Numerical Range of Two Operators in Semi-Inner Product Spaces

and Applied Analysis 3 was studied by Verma 14 . He defined the numerical range VL T of a nonlinear operator T , as VL T : { Tx, x [ Tx − Ty, x − y ‖x‖ ∥x − y∥2 : x, y ∈ D T , x / y } . 1.3 He used this concept to solve the operator equation Tx−λx y, where T is a nonlinear operator. This paper is concerned with the numerical range in a Banach space. Nanda 15 studied the numerical range for two linear operators and the coupled numerical range in a Hilbert space which was initially introduced by Amelin 3 . He also introduced the concepts of spectrum, point spectrum, approximated point spectrum, and compression spectrum for two linear operators. In Section 2, we generalize the results of Nanda 15 to semi-inner product space. Verma 14 introduced the numerical range of a nonlinear operator in a Banach space using the generalized Lipschitz norm. In Section 3, we generalize the numerical range of Verma 14 for two nonlinear operators using the generalized Lipschitz norm. We also give examples of operators in semi-inner product spaces and compute their numerical range and numerical radius. 2. Numerical Range of Two Linear Operators Let T andA be two linear operators on a uniformly convex smooth Banach spaceX. To study the properties of the numerical range, coupled numerical range for the two operators T andA, and to discuss the results of the classical spectral theory associated with the numerical range, we need the following definitions in the sequel. Numerical RangeW T,A The numerical range W T,A of the two linear 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, respectively. The numerical radiusw T,A is defined asw T,A sup{|λ| : λ ∈ W T,A }. Spectrum σ T,A The spectrum σ T,A of the two linear operators T and A is defined as σ T,A : {λ ∈ C : T − λA is not invertible}. 2.1 The spectral 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 linear operators T and A is defined as e T,A : {λ ∈ C : Tx λAx, x / 0}. 2.2 4 Abstract and Applied Analysis Approximate Point Spectrum π T,A The approximate point spectrum π T,A of two linear operators T and A is defined as π T,A : {λ ∈ C such that there exists a sequence xn in X with ‖Axn‖ 1 and ‖Txn−λAxn‖ → 0 as n → ∞}. Compression Spectrum σ0 T,A The compression spectrum σ0 T,A of two linear operators T and A is defined as σ0 T,A : { λ ∈ C : Range T − λA is not dense in X. 2.3 Coupled Numerical RangeWA T The coupled numerical range WA T of T with respect to A is defined as WA T : { ATx, x Ax, x : ‖x‖ 1, Ax, x / 0 } . 2.4 We can easily prove the following properties of the numerical range of two linear operators. Theorem 2.1. Let T1, T2, T, and A be linear operators and α, μ, and λ be scalars. Then i W T1 T2, A ⊆ W T1, A W T2, 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 T1 T2, A ≤ w T1, A w T2, A ; vi w λT,A |λ|w T,A . Theorem 2.2. For the coupled numerical range we have the following properties: i WA T1 T2 ⊆ WA T1 WA T2 ; ii WA αT αWA T ; iii WαA T WA T . We establish the following theorems which generalize the classical spectral theory results. Theorem 2.3. The approximate point spectrum π T,A is contained in the closure of the numerical range W T,A . Proof. Let λ ∈ π T,A . Then there exists a sequence xn in X such that Axn,Axn 1 and ‖ T − λA xn‖ → 0 as n → ∞. Now | Txn,Axn − λ| | T − λA xn,Axn | ≤ ‖ T − λA xn‖‖Axn‖ −→ 0 as n −→ ∞. 