Research Article Numerical Range of Two Operators in Semi-Inner Product Spaces

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1 Abstract and Appied Anaysis Voume 01, Artice ID , 13 pages doi: /01/ Research Artice Numerica Range of Two Operators in Semi-Inner Product Spaces N. K. Sahu, 1 C. Nahak, 1 and S. Nanda 1 Department of Mathematics, Indian Institute of Technoogy, Kharagpur 7130, India KIIT University, Bhubaneswar 75104, India Correspondence shoud be addressed to C. Nahak, cnahak@maths.iitkgp.ernet.in Received 9 January 01; Revised 18 January 01; Accepted 1 February 01 Academic Editor: Michie Bertsch Copyright q 01 N. K. Sahu et a. This is an open access artice distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the origina work is propery cited. In this paper, the numerica range for two operators both inear and noninear have been studied in semi-inner product spaces. The incusion reations between numerica range, approximate point spectrum, compression spectrum, eigenspectrum, and spectrum have been estabished for two inear operators. We aso show the incusion reation between approximate point spectrum and cosure of the numerica range for two noninear operators. An approximation method for soving the operator equation invoving two noninear operators is aso estabished. 1. Introduction Lumer 1 introduced the concept of semi-inner product. He defined semi-inner product as foows. Semi-Inner Product Let X be a vector space over the fied F of rea or compex numbers. A functiona : X X F is caed a semi-inner product if it satisfies the foowing conditions: i x y, z x, z y, z, forax, y, z X; ii λx, y λ x, y, foraλ F and x, y X; iii x, x > 0, for x / 0; iv x, y x, x y, y, forax, y X. The pair X,, is caed a semi-inner product space. A semi-inner product space is a normed inear space with the norm x x, x 1/. Every normed inear space can be made into a semi-inner product space in infinitey many

2 Abstract and Appied Anaysis different ways. Gies had shown that if the underying space is a uniformy convex smooth Banach space, then it is possibe to define a semi-inner product uniquey. Aso the unique semi-inner product has the foowing nice properties: i x, λy λ x, y, for a scaars λ; ii x, y 0 if and ony if y is orthogona to x, that is, if and ony if y y λx,for a scaars λ; iii generaized Riesz representation theorem: If f is a continuous inear functiona on X then there is a unique vector y X such that f x x, y, forax X; iv the semi-inner product is continuous. The sequence space p,p > 1 and the function space L p,p > 1 are uniformy convex smooth Banach spaces. So one can define semi-inner product on these spaces uniquey. Gies had shown that the functions space L p,p >1 are semi-inner product spaces with the semiinner product defined by [ ] 1 f, g : f x g p g x p 1 ( ) sgn g x dμ, f, g L P ( X, μ ). 1.1 X p To study the generaized eigenvaue probem Tx λax, Amein 3 introduced the concept of numerica range for two inear operators in a Hibert space. His purpose was to obtain some new resuts on the stabiity of index of a Fredhom operator perturbed by a bounded operator. Zarantoneo 4 had introduced the concept of numerica range for a noninear operator in a Hibert space. He proved that the numerica range contains the spectrum. He used this concept to sove the noninear functiona equations. A great dea of iterature on numerica range in unita normed agebras, numerica radius theorems, spatia numerica ranges, agebra numerica ranges, essentia numerica ranges, joint numerica ranges, and matrix ranges are avaiabe in Bonsa 5, 6. For recent work on numerica range, one may refer Chien and Nakazato 7, 8, Chien at a. 9, Gustafson and Rao 10, and Li and Tam 11. In 1, Lumer discussed the numerica range for a inear operator in a Banach space. Wiiams 1 studied the spectra of products of two inear operators and their numerica ranges. The numerica range of two noninear operators in a semi-inner product space was defined by Nanda 13. For two noninear operators T and A, he defined the numerica range W n T, A,as {[ ] } Tx Ty,Ax Ay W n T, A : : x Ax Ay / y, x, y D T D A, 1. where D T and D A denote the domains of the operators T and A, respectivey. The numerica radius w n T, A is defined as w n T, A sup{ λ : λ W n T, A }. W n T, A may not be convex. If T and A are continuous and D T,D A are connected, then W n T, A is connected. The numerica range of a noninear operator using the generaized Lipschitz norm

