Research Article Norm Estimates for Solutions of Polynomial Operator Equations

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1 Matheatics Volue 5, Article ID 5489, 7 pages Research Article Nor Estiates for Solutions of Polynoial Operator Equations Michael Gil Departent of Matheatics, Ben Gurion University of the Negev, PO Box 653, 845 Beer-Sheva, Israel Correspondence should be addressed to Michael Gil ; gili@bezeqintnet Received July 5; Accepted 3 August 5 Acadeic Editor: Willia E Fitzgibbon Copyright 5 Michael Gil This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited We consider the equations c ka k XB k =Cand c ka k XB k =C,c k C (k=,,), c =, A, B, C are given linear bounded operators in a Banach space and X is to be found Representations of solutions are derived In the cases of Euclidean and Hilbert spaces, nor estiates for the solutions are suggested Introduction and Stateent of the Main Result Theobjectsofthispaperaretheequations c k A k XB k =C, () c k A k XB k =C, () c =, c k C (k =,,), A, B, C are given linear bounded operators in a Banach space E with a nor and X is to be found There is a long tradition of finding different expressions for the solution of an operator equation in the for of operator integrals and series, soe proinent exaples of which occur in the works of E Heinz, M G Krein, M RosenbluandRBhatia,CDavis,andAMcIntoshA coprehensive suary of these is contained in [] and [] This tradition was continued in [3], which deals with the equation A k XB k =C (3) Clearly, (3) is a particular case of () The special case AX XB = C (4) of () is the uch studied Sylvester equation, of great interest in operator theory, nuerical analysis, and engineering; copare [4 6] and references therein Let A be the operator adjoint to ATheequation AX+XA =C (5) is the faous Lyapunov equation playing an essential role in the stability theory of differential equations The equation X AXB=C (6) is an exaple of (); in particular, the discrete Lyapunov equation X AXA =C (7) is an iportant tool in the theory of difference equations About other operator equations see [7 9] and references therein In particular, [9] deals with necessary and sufficient conditions for the existence of solution to the systes of the general solution to a syste of adjointable operator equations over Hilbert C odules In [8] the nonlinear operator equations of the for ABA = A and BAB = B are considered It should be noted that in the finite diensional case the nuerical ethods are well developed Besides, the traditional ethods convert atrix equations into their equivalent fors by using the Kronecker product, which involve the inversion of the associated large atrix and result in increasing coputation and excessive coputer eory

2 Matheatics The recently suggested gradient based iterative ethods are ore powerful; copare [ 3] and references therein The ai of the present paper is to derive representations of solutions to () and () and to estiate the nors of these solutions Such estiates are iportant, in particular, for the investigations of linear and perturbations of nonlinear differential and difference equations Nor estiates for solutions of the Sylvester equation whose coefficients are noral operators can be found in [] In the finite diensional case solution estiates for (4) and (6) have been established in [4] In this paper, in particular, we considerably generalize the ain results fro [4] Denote by σ(a) and r s (A) the spectru and spectral radius of A,respectively Theore Let x k (k =,,)be the roots taken with the ultiplicities