Numerical Range of Two Operators in Semi-Inner Product Spaces

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.

1. Introduction

Lumer [1] introduced the concept of semi-inner product. He defined semi-inner product as follows.

Semi-Inner Product
Let 𝑋 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: If 𝑓 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 space 𝑙^𝑝,𝑝>1 and the function space 𝐿^𝑝,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 𝐿^𝑝,𝑝>1 are semi-inner product spaces with the semi-inner product defined by1𝑓,𝑔∶=𝑔_𝑝^𝑝−2_𝑋𝑓(𝑥)𝑔(𝑥)^𝑝−1sgn𝑔(𝑥)𝑑𝜇,∀𝑓,𝑔∈𝐿^𝑃𝑋,𝜇. (1.1)
To study the generalized eigenvalue problem 𝑇𝑥=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 𝑊_𝑛𝑙(𝑇,𝐴), as 𝑊_𝑛𝑙(𝑇,𝐴)∶=𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦𝐴𝑥−𝐴𝑦²∶𝑥≠𝑦,𝑥,𝑦∈𝐷(𝑇)∩𝐷(𝐴), (1.2) where 𝐷(𝑇) and 𝐷(𝐴) denote the domains of the operators 𝑇 and 𝐴, respectively. The numerical radius 𝑤_𝑛𝑙(𝑇,𝐴) is defined as 𝑤_𝑛𝑙(𝑇,𝐴)=sup{|𝜆|∶𝜆∈𝑊_𝑛𝑙(𝑇,𝐴)}. 𝑊_𝑛𝑙(𝑇,𝐴) may not be convex. If 𝑇 and 𝐴 are continuous and 𝐷(𝑇),𝐷(𝐴) are connected, then 𝑊_𝑛𝑙(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 𝑉_𝐿(𝑇) of a nonlinear operator 𝑇, as 𝑉_𝐿(𝑇)[]+∶=𝑇𝑥,𝑥𝑇𝑥−𝑇𝑦,𝑥−𝑦‖𝑥‖²+𝑥−𝑦²∶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 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 Range 𝑊(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 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 linear operators 𝑇 and 𝐴 is defined as 𝜎(𝑇,𝐴)∶={𝜆∈ℂ∶(𝑇−𝜆𝐴)isnotinvertible}. (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 as𝑒(𝑇,𝐴)∶={𝜆∈ℂ∶𝑇𝑥=𝜆𝐴𝑥,𝑥≠0}. (2.2)

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 sequence𝑥_𝑛 in𝑋with ‖𝐴𝑥_𝑛‖=1 and ‖𝑇𝑥_𝑛−𝜆𝐴𝑥_𝑛‖→0as𝑛→∞}.

Compression Spectrum 𝜎₀(𝑇,𝐴)
The compression spectrum 𝜎₀(𝑇,𝐴) of two linear operators 𝑇and𝐴 is defined as 𝜎₀(𝑇,𝐴)∶=𝜆∈ℂ∶Range(𝑇−𝜆𝐴)isnotdensein𝑋. (2.3)

Coupled Numerical Range 𝑊_𝐴(𝑇)
The coupled numerical range 𝑊_𝐴(𝑇) of 𝑇 with respect to 𝐴 is defined as 𝑊_𝐴(𝑇)[]∶=𝐴𝑇𝑥,𝑥[]∶‖𝑥‖[].𝐴𝑥,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,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)=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)𝑊_𝐴(𝑇₁+𝑇₂)⊆𝑊_𝐴(𝑇₁)+𝑊_𝐴(𝑇₂);
(ii)𝑊_𝐴(𝛼𝑇)=𝛼𝑊_𝐴(𝑇);
(iii)𝑊_𝛼𝐴(𝑇)=𝑊_𝐴(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 𝑥_𝑛 in 𝑋 such that [𝐴𝑥_𝑛,𝐴𝑥_𝑛]=1 and ‖(𝑇−𝜆𝐴)𝑥_𝑛‖→0 as 𝑛→∞.
Now𝑇𝑥_𝑛,𝐴𝑥_𝑛−𝜆=(𝑇−𝜆𝐴)𝑥_𝑛,𝐴𝑥_𝑛≤(𝑇−𝜆𝐴)𝑥_𝑛𝐴𝑥_𝑛⟶0as𝑛⟶∞. (2.5)This implies that [𝑇𝑥_𝑛,𝐴𝑥_𝑛]→𝜆 as 𝑛→∞.
Hence, 𝜆∈𝑊(𝑇,𝐴), and consequently 𝜋(𝑇,𝐴)⊂𝑊(𝑇,𝐴).

