# Numerical Range of Two Operators in Semi-Inner

Authors: N. K. Sahu; C. Nahak; S. Nanda
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

In this paper, the numerical range for two operators (both linear and nonlinear) have
been studied in semi-inner product spaces. The inclusion relations between numerical
range, approximate point spectrum, compression spectrum, eigenspectrum, and spectrum
have been established for two linear operators. We also show the inclusion relation between
approximate point spectrum and closure of the numerical range for two nonlinear
operators. An approximation method for solving the operator equation involving two
nonlinear operators is also established.


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## Body

## 1. Introduction

Lumer [1] introduced the concept of semi-inner product. He defined semi-inner product as follows.Semi-Inner Product
LetX be a vector space over the field F of real or complex numbers. A functional []:X×X→F is called a semi-inner product if it satisfies the following conditions:
(i)[x+y,z]=[x,z]+[y,z], for all x,y,z∈X;
(ii)[λx,y]=λ[x,y], for all λ∈F and x,y∈X;
(iii)[x,x]>0, for x≠0;
(iv)|[x,y]|2≤[x,x][y,y], for all x,y∈X.
The pair(X,[,]) is called a semi-inner product space.
A semi-inner product space is a normed linear space with the norm∥x∥=[x,x]1/2. Every normed linear space can be made into a semi-inner product space in infinitely many different ways. Giles [2] had shown that if the underlying space is a uniformly convex smooth Banach space, then it is possible to define a semi-inner product uniquely. Also the unique semi-inner product has the following nice properties:
(i)
[x,λy]=λ¯[x,y], for all scalars λ;(ii)
[x,y]=0 if and only if y is orthogonal to x, that is, if and only if ∥y∥≤∥y+λx∥, for all scalars λ;(iii)
generalized Riesz representation theorem: Iff is a continuous linear functional on X then there is a unique vector y∈X such that f(x)=[x,y], for all x∈X;(iv)
the semi-inner product is continuous.
The sequence spacelp,p>1 and the function space Lp,p>1 are uniformly convex smooth Banach spaces. So one can define semi-inner product on these spaces uniquely. Giles [2] had shown that the functions space Lp,p>1 are semi-inner product spaces with the semi-inner product defined by(1.1)[f,g]:=1‖g‖pp-2∫Xf(x)|g(x)|p-1sgn⁡(g(x))dμ,∀f,g∈LP(X,μ).
To study the generalized eigenvalue problemTx=λAx, Amelin [3] introduced the concept of numerical range for two linear operators in a Hilbert space. His purpose was to obtain some new results on the stability of index of a Fredholm operator perturbed by a bounded operator. Zarantonello [4] had introduced the concept of numerical range for a nonlinear operator in a Hilbert space. He proved that the numerical range contains the spectrum. He used this concept to solve the nonlinear functional equations. A great deal of literature on numerical range in unital normed algebras, numerical radius theorems, spatial numerical ranges, algebra numerical ranges, essential numerical ranges, joint numerical ranges, and matrix ranges are available in Bonsall [5, 6]. For recent work on numerical range, one may refer Chien and Nakazato [7, 8], Chien at al. [9], Gustafson and Rao [10], and Li and Tam [11].
In [1], Lumer discussed the numerical range for a linear operator in a Banach space. Williams [12] studied the spectra of products of two linear operators and their numerical ranges. The numerical range of two nonlinear operators in a semi-inner product space was defined by Nanda [13]. For two nonlinear operators T and A, he defined the numerical range Wnl(T,A), as
(1.2)Wnl(T,A):={[Tx-Ty,Ax-Ay]‖Ax-Ay‖2:x≠y,x,y∈D(T)∩D(A)},
where D(T) and D(A) denote the domains of the operators T and A, respectively. The numerical radius wnl(T,A) is defined as wnl(T,A)=sup{|λ|:λ∈Wnl(T,A)}. Wnl(T,A) may not be convex. If T and A are continuous and D(T),D(A) are connected, then Wnl(T,A) is connected. The numerical range of a nonlinear operator using the generalized Lipschitz norm was studied by Verma [14]. He defined the numerical range VL(T) of a nonlinear operator T, as
(1.3)VL(T):={[Tx,x]+[Tx-Ty,x-y]‖x‖2+‖x-y‖2:x,y∈D(T),x≠y}.
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