3 Abstract and Appied Anaysis 3 was studied by Verma 14. He defined the numerica range V L T of a noninear operator T, as { [ ] } Tx,x Tx Ty,x y V L T : x : x, y D T,x/ y. 1.3 x y He used this concept to sove the operator equation Tx λx y, where T is a noninear operator. This paper is concerned with the numerica range in a Banach space. Nanda 15 studied the numerica range for two inear operators and the couped numerica range in a Hibert space which was initiay introduced by Amein 3. He aso introduced the concepts of spectrum, point spectrum, approximated point spectrum, and compression spectrum for two inear operators. In Section, we generaize the resuts of Nanda 15 to semi-inner product space. Verma 14 introduced the numerica range of a noninear operator in a Banach space using the generaized Lipschitz norm. In Section 3, we generaize the numerica range of Verma 14 for two noninear operators using the generaized Lipschitz norm. We aso give exampes of operators in semi-inner product spaces and compute their numerica range and numerica radius.. Numerica Range of Two Linear Operators Let T and A be two inear operators on a uniformy convex smooth Banach space X.Tostudy the properties of the numerica range, couped numerica range for the two operators T and A, and to discuss the resuts of the cassica spectra theory associated with the numerica range, we need the foowing definitions in the seque. Numerica Range W T, A The numerica range W T, A of the two inear operators T and A is defined as W T, A : { Tx,Ax : Ax 1,x D T D A }, where D T and D A are denoted as the domain of T and the domain of A, respectivey. The numerica radius w T, A is defined as w T, A sup{ λ : λ W T, A }. Spectrum σ T, A The spectrum σ T, A of the two inear operators T and A is defined as σ T, A : {λ C : T λa is not invertibe}..1 The spectra radius r T, A is defined as r T, A sup{ λ : λ σ T, A }. Eigenspectrum e T, A The eigenspectrum or point spectrum e T, A of two inear operators T and A is defined as e T, A : {λ C : Tx λax, x / 0}..

4 4 Abstract and Appied Anaysis Approximate Point Spectrum π T, A The approximate point spectrum π T, A of two inear operators T and A is defined as π T, A : {λ C such that there exists a sequence x n in X with Ax n 1and Tx n λax n 0asn }. Compression Spectrum σ 0 T, A The compression spectrum σ 0 T, A of two inear operators T and A is defined as σ 0 T, A : { λ C : Range T λa is not dense in X }..3 Couped Numerica Range W A T The couped numerica range W A T of T with respect to A is defined as W A T : { } ATx, x Ax, x : x 1, Ax, x / 0..4 We can easiy prove the foowing properties of the numerica range of two inear operators. Theorem.1. Let T 1,T,T, and A be inear operators and α, μ, and λ be scaars. Then i W T 1 T,A W T 1,A W T,A ; ii W αt, A αw T, A ; iii W T, μa μw T, A ; iv W T λa, A W T, A {λ}; v w T 1 T,A w T 1,A w T,A ; vi w λt, A λ w T, A. Theorem.. For the couped numerica range we have the foowing properties: i W A T 1 T W A T 1 W A T ; ii W A αt αw A T ; iii W αa T W A T. We estabish the foowing theorems which generaize the cassica spectra theory resuts. Theorem.3. The approximate point spectrum π T, A is contained in the cosure of the numerica range W T, A. Proof. Let λ π T, A. Then there exists a sequence x n in X such that Ax n,ax n 1and T λa x n 0asn. Now Tx n,ax n λ T λa x n,ax n T λa x n Ax n 0 as n..5 This impies that Tx n,ax n λ as n. Hence, λ W T, A, and consequenty π T, A W T, A.