2.5 This implies that Txn,Axn → λ as n → ∞. Hence, λ ∈ W T,A , and consequently π T,A ⊂ W T,A . Abstract and Applied Analysis 5 Theorem 2.4. Eigenspectrum e T,A is contained in the spectrum σ T,A . Proof. Let λ ∈ e T,A . Then there exists x0 / 0 such that T − λA x0 0. Thus T − λA −1 does not exist otherwise T − λA −1 T − λA x0 T − λA −10 0. That is Ix0 x0 0, which is a contradiction to the fact that x0 / 0. Hence λ ∈ σ T,A and consequently, e T,A ⊂ σ T,A . In the following theorems, we assume that the linear operator A is invertible. Theorem 2.5. Compression spectrum σ0 T,A is contained in the numerical range W T,A . Proof. Let λ ∈ σ0 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 orthogonal to the range of T − λA . This implies 0 T − λA y,Ay Ty,Ay − λ. So λ Ty,Ay ∈ W T,A , and consequently σ0 T,A ⊂ W T,A . The generalized Riesz representation theorem asserts that one can define semi-inner product using bounded linear functionals. In the following theorem, we denote that x, φ y x, y for all x, y ∈ X and φ ∈ X∗. Theorem 2.6. Spectrum σ T,A is contained in the closure of the numerical 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 haveand Applied Analysis 5 Theorem 2.4. Eigenspectrum e T,A is contained in the spectrum σ T,A . Proof. Let λ ∈ e T,A . Then there exists x0 / 0 such that T − λA x0 0. Thus T − λA −1 does not exist otherwise T − λA −1 T − λA x0 T − λA −10 0. That is Ix0 x0 0, which is a contradiction to the fact that x0 / 0. Hence λ ∈ σ T,A and consequently, e T,A ⊂ σ T,A . In the following theorems, we assume that the linear operator A is invertible. Theorem 2.5. Compression spectrum σ0 T,A is contained in the numerical range W T,A . Proof. Let λ ∈ σ0 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 orthogonal to the range of T − λA . This implies 0 T − λA y,Ay Ty,Ay − λ. So λ Ty,Ay ∈ W T,A , and consequently σ0 T,A ⊂ W T,A . The generalized Riesz representation theorem asserts that one can define semi-inner product using bounded linear functionals. In the following theorem, we denote that x, φ y x, y for all x, y ∈ X and φ ∈ X∗. Theorem 2.6. Spectrum σ T,A is contained in the closure of the numerical 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. 2.6 Hence T − λA is one-to-one with a closed range. Again for φ ∈ X∗, X∗ being the dual space of X, we have ∥∥A−1 ∥∥ ∥∥ T − λA ∗φ Ax ∥∥ ≥ ∣x, T − λA ∗φ Ax )∣∣ | T − λA x,Ax | ≥ δ. 2.7 Hence T − λA ∗ is bounded below on the range of φ and since this is dense in X∗, T −λA ∗ is bounded below, and it is one-to-one. This implies that T −λA has a dense range. By openmapping 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 consequently σ T,A ⊂ W T,A . Remark 2.7. Theorem 2.6 is a generalization of a known result for Hilbert space operators to Banach space operators. Here, T and A are bounded linear operators on a Banach space X. If A is invertible, then the spectrum σ T,A coincides with the classical spectrum σ TA−1 of TA−1. The numerical rangeW T,A coincides with the classical numerical rangeW TA−1 of TA−1. So the assertion of Theorem 2.6 can also be deduced from a classical result on Banach space. 