LetT and A be two linear operators on a uniformly convex smooth Banach space X. To study the properties of the numerical range, coupled numerical range for the two operators T and A, 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 rangeW(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 radius w(T,A) is defined as w(T,A)=sup{|λ|:λ∈W(T,A)}.Spectrumσ(T,A)
The spectrumσ(T,A) of the two linear operators T and A is defined as
(2.1)σ(T,A):={λ∈C:(T-λA)isnotinvertible}.
The spectral radius r(T,A) is defined as r(T,A)=sup{|λ|:λ∈σ(T,A)}.Eigenspectrume(T,A)
The eigenspectrum or point spectrume(T,A) of two linear operators T and A is defined as(2.2)e(T,A):={λ∈C:Tx=λAx,x≠0}.Approximate Point Spectrumπ(T,A)
The approximate point spectrumπ(T,A) of two linear operators T and A is defined as π(T,A):={λ∈ℂ such that there exists a sequencexn inXwith ∥Axn∥=1 and ∥Txn-λAxn∥→0asn→∞}.Compression Spectrumσ0(T,A)
The compression spectrumσ0(T,A) of two linear operators TandA is defined as
(2.3)σ0(T,A):={λ∈C:Range(T-λA)isnotdenseinX}.Coupled Numerical RangeWA(T)
The coupled numerical rangeWA(T) of T with respect to A is defined as
(2.4)WA(T):={[ATx,x][Ax,x]:‖x‖=1,[Ax,x]≠0}.
We can easily prove the following properties of the numerical range of two linear operators.Theorem 2.1.
LetT1,T2,T,andA be linear operators and α,μ,andλ 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)=μ¯W(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 spectrumπ(T,A) is contained in the closure of the numerical range W(T,A).Proof.
Letλ∈π(T,A). Then there exists a sequence xn in X such that [Axn,Axn]=1 and ∥(T-λA)xn∥→0 as n→∞.
Now(2.5)|[Txn,Axn]-λ|=|[(T-λA)xn,Axn]|≤‖(T-λA)xn‖‖Axn‖⟶0asn⟶∞.This implies that [Txn,Axn]→λ as n→∞.
Hence,λ∈W(T,A),¯ and consequently π(T,A)⊂W(T,A)¯.Theorem 2.4.
Eigenspectrume(T,A) is contained in the spectrum σ(T,A).Proof.
Letλ∈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 isIx0=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 operatorA is invertible.Theorem 2.5.
Compression spectrumσ0(T,A) is contained in the numerical range W(T,A).Proof.
Letλ∈σ0(T,A), then range(T-λA) is not dense in X.
So we can find ay in X with ∥Ay∥=1 such that Ay is orthogonal to the range of (T-λA).
This implies0=[(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.
Spectrumσ(T,A) is contained in the closure of the numerical range W(T,A).Proof.
Letλ∈σ(T,A). To show that λ∈W(T,A)¯.
Suppose thatλ∉W(T,A)¯, then d(λ,W(T,A)¯)=δ>0.
For∥Ax∥=1, we have(2.6)‖(T-λA)x‖≥|[(T-λA)x,Ax]|=|[Tx,Ax]-λ|≥δ>0.
Hence(T-λA) is one-to-one with a closed range. Again for ϕ∈X*, X* being the dual space of X, we have(2.7)‖A-1‖‖(T-λA)*ϕ(Ax)‖≥|(x,(T-λA)*ϕ(Ax))|=|[(T-λA)x,Ax]|≥δ.
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 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 consequently σ(T,A)⊂W(T,A)¯.Remark 2.7.
Theorem2.6 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 range W(T,A) coincides with the classical numerical range W(TA-1) of TA-1. So the assertion of Theorem 2.6 can also be deduced from a classical result on Banach space.Theorem 2.8.
LetT and A be two linear operators on a semi-inner product space X, so that w(T,A)<1. If A is invertible, then A-T is invertible, and ∥A(A-T)-1∥≤1/(1-w(T,A)).Proof.
We haver(T,A)≤w(T,A)<1. For ∥Ax∥=1, we have
(2.8)‖(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.
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(2.9)1≥(1-w(T,A))‖A(A-T)-1y‖⇒‖A(A-T)-1y‖≤(1-w(T,A))-1‖y‖⇒‖A(A-T)-1‖≤11-w(T,A).