5 Abstract and Appied Anaysis 5 Theorem.4. Eigenspectrum e T, A is contained in the spectrum σ T, A. Proof. Let λ e T, A. Then there exists x 0 / 0 such that T λa x 0 0. Thus T λa 1 does not exist otherwise T λa 1 T λa x 0 T λa That is Ix 0 x 0 0, which is a contradiction to the fact that x 0 / 0. Hence λ σ T, A and consequenty, e T, A σ T, A. In the foowing theorems, we assume that the inear operator A is invertibe. Theorem.5. Compression spectrum σ 0 T, A is contained in the numerica range W T, A. Proof. Let λ σ 0 T, A, then range T λa is not dense in X. So we can find a y in X with Ay 1 such that Ay is orthogona to the range of T λa. This impies 0 T λa y, Ay Ty,Ay λ. Soλ Ty,Ay W T, A, and consequenty σ 0 T, A W T, A. The generaized Riesz representation theorem asserts that one can define semi-inner product using bounded inear functionas. In the foowing theorem, we denote that x, φ y x, y for a x, y X and φ X. Theorem.6. Spectrum σ T, A is contained in the cosure of the numerica range W T, A. Proof. Let λ σ T, A. To show that λ W T, A. Suppose that λ/ W T, A, then d λ, W T, A δ>0. For Ax 1, we have T λa x T λa x, Ax Tx,Ax λ δ>0..6 Hence T λa is one-to-one with a cosed range. Again for φ X, X being the dua space of X, we have A 1 T λa φ Ax ( x, T λa φ Ax ) T λa x, Ax δ..7 Hence T λa is bounded beow on the range of φ and since this is dense in X, T λa is bounded beow, and it is one-to-one. This impies that T λa has a dense range. By open mapping theorem T λa has a bounded inverse, which is a contradiction to the fact that λ σ T, A. Therefore, λ W T, A, and consequenty σ T, A W T, A. Remark.7. Theorem.6 is a generaization of a known resut for Hibert space operators to Banach space operators. Here, T and A are bounded inear operators on a Banach space X. If A is invertibe, then the spectrum σ T, A coincides with the cassica spectrum σ TA 1 of TA 1. The numerica range W T, A coincides with the cassica numerica range W TA 1 of TA 1. So the assertion of Theorem.6 can aso be deduced from a cassica resut on Banach space.

6 6 Abstract and Appied Anaysis Theorem.8. Let T and A be two inear operators on a semi-inner product space X, so that w T, A < 1. IfA is invertibe, then A T is invertibe, and A A T 1 1/ 1 w T, A. Proof. We have r T, A w T, A < 1. For Ax 1, we have ( ) A T x A I A 1 T x ( ) A I A 1 T x Ax [ ( ) A I A 1 T x, Ax].8 Ax, Ax Tx,Ax 1 w T, A > 0. This impies that A T is invertibe in its range. Again A T x 1 w T, A Ax. Setting x A 1 I A 1 T 1 y with y 1, we get 1 1 w T, A A A T 1 y A A T 1 y 1 w T, A 1 y A A T w T, A Numerica Range of Two Noninear Operators Let Lip X denote the set of a Lipschitz operators on X. Suppose that T Lip X, and x, y Dom T with x / y. The generaized Lipschitz norm T L of a noninear operator T on a Banach space X is defined as T L T T, where T sup x Tx / x and T sup x / y Tx Ty / x y. If there exists a finite constant M such that T L <M, then the operator T is caed the generaized Lipschitz operator Verma 14. LetG L X be the cass of a generaized Lipschitz operators. Now we define the concepts of resovent set, spectrum, eigenspectrum, and point spectrum for a noninear operator with respect to another noninear operator, which generaize the concepts of the cassica spectra theory. A-Resovent Set A-resovent set ρ A T of a noninear operator T with respect to another operator A is defined as ρ A T : { } λ C : T λa 1 exists and is generaized Lipschitzian. 3.1