of the polynoial p (x) := c k x k (8) and let r s (A) r s (B) < in k x k (9) Then () has a unique solution X which can be represented by X=( ) j,j,,j = x j + x j + x j A j +j ++j CB j +j ++j, + and the series strongly converges () The proof of this theore is presented in the next section Proof of Theore Following [, Section I3], introduce the operators A l and B r by A l X:=AXand B r X:=XB, respectively Besides, A l and B r coute Let f(z, w) be a scalar function regular on σ(a) σ(b) Define the operator valued function Φ (f, A, B) := 4π L B L A f (z, w) R z (A l )R w (B r )dwdz, () L A, L B are closed Jordan contours surrounding σ(a) and σ(b), respectively So for any bounded linear operator C Φ (f, A, B) C Besides, = 4π L B L A f (z, w) R z (A) CR w (B) dw dz () Φ(f,A,B)Φ(f,A,B)=Φ(f f,a,b), (3) for functions f, f regular on σ(a) σ(b) Takef(z, w) = p(zw)if p (zw) = (z σ(a), w σ(b)), (4) then due to forula (35) fro [, Section I3] () can be written as Since Φ(p(zw),A,B)X=C (5) p (x) = k= akinguseof(3),wecanwrite Φ (zw x,a,b) (x x k ), (6) Φ (zw x,a,b)φ(zw x,a,b)x=c But Φ(zw x k,a,b)=a l B r x k IThus, k= and therefore we get the following result (7) (A l B r x k I) X=C, (8) Lea Under condition (4) equation () has a unique solution defined by X= k= Furtherore, for a C, satisfying consider the operator (A l B r x k I) C (9) a >r s (A) r s (B), () Z= a k+ Ak l Bk r C= a k+ Ak CB k () The series converges and A l B r Z=AZB= a k+ Ak+ CB k+ = a k Ak CB k +C=aZ+C We thus get the following result () Lea 3 Let condition () hold Then operator Z defined by ()istheuniquesolutiontotheequationazb az = C and Z=(A l B r ai) C

3 Matheatics 3 Proof of Theore Put Y =(A l B r x I) C, Y =(A l B r x I) Y, (3) Condition(7)doesnotallowustoconsidertheLyapunov equation, since r s (A) = r s (A )Becauseofthisweare going to derive the representation of solutions to () under other conditions To this end put α (A) = sup Re σ (A), β (A) = inf Re σ (A) (9) Y k =(A l B r x k I) Y k Lea iplies X=Y DuetothepreviousleaY k are solutions to the equations AY k B x k Y k =Y k (k=,,), According to (9) for a solution to (6) we have provided X= k= (A l B r x k I) C, (3) AY B x Y =C, provided (9) holds So Y = Y = = j= j, x k+ x j+ A k CB k, Y j = A j Y B j = x k+ x j+ j= x k+ j x j+ A k+j CB k+j A k Y k B k, A j x k+ A k CB k B j (4) (5) Here B r p( z w ) = (z σ(a), w σ(b)) (3) is defined as B r C=CB Thus X= = k= k= (A l B r x k I) B r B C r (A l x k B r ) C (3) Now the invertibility condition for B can be reovedwe thus have proved the following result Lea 5 Let the condition z x k w = (k=,,; z σ(a), w σ(b)) (33) Continuing this process for j = 3,,, according to Lea, we prove the theore 3 Representations of Solutions to () Assue that B is invertible; then fro () we have c k A k XB k = C :=CB (6) Let r l (B) denote the lower spectral radius of B: r l (B) = in λ σ(b) λ Taking into account the fact that r l (B) = /r s (B ) and applying now Theore to (6) with B instead of B,wegetthefollowingresult Corollary 4 Let r s (A) <r l (B) in k x k (7) Then () has a unique solution X which can be represented by X=( ) j,j,,j = x j + x j + x j A j +j ++j CB j j j, + and the series strongly converges (8) hold Then () has a unique solution X which can be represented by (3) Lea 6 Let the condition β (A) >α(b) (34) hold Then (4) has a unique solution X S,whichcanberepresented as X S = e At Ce Bt dt (35) Proof For the brevity put b=α(b)and c = (β(a) α(b))/ Then β(a) = b + c and α(ab I(b+c)) = cmoreover, α( (A I(b + c))) = α( A) + b + c = β(a) + b + c = c Equation (4) is equivalent to the following one: (A (α (B) +c) I) X X(B (α (B) +c) I) =C (36) Due to Theore 9 fro [] a solution of (36) is defined by the equality X S = e (A I(α(B)+c))t Ce (B (α(b)+c)i)t dt as claied = e At Ce Bt dt, (37)