## 3. Numerical Range of Two Nonlinear Operators

LetLip(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.Now we 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 setρA(T) of a nonlinear operator T with respect to another operator A is defined as(3.1)ρA(T):={λ∈C:(T-λA)-1existsandisgeneralizedLipschitzian}.A-Spectrum ofT
A-spectrum ofT, σA(T) is the complement of the A-resolvent set of T.Numerical Range of Two Nonlinear Operators
The numerical rangeVL(T,A) of two nonlinear operators T and A is defined as
(3.2)VL(T,A):={[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2:x,y∈D(T)∩D(A),x≠y},
where D(T) and D(A) are the domains of the operators T and A, 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 spacelp,p>1. Let x=(x1,x2,…),y=(y1,y2,…)∈lp. Consider the two nonlinear operators T,A:lp→lp defined by Tx=(∥x∥,x1,x2,…) and Ax=(∥x∥,0,0,…). The semi-inner product on the real sequence space lp is defined as [x,y]=(1/∥y∥p-2)∑n=1∞|yn|p-2ynxn,∀x={xn},y={yn}∈lp. 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∥)2}=1/|(∥x∥-∥y∥)|p-2|(∥x∥-∥y∥)|p=|(∥x∥-∥y∥)|2.
We can calculate(3.3)[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=‖x‖2+|(‖x‖-‖y‖)|2‖x‖2+|(‖x‖-‖y‖)|2=1,∀x,y∈lp.
Therefore, VL(T,A)={1}, and wL(T,A)=1.Example 3.2.
Consider the real sequence spacelp,p>1. Let x=(x1,x2,…)andy=(y1,y2,…)∈lp. Consider the two nonlinear operators T,A:lp→lp 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|p and(3.4)[Tx-Ty,Ax-Ay]=1‖Ax-Ay‖p-2∑n=1∞|xn-yn|p=‖Ax-Ay‖p‖Ax-Ay‖p-2=‖Ax-Ay‖2.
For any x,y∈lp, we have
(3.5)[Tx-Ty,Ax-Ay]‖Ax-Ay‖2=‖Ax-Ay‖2‖Ax-Ay‖2=1.
Therefore, the numerical range of two nonlinear operators T and A in the sense of Nanda [13] is Wnl(T,A)={1} and the numerical radius wnl(T,A)=1.We have the following elementary properties for the numerical range of two nonlinear operators.Theorem 3.3.
LetX be a Banach space over ℂ. If T,A,T1,andT2 are nonlinear operators defined on X, and λandμ 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):(3.6)[λTx,Ax]+[λTx-λTy,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=λ[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2.
Hence, VL(λT,A)=λVL(T,A).
To show (ii):(3.7)[Tx,μAx]+[Tx-Ty,μAx-μAy]‖μAx‖2+‖μAx-μAy‖2=μ¯[Tx,Ax]+μ¯[Tx-Ty,Ax-Ay]|μ|2(‖Ax‖2+‖Ax-Ay‖2)=μ¯|μ|2[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=1μ[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2.
Hence VL(T,μA)=(1/μ)VL(T,A).
Letx,y∈D(T1)∩D(T2).
Then,(3.8)[(T1+T2)x,Ax]+[(T1+T2)x-(T1+T2)y,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=[T1x,Ax]+[T2x,Ax]+[T1x-T1y,Ax-Ay]+[T2x-T2y,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=[T1x,Ax]+[T1x-T1y,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2+[T2x,Ax]+[T2x-T2y,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2.
Therefore, VL(T1+T2,A)⊆VL(T1,A)+VL(T2,A). Thus, (iii) is proved.
Finally, to prove (iv):(3.9)[(T-λA)x,Ax]+[(T-λA)x-(T-λA)y,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2=[Tx,Ax]-λ‖Ax‖2+[Tx-Ty,Ax-Ay]-λ‖Ax-Ay‖2‖Ax‖2+‖Ax-Ay‖2=[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2-λ.