7 Abstract and Appied Anaysis 7 A-Spectrum of T A-spectrum of T, σ A T is the compement of the A-resovent set of T. Numerica Range of Two Noninear Operators The numerica range V L T, A of two noninear operators T and A is defined as { [ ] } Tx,Ax Tx Ty,Ax Ay V L T, A : Ax : x, y D T D A, x/ y, 3. Ax Ay where D T and D A are the domains of the operators T and A, respectivey. The numerica radius w L T, A is defined as w L T, A {sup λ : λ V L T, A }. We give exampes of two noninear operators in a semi-inner product space and compute their numerica range and numerica radius. Exampe 3.1. Consider the rea sequence space p,p > 1. Let x x 1,x,...,y y 1,y,... p. Consider the two noninear operators T, A : p p defined by Tx x,x 1,x,... and Ax x, 0, 0,.... The semi-inner product on the rea sequence space p is defined as x, y 1/ y p n 1 y n p y n x n, x {x n },y {y n } p. One can easiy compute that Ax x, Ax Ay x y, Tx,Ax x and Tx Ty,Ax Ay 1/ Ax Ay p { x y p x y } 1/ x y p x y p x y. We can cacuate Tx,Ax [ Tx Ty,Ax Ay ] Ax Ax Ay x ( x y ) x ( x y ) 1, x, y p. 3.3 Therefore, V L T, A {1}, andw L T, A 1. Exampe 3.. Consider the rea sequence space p,p > 1. Let x x 1,x,... and y y 1,y,... p. Consider the two noninear operators T, A : p p defined by Tx x,x 1,x,... and Ax α, x 1,x,..., where α R is any constant. One can easiy compute Ax Ay p n 1 x n y n p and [ ] 1 Tx Ty,Ax Ay Ax Ay p xn y n p n 1 Ax Ay p Ax Ay p Ax Ay. 3.4 For any x, y p, we have [ Tx Ty,Ax Ay ] Ax Ay Ax Ay Ax Ay Therefore, the numerica range of two noninear operators T and A inthesenseofnanda 13 is W n T, A {1} and the numerica radius w n T, A 1.

8 8 Abstract and Appied Anaysis We have the foowing eementary properties for the numerica range of two noninear operators. Theorem 3.3. Let X be a Banach space over C. IfT, A, T 1, and T are noninear operators defined on X, and λ and μ are scaars, then i V L λt, A λv L T, A ; ii V L T, μa 1/μ V L T, A ; iii V L T 1 T,A V L T 1,A V L T,A ; iv V L T λa, A V L T, A {λ}. Proof. To prove i : λtx, Ax [ λtx λty, Ax Ay ] Ax Ax Ay λ Tx,Ax [ Tx Ty,Ax Ay ] Ax Ax Ay. 3.6 Hence, V L λt, A λv L T, A. To show ii : [ Tx,μAx ] [ Tx Ty,μAx μay ] μax μax μay μ Tx,Ax μ[ Tx Ty,Ax Ay ] μ ( Ax Ax Ay ) μ [ ] μ Tx,Ax Tx Ty,Ax Ay Ax Ax Ay 1 Tx,Ax [ Tx Ty,Ax Ay ] μ Ax. Ax Ay 3.7 Hence V L T, μa 1/μ V L T, A. Let x, y D T 1 D T. Then, T 1 T x, Ax [ T 1 T x T 1 T y, Ax Ay ] Ax Ax Ay T 1x, Ax T x, Ax [ T 1 x T 1 y, Ax Ay ] [ T x T y, Ax Ay ] T 1x, Ax [ T 1 x T 1 y, Ax Ay ] Ax Ax Ay Ax Ax Ay T x, Ax [ T x T y, Ax Ay ] Ax. Ax Ay 3.8 Therefore, V L T 1 T,A V L T 1,A V L T,A.Thus, iii is proved.