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 have𝐴⁻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 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. 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 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. Let 𝑇 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 have 𝑟(𝑇,𝐴)≤𝑤(𝑇,𝐴)<1. For ‖𝐴𝑥‖=1, we have ‖(𝐴−𝑇)𝑥‖=𝐴𝐼−𝐴⁻¹𝑇𝑥=𝐴𝐼−𝐴⁻¹𝑇𝑥‖≥𝐴‖𝐴𝑥𝐼−𝐴⁻¹𝑇≥[]−()𝑥,𝐴𝑥𝐴𝑥,𝐴𝑥|[𝑇𝑥,𝐴𝑥]|≥1−𝑤𝑇,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 get1≥(1−𝑤(𝑇,𝐴))𝐴(𝐴−𝑇)⁻¹𝑦⇒𝐴(𝐴−𝑇)⁻¹𝑦≤(1−𝑤(𝑇,𝐴))⁻¹𝑦⇒𝐴(𝐴−𝑇)⁻¹≤1().1−𝑤𝑇,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‖=sup_𝑥‖𝑇𝑥‖/‖𝑥‖ and ‖𝑇‖_𝑙=sup_𝑥≠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 𝐺_𝐿(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 set 𝜌_𝐴(𝑇) of a nonlinear operator 𝑇 with respect to another operator 𝐴 is defined as

𝜌_𝐴(𝑇)∶=𝜆∈ℂ∶(𝑇−𝜆𝐴)⁻¹existsandisgeneralizedLipschitzian. (3.1)

A-Spectrum of 𝑇
A-spectrum of 𝑇, 𝜎_𝐴(𝑇) is the complement of the A-resolvent set of 𝑇.

Numerical Range of Two Nonlinear Operators
The numerical range 𝑉_𝐿(𝑇,𝐴) of two nonlinear operators 𝑇 and 𝐴 is defined as 𝑉_𝐿()[]+𝑇,𝐴∶=𝑇𝑥,𝐴𝑥𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²∶𝑥,𝑦∈𝐷(𝑇)∩𝐷(𝐴),𝑥≠𝑦, (3.2) where 𝐷(𝑇) and 𝐷(𝐴) are the domains of the operators 𝑇 and 𝐴, respectively. The numerical radius 𝑤_𝐿(𝑇,𝐴) is defined as 𝑤_𝐿(𝑇,𝐴)={sup|𝜆|∶𝜆∈𝑉_𝐿(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,𝑝>1. Let 𝑥=(𝑥₁,𝑥₂,…),𝑦=(𝑦₁,𝑦₂,…)∈𝑙^𝑝. Consider the two nonlinear operators 𝑇,𝐴∶𝑙^𝑝→𝑙^𝑝 defined by 𝑇𝑥=(‖𝑥‖,𝑥₁,𝑥₂,…) and 𝐴𝑥=(‖𝑥‖,0,0,…). The semi-inner product on the real sequence space 𝑙^𝑝 is defined as [𝑥,𝑦]=(1/‖𝑦‖^𝑝−2)∑^∞_𝑛=1|𝑦_𝑛|^𝑝−2𝑦_𝑛𝑥_𝑛,∀𝑥={𝑥_𝑛},𝑦={𝑦_𝑛}∈𝑙^𝑝. 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[𝑇𝑥,𝐴𝑥]+𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=‖𝑥‖²+‖𝑥‖−𝑦²‖𝑥‖²+‖𝑥‖−𝑦²=1,∀𝑥,𝑦∈𝑙^𝑝. (3.3) Therefore, 𝑉_𝐿(𝑇,𝐴)={1}, and 𝑤_𝐿(𝑇,𝐴)=1.

Example 3.2. Consider the real sequence space 𝑙^𝑝,𝑝>1. Let 𝑥=(𝑥₁,𝑥₂,…)and𝑦=(𝑦₁,𝑦₂,…)∈𝑙^𝑝. Consider the two nonlinear operators 𝑇,𝐴∶𝑙^𝑝→𝑙^𝑝 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|𝑥_𝑛−𝑦_𝑛|^𝑝 and=1𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦𝐴𝑥−𝐴𝑦^∞𝑝−2_𝑛=1𝑥_𝑛−𝑦_𝑛^𝑝=𝐴𝑥−𝐴𝑦^𝑝𝐴𝑥−𝐴𝑦^𝑝−2=𝐴𝑥−𝐴𝑦². (3.4) For any 𝑥,𝑦∈𝑙^𝑝, we have 𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦𝐴𝑥−𝐴𝑦²=𝐴𝑥−𝐴𝑦²𝐴𝑥−𝐴𝑦²=1. (3.5) Therefore, the numerical range of two nonlinear operators 𝑇 and 𝐴 in the sense of Nanda [13] is 𝑊_𝑛𝑙(𝑇,𝐴)={1} and the numerical radius 𝑤_𝑛𝑙(𝑇,𝐴)=1.