This implies that VL(T-λA,A)=VL(T,A)-{λ}.Approximate Point Spectrum of Two Nonlinear Operators
Approximate point spectrumπ(T,A) of two nonlinear operators T and A is defined asπ(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 operatorT: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 spectrumπ(T,A) of two nonlinear operators T and A is contained in the closure of their numerical range VL(T,A).Proof.
Letλ∈π(T,A). Now
(3.10)|[Txn,Axn]+[Txn-Tyn,Axn-Ayn]‖Axn‖2+‖Axn-Ayn‖2-λ|=|[(T-λA)xn,Axn]+[(T-λA)xn-(T-λA)yn,Axn-Ayn]|‖Axn‖2+‖Axn-Ayn‖2≤‖(T-λA)xn‖‖Axn‖+‖(T-λA)xn-(T-λA)yn‖‖Axn-Ayn‖‖Axn‖2+‖Axn-Ayn‖2.
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.
Letμ 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.Proof.
We have(3.11)0<d≤|[Tx,Ax]+[Tx-Ty,Ax-Ay]‖Ax‖2+‖Ax-Ay‖2-μ|=|[(T-μA)x,Ax]+[(T-μA)x-(T-μA)y,Ax-Ay]|‖Ax‖2+‖Ax-Ay‖2.
This implies that
(3.12)|[(T-μA)x,Ax]+[(T-μA)x-(T-μA)y,Ax-Ay]|≥d(‖Ax‖2+‖Ax-Ay‖2).
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.
LetX 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 in GL(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∥≤∥A-1T∥ln∥A-1Tx∥(1-∥A-1T∥l)-1, for x∈X, n=0,1,2,….Proof.
For eachx,y∈X with Ax≠Ay, we have
(3.13)‖(A-T)x-(A-T)y‖≥‖Ax-Ay‖-‖Tx-Ty‖≥‖Ax-Ay‖-K‖Ax-Ay‖=(1-K)‖Ax-Ay‖>0,
since K≠1, and Ax≠Ay. This implies that (A-T) is injective.
Next ifu,v∈R(A-T), then(3.14)‖(A-T)-1‖=‖A-1(I-A-1T)-1‖≤‖A-1‖‖(I-A-1T)-1‖≤‖A-1‖(1-‖A-1T‖)-1.
Similarly, we have ∥(A-T)-1∥l≤∥A-1∥l(1-∥A-1T∥l)-1.
Now(3.15)‖(A-T)-1‖+‖(A-T)-1‖l≤‖A-1‖1-‖A-1T‖+‖A-1‖l1-‖A-1T‖l=‖A-1‖(1-‖A-1T‖l)+‖A-1‖(1-‖A-1T‖l)(1-‖A-1T‖)(1-‖A-1T‖l)≤‖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).
To prove the second part, consider the sequence of approximating operators
(3.16)B0=I,Bn=I+(A-1T)Bn-1,forn=1,2,3,….
We claim that(3.17)‖Bn+1x-Bnx‖≤‖A-1T‖ln‖A-1Tx‖.
Forn=0,∥B1x-B0x∥=∥(I+A-1T)Ix-Ix∥=∥A-1Tx∥.
Forn=1,∥B2x-B1x∥=∥(I+(A-1T)B1)x-(I+(A-1T)B0)x∥≤∥A-1T∥l1∥A-1Tx∥.
Assume that (3.17) is true for n=k-1, that is ∥Bkx-Bk-1x∥≤∥A-1T∥lk-1∥A-1Tx∥.
Now forn=k,(3.18)‖Bk+1x-Bkx‖=‖(I+(A-1T)Bk)x-(I+(A-1T)Bk-1)x‖=‖(A-1T)Bkx-(A-1T)Bk-1x‖≤‖A-1T‖l‖Bkx-Bk-1x‖≤‖A-1T‖lk‖A-1Tx‖.Now for a positive integer p,
(3.19)‖Bn+px-Bnx‖=‖∑k=0p-1Bn+k+1x-Bn+kx‖≤∑k=0p-1‖Bn+k+1x-Bn+kx‖≤∑k=0p-1‖A-1T‖ln+k‖A-1Tx‖=‖A-1T‖ln‖A-1Tx‖{1+‖A-1T‖l+‖A-1T‖l2+⋯+‖A-1T‖lp-1}≤‖A-1T‖ln‖A-1Tx‖(1-‖A-1T‖l)-1.Since ∥A-1T∥l<1, the sequence {Bnx} is a Cauchy sequence. Again since X is complete, we have limm→∞Bm(x)=Ex exists for all x∈X.
Form=n+p,(3.20)‖Ex-Bnx‖=lim⁡p→∞‖Bn+px-Bnx‖≤‖A-1T‖ln‖A-1Tx‖(1-‖A-1T‖l)-1.As A-1T is continuous, we have
(3.21)Ex=lim⁡n→∞Bn(x)=lim⁡n→∞(I+(A-1T)Bn-1)x=(I+(A-1T)E)x⟹Ex-(A-1T)Ex=Ix⟹E=(I-A-1T)-1.Now multiplying A-1 we get (A-T)-1=A-1E.Using this technique, one can solve an operator equation involving two nonlinear operators under the condition that one of the operator is invertible.



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*Source: 846396-2012-04-04.xml*