6 Abstract and Applied Analysis Theorem 2.8. Let T andA be two linear operators on a semi-inner product spaceX, so thatw T,A < 1. If A is invertible, then A − T is invertible, 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−1T ) x ∥∥ ∥∥A ( I −A−1T ) x ∥∥‖Ax‖ ≥ ∣∣ [ A ( I −A−1T ) x,Ax ∣∣ ≥ Ax,Ax − | Tx,Ax | ≥ 1 −w T,A > 0. 2.8 This implies that A − T is invertible in its range. Again ‖ A − T x‖ ≥ 1 −w T,A ‖Ax‖. Setting x A−1 I −A−1T −1y with ‖y‖ 1, we get 1 ≥ 1 −w T,A ∥∥A A − T −1y ∥∥ ⇒ ∥∥A A − T −1y ∥∥ ≤ 1 −w T,A −1 ∥y ∥∥ ⇒ ∥∥A A − T −1 ∥∥ ≤ 1 1 −w T,A . 2.9 3. Numerical Range of Two Nonlinear Operators Let Lip X denote the set of all Lipschitz operators on X. Suppose that T ∈ Lip X , and x, y ∈ Dom T with x / y. The generalized Lipschitz norm ‖T‖L of a nonlinear operator T on a Banach space X is defined as ‖T‖L ‖T‖ ‖T‖l, where ‖T‖ supx‖Tx‖/‖x‖ and ‖T‖l supx / y‖Tx − Ty‖/‖x − y‖. If there exists a finite constant M such that ‖T‖L < M, then the operator T is called the generalized Lipschitz operator Verma 14 . Let GL X be the class of all generalized Lipschitz operators. Nowwe define the concepts of resolvent set, spectrum, eigenspectrum, and point spectrum for a nonlinear operator with respect to another nonlinear operator, which generalize the concepts of the classical spectral theory. A-Resolvent Set A-resolvent set ρA T of a nonlinear operator T with respect to another operator A is defined as ρA T : { λ ∈ C : T − λA −1 exists and is generalized Lipschitzian } . 3.1 Abstract and Applied Analysis 7 A-Spectrum of T A-spectrum of T , σA T is the complement of the A-resolvent set of T .and Applied Analysis 7 A-Spectrum of T A-spectrum of T , σA T is the complement of the A-resolvent set of T . Numerical Range of Two Nonlinear Operators The numerical range VL T,A of two nonlinear operators T and A is defined as VL T,A : { Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 : x, y ∈ D T ∩D A , x / y } , 3.2 whereD T andD A are the domains of the operators T andA, respectively. The numerical radius wL T,A is defined as wL T,A {sup |λ| : λ ∈ VL T,A }. We give examples of two nonlinear operators in a semi-inner product space and compute their numerical range and numerical radius. Example 3.1. Consider the real sequence space l, p > 1. Let x x1, x2, . . . , y y1, y2, . . . ∈ l. Consider the two nonlinear operators T,A : l → l defined by Tx ‖x‖, x1, x2, . . . and Ax ‖x‖, 0, 0, . . . . The semi-inner product on the real sequence space l is defined as x, y 1/‖y‖p−2 ∞n 1 |yn|p−2ynxn, ∀x {xn}, y {yn} ∈ l. One can easily compute that ‖Ax‖ ‖x‖, ‖Ax − Ay‖ |‖x‖ − ‖y‖|, Tx,Ax ‖x‖2 and Tx − Ty,Ax − Ay 1/‖Ax − Ay‖p−2 {| ‖x‖ − ‖y‖ |p−2 ‖x‖ − ‖y‖ } 1/| ‖x‖ − ‖y‖ |p−2| ‖x‖ − ‖y‖ |p | ‖x‖ − ‖y‖ |2. We can calculate Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 ‖x‖ ∣‖x‖ − ∥y∥∣2 ‖x‖ ∣‖x‖ − ∥y∥∣2 1, ∀x, y ∈ l. 3.3 Therefore, VL T,A {1}, and wL T,A 1. Example 3.2. Consider the real sequence space l, p > 1. Let x x1, x2, . . . and y y1, y2, . . . ∈ l. Consider the two nonlinear operators T,A : l → l defined by Tx ‖x‖, x1, x2, . . . and Ax α, x1, x2, . . . , where α ∈ R is any constant. One can easily compute ‖Ax −Ay‖p ∞n 1 |xn − yn| and [ Tx − Ty,Ax −Ay 1 ∥Ax −Ay∥p−2 ∞ ∑ n 1 ∣xn − yn ∣∣p ∥Ax −Ay∥p ∥Ax −Ay∥p−2 ∥Ax −Ay∥2. 3.4 For any x, y ∈ l, we have [ Tx − Ty,Ax −Ay ∥Ax −Ay∥2 ∥Ax −Ay∥2 ∥Ax −Ay∥2 1. 3.5 Therefore, the numerical range of two nonlinear operators T and A in the sense of Nanda 13 isWnl T,A {1} and the numerical radius wnl T,A 1. 8 Abstract and Applied Analysis We have the following elementary properties for the numerical range of two nonlinear operators. Theorem 3.3. Let X be a Banach space over C. If T,A, T1, and T2 are nonlinear operators defined on X, and λ and μ are scalars, then i VL λT,A λVL T,A ; ii VL T, μA 1/μ VL T,A ; iii VL T1 T2, A ⊆ VL T1, A VL T2, A ; iv VL T − λA,A VL T,A − {λ}. Proof. To prove i : λTx,Ax [ λTx − λTy,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 λ Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 . 3.6 Hence, VL λT,A λVL T,A . To show ii : [ Tx, μAx ] [ Tx − Ty, μAx − μAy ∥μAx ∥∥2 ∥μAx − μAy∥2 μ Tx,Ax μ [ Tx − Ty,Ax −Ay ∣μ ∣∣2 ( ‖Ax‖ ∥Ax −Ay∥2 ) μ ∣μ ∣∣2 Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 1 μ Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 . 3.7 Hence VL T, μA 1/μ VL T,A . Let x, y ∈ D T1 ∩D T2 . Then, T1 T2 x,Ax [ T1 T2 x − T1 T2 y,Ax −Ay ] ‖Ax‖ ∥Ax −Ay∥2 T1x,Ax T2x,Ax [ T1x − T1y,Ax −Ay ] [ T2x − T2y,Ax −Ay ] ‖Ax‖ ∥Ax −Ay∥2 T1x,Ax [ T1x − T1y,Ax −Ay ] ‖Ax‖ ∥Ax −Ay∥2 T2x,Ax [ T2x − T2y,Ax −Ay ] ‖Ax‖ ∥Ax −Ay∥2 . 3.8 Therefore, VL T1 T2, A ⊆ VL T1, A VL T2, A . Thus, iii is proved. Abstract and Applied Analysis 9 Finally, to prove iv :and Applied Analysis 9 Finally, to prove iv : T − λA x,Ax [ T − λA x − T − λA y,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 Tx,Ax − λ‖Ax‖ Tx − Ty,Ax −Ay − λ∥Ax −Ay∥2 ‖Ax‖ ∥Ax −Ay∥2 Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 − λ. 3.9 This implies that VL T − λA,A VL T,A − {λ}. Approximate Point Spectrum of Two Nonlinear Operators Approximate point spectrum π T,A of two nonlinear operators T and A is defined as π T,A : {λ ∈ C: there exist sequences {xn} and {yn} such that ‖Axn‖ 1, ‖Axn −Ayn‖ 1, ‖ T − λA xn‖ → 0 and ‖ T − λA xn − T − λA yn‖ → 0 as n → ∞}. Strongly Generalized A-Monotone Operator A nonlinear operator T : D T ⊂ X → X is called strongly generalized A-monotone operator if there is a constant C > 0, such that | Tx,Ax Tx−Ty,Ax−Ay | ≥ C ‖Ax‖2 ‖Ax−Ay‖2 . Theorem 3.4. Approximate point spectrum π T,A of two nonlinear operators T andA is contained in the closure of their numerical range VL T,A . Proof. Let λ ∈ π T,A . Now ∣∣∣∣ Txn,Axn [ Txn − Tyn,Axn −Ayn ] ‖Axn‖ ∥Axn −Ayn ∥∥2 − λ ∣∣∣∣ ∣∣ T − λA xn,Axn [ T − λA xn − T − λA yn,Axn −Ayn ]∣∣ ‖Axn‖ ∥Axn −Ayn ∥∥2 ≤ ‖ T − λA xn‖‖Axn‖ ∥∥ T − λA xn − T − λA yn ∥∥Axn −Ayn ∥∥ ‖Axn‖ ∥Axn −Ayn ∥∥2 . 