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,and𝑇₂ are nonlinear operators defined on 𝑋, and 𝜆and𝜇 are scalars, then
(i)𝑉_𝐿(𝜆𝑇,𝐴)=𝜆𝑉_𝐿(𝑇,𝐴);
(ii)𝑉_𝐿(𝑇,𝜇𝐴)=(1/𝜇)𝑉_𝐿(𝑇,𝐴);
(iii)𝑉_𝐿(𝑇₁+𝑇₂,𝐴)⊆𝑉_𝐿(𝑇₁,𝐴)+𝑉_𝐿(𝑇₂,𝐴);
(iv)𝑉_𝐿(𝑇−𝜆𝐴,𝐴)=𝑉_𝐿(𝑇,𝐴)−{𝜆}.

Proof. To prove (i): [𝜆𝑇𝑥,𝐴𝑥]+𝜆𝑇𝑥−𝜆𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²[]+=𝜆𝑇𝑥,𝐴𝑥𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦². (3.6) Hence, 𝑉_𝐿(𝜆𝑇,𝐴)=𝜆𝑉_𝐿(𝑇,𝐴).
To show (ii):𝑇𝑥,𝜇𝐴𝑥+𝑇𝑥−𝑇𝑦,𝜇𝐴𝑥−𝜇𝐴𝑦𝜇𝐴𝑥²+𝜇𝐴𝑥−𝜇𝐴𝑦²=𝜇[𝑇𝑥,𝐴𝑥]+𝜇𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦𝜇²‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=𝜇𝜇²[𝑇𝑥,𝐴𝑥]+𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=1𝜇[𝑇𝑥,𝐴𝑥]+𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦². (3.7) Hence 𝑉_𝐿(𝑇,𝜇𝐴)=(1/𝜇)𝑉_𝐿(𝑇,𝐴).
Let 𝑥,𝑦∈𝐷(𝑇₁)∩𝐷(𝑇₂).
Then,𝑇₁+𝑇₂𝑥,𝐴𝑥+𝑇₁+𝑇₂𝑥−𝑇₁+𝑇₂𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=𝑇₁𝑥,𝐴𝑥+𝑇₂𝑥,𝐴𝑥+𝑇₁𝑥−𝑇₁𝑦,𝐴𝑥−𝐴𝑦+𝑇₂𝑥−𝑇₂𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=𝑇₁𝑥,𝐴𝑥+𝑇₁𝑥−𝑇₁𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²+𝑇₂𝑥,𝐴𝑥+𝑇₂𝑥−𝑇₂𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦². (3.8) Therefore, 𝑉_𝐿(𝑇₁+𝑇₂,𝐴)⊆𝑉_𝐿(𝑇₁,𝐴)+𝑉_𝐿(𝑇₂,𝐴). Thus, (iii) is proved.
Finally, to prove (iv):[(𝑇−𝜆𝐴)𝑥,𝐴𝑥]+(𝑇−𝜆𝐴)𝑥−(𝑇−𝜆𝐴)𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=[𝑇𝑥,𝐴𝑥]−𝜆‖𝐴𝑥‖²+𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦−𝜆𝐴𝑥−𝐴𝑦²‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=[𝑇𝑥,𝐴𝑥]+𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²−𝜆. (3.9) This implies that 𝑉_𝐿(𝑇−𝜆𝐴,𝐴)=𝑉_𝐿(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 as𝜋(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‖→0 and ‖(𝑇−𝜆𝐴)𝑥_𝑛−(𝑇−𝜆𝐴)𝑦_𝑛‖→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 and 𝐴 is contained in the closure of their numerical range 𝑉_𝐿(𝑇,𝐴).