4 4 Matheatics Corollary 7 Let the condition β (A) >α(ab) (a C) (38) hold Then the equation AX axb = C (39) has a unique solution X a,whichcanberepresentedas X a = e At Ce abt dt (4) Asolutionof(39)isalsogivenbyX a =(A l ab r ) CSo under condition (38) we have (A l ab r ) C= e At Ce abt dt (4) Now assue that β (A) > ax α(x k B) k (4) Then by (4) (A l x k B r ) = e At Ce xkbt dt (43) Let x k =a k +ib k Thenα(x k B) = sup s σ(b) (a k Re s b k I s)so condition (4) eans that β(a) > sup s σ(b) (a k Re s b k I s) Put W =(A l x B r ) C, W =(A l x B r ) Y, W k =(A l x k B r ) W k (44) Then X=W DuetothepreviousleaY k are solutions to the equations AW k x k B=W k (k=,,), AY B x Y =C, W j = e At W j e xkbt dt provided (4) holds So W = e At Ce x Bt dt, W = e At W e x Bt dt = e (t +t )A Ce (x t +x t )B dt dt (45) (46) Continuing this process for j = 3,,, according to Lea 5, we obtain X = e (t ++t )A Ce (x t ++x t )B dt dt We arrive at the following result (47) Theore 8 Let x k (k=,,)be the roots of p(x) and let condition (4) hold Then () has a unique solution X, which can be represented by (47) 4 Solution Estiates in the Finite Diensional Case 4 Equation () In this section E = C n is a Euclidean space Let λ k (A) (k =,,n) be the eigenvalues of a atrix A counted with their ultiplicities The following quantity (the departure fro norality of A) plays a key role hereafter: g (A) =[N (A) n k= λ k (A) ] /, (48) N (A) = (Trace AA ) / is the Frobenius (Hilbert- Schidt nor) of A The following relations are checked in [5, Section ]: g (A) N (A) (A) Trace A, g (A) N (A I), (49) A I = (A A )/i IfA is a noral atrix: AA = A A,theng(A) = IfA and A are couting atrices, then g(a +A ) g(a )+g(a ) By the inequality between geoetric and arithetic ean values we have ( n n k= λ k (A) ) n ( n k= λ k (A) ) (5) So g (A) N (A) n(det A) /n Obviouslyg(aA) = a g(a) (a C) By Corollary 7 fro [5], one has n Aw w!g k (A) r w k s (A) (w =,, ) (5) (w k)! (k!) 3/ Note that /(w k)! = (s < k) Furtherore, due to Exaple3fro[5]wehave n t k g k (A) eat eα(a)t (t ) (5) (k!) 3/ Fro () and Lea 3 we have (A l B r ai) C= a w+ Aw CB w w= (s=,,), (53) provided condition () holds Hence, and fro (5), it directly follows (A lb r x s I) J(A, B, x s), (54)

5 Matheatics 5 J (A, B, a) = n g k (A) g j (B) j, 3/ w= (w!) r w k s (A) r w j s (B) a w+ (w k)! (w j)! Now Lea and (54) iply the following result (55) Corollary 9 Let condition (9) hold Then a unique solution X to () in C n satisfies the inequality X C J (A, B, x ) J (A, B, x ) J(A, B, x ) (56) w= If A is noral, then g(a) = and n g j (B) J (A, B, a) = j= (j!) 3/ But w!r w s (A) rw j s (B) a w j (w j)! = rj s (A) a j+ = rj s (A) a j+ w= w! (r s (A) r s (B)) w j a w j (w j)! w= d j w!r w s (A) rw j s (B) a w+ (57) (w j)! dx j x w =j! rj s (A) ( x) j j+ a w= (58) (x = a r s (A) r s (B)) Thus, we have n g j (B) r j s J (A, B, a) (A), j= (j!) / j+ (59) ( a r s (A) r s (B)) provided A is noral If both A and B are noral, then J (A, B, a) a r s (A) r s (B) (6) Furtherore, obviously, (w!) b w (w k)! (w j)! t!b t w!b w (t k)! (w j)! w= Taking t= w= =b k+j (t k)! (w j)! t= t!b t k w= j!k!b k+j = ( b) k+ ( b) j+ w!b w j ( <b<) (6) b=b(a, B, a) = r s (A) r s (B), (6) a we get J (A, B, a) = a n j, J (A, B, a) J (A, B, a), (63) g k (A) g j (B) b k+j (64) (A, B, a) / ( b(a, B, a)) k+ ( b(a, B, a)) j+ Now Corollary 9 iplies the following result Corollary Let condition (9) hold Then a unique solution X of () in C n satisfies the inequality X C J (A, B, x ) J (A, B, x ) J (A, B, x ) (65) Theore and siple calculations iply the following result Corollary Let r s (A) < Then(7)hasauniquesolution X L, which satisfies the inequalities X L C = n ( g k (A)!r k s (k!) 3/ ( k)! Fro Corollary and (63) it follows that (A) ) (66) n X L C ( g k (A) (k!) / ( r s (A)) ) (67) k+ 4 Equation () Dueto(4)and(5)under(38)wehave (A l ab r ) e As eabs ds n sc s k+j g k (A) g j (ab) e ds, j, 3/ (68) c = β(a) α(ab)but s k+j e cs (k + j)! ds = c (69) k+j+ Thus γ (a, A, B) := (A l ab r ) γ(a, A, B), (7) n j, (k + j)!g k (A) g j (ab), (β (A) α(ab)) k+j+ 3/ (7) provided β(a) > α(ab) Due to Lea 5 we arrive at the following result Corollary Let condition (4) hold Then a unique solution X to () in C n satisfies the inequality X C γ (A, B, x ) γ (A, B, x )γ(a,b,x ) (7)

6 6 Matheatics If A is noral, then g(a) = and therefore n a j g j (B) γ (a, A, B) := j= (β (A) α(ab)) j+ / (73) (j!) If both A and B are noral, then γ (a, A, B) β (A) α(ab) (74) For the Sylvester equation we have the following result Corollary 3 Let α(b) < β(a) Then (4) in C n has a unique solution Y S, which satisfies the inequality Y S n j, (k + j)!g k (A) g j (B) (β (A) α(b)) k+j+ 3/ (75) Consider the Lyapunov equation (5) Taking in Corollary 3 A instead of A and A instead of B, since α(a ) = α(a) and β( A) = α(a), wegetthefollowing result Corollary 4 Let α(a) < Then(5)inC n has a unique solution Y L, which satisfies the inequality Y L C n j, 5 Solution Estiates in the Infinite-Diensional Case (k+j)!g k+j (A) ( α (A) ) k+j+ 3/ (76) 5 Equation () In this section E =H,a separable Hilbert space It is assued that Put A A and B B are Hilbert-Schidt operators (77) g I (A) = ( N (A A ) k= I λ k (A) ) / (78) Recall that N (A) = (Trace AA ) / and λ k (A) are the eigenvalues of a atrix A counted with their ultiplicities If A is a Hilbert-Schidt operator, then g I (A) = g(a), g (A) =(N (A A ) k= λ k (A) ) / ; (79) copare Lea 65 [5] Due to Exaple 7 fro [5] we can write Aw w!g k I (A) rw k s (A) (w k)! (k!) 3/ (w=,,) (8) As it was shown in Section 4 (A l B r x s I) C= w=xs A w CB w (8) (s=,,), provided (9) holds Hence, and fro (8), it directly follows that J H (A, B, a) = (A lb r x s I) J H (A, B, x k ), (8) g k I (A) gj I (B) j, 3/ w= (w!) r w k s (A) r w j s (B) a w+ (w k)!(w j)! Now Lea iplies the following result (83) Corollary 5 Let conditions (9) and (77) hold Then the unique solution X to () in H satisfies the inequality X (84) C J H (A, B, x )J H (A, B, x )J H (A, B, x ) If A is noral, then g(a I )=andaccording to (59) g j I J H (A, B, a) (B) rj s (A) j=(j!) / (85) j+ ( a r s (A) r s (B)) If both A and B are noral, then J H (A, B, a) a r s (A) r s (B) (86) Furtherore, according to (63), we get J H (A,B,a) J H (A, B, a), J H (A, B, a) = a j, g k I (A) gj I (B) bk+j (A, B, a) / ( b(a, B, a)) k+ ( b(a, B, a)), j+ (87) b(a, B, a) isdefinedby(6)nowcorollary5iplies the following result Corollary 6 Let conditions (9) and (77) hold Then a unique solution X of () in H satisfies the inequality X (88) C J H (A, B, x ) J H (A, B, x ) J H (A,B,x ) By virtue of Corollary 5 we can assert that a unique solution X L of (7) satisfies the inequality X L C = ( g k I (A)!r k s (A) ) 3, (89) (k!) ( k)! provided A satisfies conditions (77) and r s (A) < Fro Corollary 6 it follows that X L C ( g k I (A) (k!) / ( r s (A)) ) (9) k+

