency indices -1 are (m,0) presents nothing new, since , in So = i ( I + V o ) ( I – Vo ) , S o is defined by

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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 fact that the Cayley transform of – S is obviously equal to the invese of that of S. recent work, that is to say the Cayley transform of accretive and dissipative Remark 2: the real symmetric operators and purely maximal of these , the transformations of the space L 2 always symmetric , bounded , self adjoint and unitary have equal deficiency indices , hence they operatiors , the second direction is a P-matrix are either self–adjoint or possess self– tranformaition. adjoint extensions. REFERENCES: The transformation S is said to be real if 1. Sz Nagy, B., Foias, C., Bercovici, H., its domain contain with a function f(x) its Kérchy, L., (1966). Harmonic Analysis conjugate f(x) , and if in addition S f (x) of operators on Hilbert space . Springer = fS (x). publications. 2. Some notes on parametrizing Our proposition is verified as follows: the representations.http://www.liegroups.org/paper

′ s/basepoint.pdf (retrieved August 2009). domains D v and the range Dv of the 3. Fallat, Shaun M., Tsatsomeros, and Cayley transform of S consist of the Michael J., (2002). On the Cayley transform of functions: positivity classes of Matrices. The Electronic Journal of Linear Algebra , 9: 190-196. u(x) = Sf(x) = if(x) and v(x) = S g(x) – 4. Frigyes Riesz and Sz. Nagy. B., (1965). ig(x), respectively , where f and g run Functional Analysis translated from the 2 nd through the domain of S. Setting French edition by LEO F. Boron Frederick g(x) = f (x) we have v(x) = u (x), UNGAR Publishing Co. NY. ′ hence D consists of the conjugates of two 5. Fokko du Cloux, Combinatorics for the orthogonal function are also orthogonal, representation theory of real reductive H–Dv consists of the conjugates of H – Dv, http://www.liegroups.org/papers/summer05/com binatorics.pdf (retrieved August 2009). operators and purely maximal of these , For further details see Frigyes and Sz. 6. Jeffery A., (1991). Lifting of Characters, Nagy (4) for positive symmetric the symmetric , bounded , self adjoint and volume 101, a Progress in Mathematic transformation, and all its extensions to unitary operatiors , the second direction is positive self–adjoint transformation. Birkhauser,a P-matrix Boston,tranformaition. Basle, Berlin.

CONCLUSIONS 7. Parameters for Representations of Real Groups, Atlas Workshop, July 2004 Now we see how the Cayley transforms http://atlas.math.umd.edu/papers/summer05/par had been rapidly developed in the more ameters.pdf (retrieved August 2009). recent work. The study showed the necessary historical 8.Hakan S., (2006). Multidimentional Cayely advances and important two directions of trasnsform and tuples of unbounded operators recent work, that is to say the Cayley theory; J. Operator 56: 317-342 transform of accretive and dissipative 30