3.10 The right-hand side goes to 0 as n → ∞. This implies that λ ∈ VL T,A , and hence π T,A ⊂ VL T,A . Theorem 3.5. Let μ be a complex number. Then μ is at a distance d > 0 from VL T,A if and only if T − μA is strongly generalized A-monotone. 10 Abstract and Applied Analysis Proof. We have 0 < d ≤ ∣∣∣∣ Tx,Ax [ Tx − Ty,Ax −Ay ‖Ax‖ ∥Ax −Ay∥2 − μ ∣∣∣∣ ∣T − μAx,Ax T − μAx − T − μAy,Ax −Ay∣ ‖Ax‖ ∥Ax −Ay∥2 . 3.11 This implies that ∣T − μAx,Ax T − μAx − T − μAy,Ax −Ay∣ ≥ d ( ‖Ax‖ ∥Ax −Ay∥2 ) . 3.12 Hence, T − μA is a strongly generalized A-monotone operator. The converse part follows easily and hence omitted. The following theorem is an approximation method for solving an operator equation involving two nonlinear operators. Theorem 3.6. Let X be a complex Banach space, T ∈ GL X , and ‖T‖L < 1. Also let A be another generalized Lipschitzian and invertible operator on X. If T is A-Lipschitz with constant K/ 1, then A−T is invertible inGL X and ‖ A−T −1‖ ≤ ‖A−1‖L 2−‖A−1T‖L / 1−‖A−1T‖ 1−‖A−1T‖l . Again if B0 I and Bn I A−1T Bn−1 for n 1, 2, 3, . . ., and ‖A−1T‖l < 1, then limn→∞Bnx I − A−1T −1x for every x ∈ X, as n → ∞ and ‖ I − A−1T −1x − Bnx‖ ≤ ‖AT‖l ‖A−1Tx‖ 1 − ‖A−1T‖l −1, for x ∈ X, n 0, 1, 2, . . .. Proof. For each x, y ∈ X with Ax/ Ay, we have ∥∥ A − T x − A − T y∥ ≥ ∥Ax −Ay∥ − ∥Tx − Ty∥ ≥ ∥Ax −Ay∥ −K∥Ax −Ay∥ 1 −K ∥Ax −Ay∥ > 0, 3.13 since K/ 1, and Ax/ Ay. This implies that A − T is injective. Next if u, v ∈ R A − T , then ∥∥ A − T −1 ∥∥ ∥∥∥∥A −1 ( I −A−1T ∥∥∥∥ ≤ ∥∥A−1 ∥∥ ∥∥∥∥ ( I −A−1T ∥∥∥∥ ≤ ∥∥A−1 ∥∥ ( 1 − ∥∥A−1T ∥∥ )−1 . 3.14 Similarly, we have ‖ A − T ‖l ≤ ‖A−1‖l 1 − ‖A−1T‖l −1. Now ∥∥ A − T −1 ∥∥ ∥∥ A − T −1 ∥∥ l ≤ ∥A−1 ∥∥ 1 − ∥A−1T∥ ∥A−1 ∥∥ l 1 − ∥∥A−1T∥∥l ∥A−1 ∥1 − ∥∥A−1T∥∥l ) ∥A−1 ∥1 − ∥∥A−1T∥∥l ) ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) Abstract and Applied Analysis 11 ≤ ∥A−1 ∥∥ L ( 1 − ∥∥A−1T∥∥l ) ∥A−1 ∥∥ L ( 1 − ∥∥A−1T∥∥l ) ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) 2 ∥A−1 ∥∥ L − ∥A−1 ∥∥ L ∥A−1T ∥∥ l − ∥A−1 ∥∥ L ∥A−1T ∥∥ ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) 2 ∥A−1 ∥∥ L − ∥A−1 ∥∥ L ∥A−1T ∥∥ L ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) , ⇒ ∥∥ A − T −1 ∥∥ L ≤ ∥A−1 ∥∥ L ( 2 − ∥∥A−1T∥∥L ) ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) . 3.15and Applied Analysis 11 ≤ ∥A−1 ∥∥ L ( 1 − ∥∥A−1T∥∥l ) ∥A−1 ∥∥ L ( 1 − ∥∥A−1T∥∥l ) ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) 2 ∥A−1 ∥∥ L − ∥A−1 ∥∥ L ∥A−1T ∥∥ l − ∥A−1 ∥∥ L ∥A−1T ∥∥ ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) 2 ∥A−1 ∥∥ L − ∥A−1 ∥∥ L ∥A−1T ∥∥ L ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) , ⇒ ∥∥ A − T −1 ∥∥ L ≤ ∥A−1 ∥∥ L ( 2 − ∥∥A−1T∥∥L ) ( 1 − ∥A−1T∥1 − ∥∥A−1T∥∥l ) . 3.15 To prove the second part, consider the sequence of approximating operators B0 I, Bn I ( A−1T ) Bn−1, for n 1, 2, 3, . . . . 3.16 We claim that ‖Bn 1x − Bnx‖ ≤ ∥∥A−1T ∥∥ n