9 Abstract and Appied Anaysis 9 Finay, to prove iv : T λa x, Ax [ T λa x T λa y, Ax Ay ] Ax Ax Ay Tx,Ax λ Ax [ Tx Ty,Ax Ay ] λ Ax Ay Ax Ax Ay Tx,Ax [ Tx Ty,Ax Ay ] Ax Ax Ay λ. 3.9 This impies that V L T λa, A V L T, A {λ}. Approximate Point Spectrum of Two Noninear Operators Approximate point spectrum π T, A of two noninear operators T and A is defined as π T, A : {λ C: there exist sequences {x n } and {y n } such that Ax n 1, Ax n Ay n 1, T λa x n 0and T λa x n T λa y n 0asn }. Strongy Generaized A-Monotone Operator A noninear operator T : D T X X is caed strongy generaized A-monotone operator if there is a constant C>0, such that Tx,Ax Tx Ty,Ax Ay C Ax Ax Ay. Theorem 3.4. Approximate point spectrum π T, A of two noninear operators T and A is contained in the cosure of their numerica range V L T, A. Proof. Let λ π T, A.Now Tx n,ax n [ ] Tx n Ty n,ax n Ay n Ax n λ Axn Ay n T λa xn,ax n [ ] T λa x n T λa y n,ax n Ay n Ax n Ax n Ay n T λa x n Ax n T λa xn T λa y n Axn Ay n Ax n. Axn Ay n 3.10 The right-hand side goes to 0 as n. This impies that λ V L T, A, and hence π T, A V L T, A. Theorem 3.5. Let μ be a compex number. Then μ is at a distance d>0 from V L T, A if and ony if T μa is strongy generaized A-monotone.

10 10 Abstract and Appied Anaysis Proof. We have Tx,Ax [ Tx Ty,Ax Ay ] 0 <d Ax μ Ax Ay [( T μa ) x, Ax ] [( T μa ) x ( T μa ) y, Ax Ay ] Ax Ax Ay This impies that [( T μa ) x, Ax ] [( T μa ) x ( T μa ) y, Ax Ay ] d ( Ax ) Ax Ay. 3.1 Hence, T μa is a strongy generaized A-monotone operator. The converse part foows easiy and hence omitted. The foowing theorem is an approximation method for soving an operator equation invoving two noninear operators. Theorem 3.6. Let X be a compex Banach space, T G L X, and T L < 1. AsoetA be another generaized Lipschitzian and invertibe operator on X. IfT is A-Lipschitz with constant K / 1, then A T is invertibe in G L X and A T 1 A 1 L A 1 T L / 1 A 1 T 1 A 1 T. Again if B 0 I and B n I A 1 T B n 1 for n 1,, 3,..., and A 1 T < 1, thenim n B n x I A 1 T 1 x for every x X, asn and I A 1 T 1 x B n x A 1 T n A 1 Tx 1 A 1 T 1,forx X, n 0, 1,,... Proof. For each x, y X with Ax / Ay, we have A T x A T y Ax Ay Tx Ty Ax Ay KAx Ay 1 K Ax Ay > 0, 3.13 since K / 1, and Ax / Ay. This impies that A T is injective. Next if u, v R A T, then A T 1 A 1( 1 I A T) 1 A 1 1 (I A T) 1 A 1 (1 A 1 ) T Simiary, we have A T 1 A 1 1 A 1 T 1. Now A T 1 A T 1 A 1 1 A 1 T A 1 1 A 1 T A 1 ( 1 A 1 T ) A 1 ( 1 A 1 T ) ( 1 A 1 T )( 1 A 1 T )

11 Abstract and Appied Anaysis 11 A 1 ( L 1 A 1 T ) A 1 ( L 1 A 1 T ) ( 1 A 1 T )( 1 A 1 T ) A 1 L A 1 L A 1 T A 1 L A 1 T ( 1 A 1 T )( 1 A 1 T ) A 1 L A 1 A 1 L T ( L 1 A 1 T )( 1 A 1 T ), A T 1 ( A 1 L L A 1 T ) ( L 1 A 1 T )( 1 A 1 T ) To prove the second part, consider the sequence of approximating operators B 0 I, ( ) B n I A 1 T B n 1, for n 1,, 3, We caim that B n 1 x B n x A 1 T n A 1 Tx For n 0, B 1 x B 0 x I A 1 T Ix Ix A 1 Tx. For n 1, B x B 1 x I A 1 T B 1 x I A 1 T B 0 x A 1 T 1 A 1 Tx. Assume that 3.17 is true for n k 1, that is B k x B k 1 x A 1 T k 1 A 1 Tx. Now for n k, ( ( ) ( ( ) B k 1 x B k x I A 1 T )B k x I A 1 T )B k 1 x ( ) ( ) A 1 T B k x A 1 T B k 1 x 3.18 A 1 T B k x B k 1 x A 1 T A 1 Tx. k Now for a positive integer p, Bn p x B n x p 1 B n k 1 x B n k x k 0 p 1 B n k 1 x B n k x k 0 p 1 A 1 T k 0 n k A 1 Tx