7 Matheatics 7 5 Equation () Due to Exaple 73 fro [5] we have t k a k g k (A eaat eα(aa)t I ) (a C, t ), (k!) 3/ (9) provided A satisfies (77) Now let us consider (), assuing that condition (4) holds Due to (4) and (9), (A l ab r ) c = β(a) α(ab)thus γ H (a, A, B) := e As eabs ds e sc s k+j g k I (A) gj I (ab) ds, j, 3/ (9) (A l ab r ) γ H (A, B, a), (93) j, (k+j)!g k I (A) gj I (ab), (β (A) α(ab)) k+j+ 3/ (94) provided β(a) > α(ab) Due to Lea 5 we arrive at the following result Corollary 7 Let conditions (4) and (77) hold Then the unique solution X to () in H satisfies the inequality X C γ H (A, B, x )γ H (A,B,x )γ H (A, B, x ) (95) Let β(a) > α(ab) Then,ifA is noral, then g I (A) = and therefore a j g j I γ H (a, A, B) := (B) (β (A) α(ab)) j+ (96) / (j!) j= If both A and B are noral, then γ H (a, A, B) = β (A) α(ab) (97) According to Corollary 7 we get the following result Corollary 8 Let A satisfies conditions (77) and α(a) < Then the solution Y L tothelyapunovequation(5)inh satisfies the inequality References [] Y L Daleckii and M G Krein, Stability of Solutions of Differential Equations in Banach Space, Aerican Matheatical Society, Providence, RI, USA, 97 [] R Bhatia and P Rosenthal, How and why to solve the operator equation AX XB =Y, The Bulletin of the London Matheatical Society,vol9,no,pp,997 [3] R Bhatia and M Uchiyaa, The operator equation n i= An XB i =Y, Expositiones Matheaticae, vol7,no3, pp 5 55, 9 [4] M Dehghan and M Hajarian, The reflexive and anti-reflexive solutions of a linear atrix equation and systes of atrix equations, Rocky Mountain Matheatics,vol4,no 3,pp85 848, [5] M Konstantinov, D-W Gu, V Mehrann, and P Petkov, Perturbation Theory for Matrix Equations, vol9ofstudies in Coputational Matheatics, North-Holland,Asterda,The Netherlands, 3 [6] AGMazko, Matrixequations,spectralproblesandstability of dynaic systes, in Stability, Oscillations and Optiization of Systes, Cabridge Scientific Publishers, 8 [7] D S Djordjević, Explicit solution of the operator equation A * X+X * A = B, Coputational and Applied Matheatics,vol,no,pp7 74,7 [8] BPDuggal, OperatorequationsABA = A and BAB = B, Functional Analysis, Approxiation and Coputation,vol3,no, pp 9 8, [9] Q-W Wang and C-Z Dong, The general solution to a syste of adjointable operator equations over Hilbert C odules, Operators and Matrices,vol5,no,pp333 35, [] B Zhou, G-R Duan, and Z-Y Li, Gradient based iterative algorith for solving coupled atrix equations, Systes & Control Letters,vol58,no5,pp37 333,9 [] F Ding and T Chen, Gradient based iterative algoriths forsolvingaclassofatrixequations, IEEE Transactions on Autoatic Control,vol5,no8,pp6,5 [] L Xie, J Ding, and F Ding, Gradient based iterative solutions for general linear atrix equations, Coputers & Matheatics with Applications, vol 58, no 7, pp , 9 [3] F P A Beik, D K Salkuyeh, and M M Moghada, Gradientbased iterative algorith for solving the generalized coupled Sylvester-transpose and conjugate atrix equations over reflexive (anti-reflexive) atrices, Transactions of the Institute of Measureent and Control,vol36,no,pp99,4 [4] M I Gil, Nor estiates for solutions of atrix equations AX XB = C and X AXB=C, Discussiones Matheaticae: Differential Inclusions, Control & Optiization,vol34,no,pp 9 6, 4 [5] M I Gil, Operator Functions and Localization of Spectra, vol 83 of Lecture Notes in Matheatics, Springer, Berlin, Gerany, 3 Y l C j, Conflict of Interests (k + j)!g k+j I (A) (98) ( α (A) ) k+j+ 3/ The author declares that there is no conflict of interests regarding the publication of this paper

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