12 1 Abstract and Appied Anaysis { } A 1 T A 1 Tx 1 A 1 T A 1 T A 1 T n A 1 T n A 1 Tx ( ) 1 A 1 1. T p Since A 1 T < 1, the sequence {B n x} is a Cauchy sequence. Again since X is compete, we have im m B m x Ex exists for a x X. For m n p, Ex B n x im Bn p x B n x A 1 T p n A 1 Tx ( ) 1 A 1 1. T 3.0 As A 1 T is continuous, we have Ex im B n x im n ( Ex A 1 T E n ( ( I ) Ex Ix A 1 T ) ( ( ) ) )B n 1 x I A 1 T E x 3.1 ( ) 1. I A 1 T Now mutipying A 1 we get A T 1 A 1 E. Using this technique, one can sove an operator equation invoving two noninear operators under the condition that one of the operator is invertibe. Acknowedgments The authors are thankfu to the referees and editors for their vauabe suggestions which improved the presentation of the paper. The authors are aso thankfu to Professor R. N. Mohapatra, University of Centra Forida, USA, for his vauabe discussions and suggestions. The first author is very much thankfu to the Counci of Scientific and Industria Research CSIR, India, for its financia support in executing this study. References 1 G. Lumer, Semi-inner-product spaces, Transactions of the American Mathematica Society, vo. 100, pp. 9 43, J. R. Gies, Casses of semi-inner-product spaces, Transactions of the American Mathematica Society, vo. 19, pp , C. F. Amein, A numerica range for two inear operators, Pacific Journa of Mathematics, vo. 48, pp , E. H. Zarantoneo, The cosure of the numerica range contains the spectrum, Pacific Journa of Mathematics, vo., pp , F. F. Bonsa and J. Duncan, Numerica Ranges, vo. 1, Cambridge University Press, London, UK, F. F. Bonsa and J. Duncan, Numerica Ranges, vo., Cambridge University Press, London, UK, 1973.

13 Abstract and Appied Anaysis 13 7 M.-T. Chien and H. Nakazato, The numerica range of a tridiagona operator, Journa of Mathematica Anaysis and Appications, vo. 373, no. 1, pp , M.-T. Chien and H. Nakazato, The numerica radius of a weighted shift operator with geometric weights, Eectronic Journa of Linear Agebra, vo. 18, pp , M.-T. Chien, L. Yeh, Y.-T. Yeh, and F.-Z. Lin, On geometric properties of the numerica range, Linear Agebra and its Appications, vo. 74, pp , K.E.GustafsonandD.K.M.Rao,Numerica Range: The Fied of Vaues of Linear Operators and Matrices, Springer-Verag, New York, NY, USA, C. K. Li and T. Y. Tam, Eds., Specia issue on the numerica range and numerica radius, Linear and Mutiinear Agebra, vo. 5, no. 3-4, p. 157, J. P. Wiiams, Spectra of products and numerica ranges, Journa of Mathematica Anaysis and Appications, vo. 17, pp. 14 0, S. Nanda, Numerica range for two non-inear operators in semi-inner-product space, Journa of Nationa Academy of Mathematics, vo. 17, pp. 16 0, R. U. Verma, The numerica range of noninear Banach space operators, Acta Mathematica Hungarica, vo. 63, no. 4, pp , S. Nanda, A note on numerica ranges of operators, Proceedings of the Nationa Academy of Sciences, vo. 66, no. 1, pp. 59 6, 1996.

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