Proof. Let 𝜆∈𝜋(𝑇,𝐴). Now 𝑇𝑥_𝑛,𝐴𝑥_𝑛+𝑇𝑥_𝑛−𝑇𝑦_𝑛,𝐴𝑥_𝑛−𝐴𝑦_𝑛𝐴𝑥_𝑛²+𝐴𝑥_𝑛−𝐴𝑦_𝑛²=)𝑥−𝜆(𝑇−𝜆𝐴_𝑛,𝐴𝑥_𝑛+(𝑇−𝜆𝐴)𝑥_𝑛−(𝑇−𝜆𝐴)𝑦_𝑛,𝐴𝑥_𝑛−𝐴𝑦_𝑛𝐴𝑥_𝑛²+𝐴𝑥_𝑛−𝐴𝑦_𝑛²≤(𝑇−𝜆𝐴)𝑥_𝑛𝐴𝑥_𝑛+(𝑇−𝜆𝐴)𝑥_𝑛−(𝑇−𝜆𝐴)𝑦_𝑛𝐴𝑥_𝑛−𝐴𝑦_𝑛𝐴𝑥_𝑛²+𝐴𝑥_𝑛−𝐴𝑦_𝑛². (3.10) The right-hand side goes to 0 as 𝑛→∞. This implies that 𝜆∈𝑉_𝐿(𝑇,𝐴), and hence 𝜋(𝑇,𝐴)⊂𝑉_𝐿(𝑇,𝐴).

Theorem 3.5. Let 𝜇 be a complex number. Then 𝜇 is at a distance 𝑑>0 from 𝑉_𝐿(𝑇,𝐴) if and only if (𝑇−𝜇𝐴) is strongly generalized A-monotone.

Proof. We have []+0<𝑑≤𝑇𝑥,𝐴𝑥𝑇𝑥−𝑇𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦²=+−𝜇𝑇−𝜇𝐴𝑥,𝐴𝑥𝑇−𝜇𝐴𝑥−𝑇−𝜇𝐴𝑦,𝐴𝑥−𝐴𝑦‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦². (3.11) This implies that 𝑇−𝜇𝐴𝑥,𝐴𝑥+𝑇−𝜇𝐴𝑥−𝑇−𝜇𝐴𝑦,𝐴𝑥−𝐴𝑦≥𝑑‖𝐴𝑥‖²+𝐴𝑥−𝐴𝑦². (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∈𝐺_𝐿(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 𝐺_𝐿(𝑋) and ‖(𝐴−𝑇)⁻¹‖≤(‖𝐴⁻¹‖_𝐿(2−‖𝐴⁻¹𝑇‖_𝐿))/(1−‖𝐴⁻¹𝑇‖)(1−‖𝐴⁻¹𝑇‖_𝑙). Again if 𝐵₀=𝐼 and 𝐵_𝑛=𝐼+(𝐴⁻¹𝑇)𝐵_𝑛−1 for 𝑛=1,2,3,…, and ‖𝐴⁻¹𝑇‖_𝑙<1, then lim_𝑛→∞𝐵_𝑛𝑥=(𝐼−𝐴⁻¹𝑇)⁻¹𝑥 for every 𝑥∈𝑋, as 𝑛→∞ and ‖(𝐼−𝐴⁻¹𝑇)⁻¹𝑥−𝐵_𝑛𝑥‖≤‖𝐴⁻¹𝑇‖^𝑛_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 )()𝑦≥−≥=((𝐴−𝑇𝑥−𝐴−𝑇𝐴𝑥−𝐴𝑦𝑇𝑥−𝑇𝑦𝐴𝑥−𝐴𝑦−𝐾𝐴𝑥−𝐴𝑦1−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⁻¹𝐼−𝐴⁻¹𝑇⁻¹≤𝐴⁻¹𝐼−𝐴⁻¹𝑇⁻¹≤𝐴⁻¹1−𝐴⁻¹𝑇⁻¹. (3.14) Similarly, we have ‖(𝐴−𝑇)⁻¹‖_𝑙≤‖𝐴⁻¹‖_𝑙(1−‖𝐴⁻¹𝑇‖_𝑙)⁻¹.
Now(𝐴−𝑇)⁻¹+(𝐴−𝑇)⁻¹_𝑙≤𝐴⁻¹1−𝐴⁻¹𝑇+𝐴⁻¹_𝑙1−𝐴⁻¹𝑇_𝑙=𝐴⁻¹1−𝐴⁻¹𝑇_𝑙+𝐴⁻¹1−𝐴⁻¹𝑇_𝑙1−𝐴⁻¹𝑇1−𝐴⁻¹𝑇_𝑙≤𝐴⁻¹_𝐿1−𝐴⁻¹𝑇_𝑙+𝐴⁻¹_𝐿1−𝐴⁻¹𝑇_𝑙1−𝐴⁻¹𝑇1−𝐴⁻¹𝑇_𝑙=2𝐴⁻¹_𝐿−𝐴⁻¹_𝐿𝐴⁻¹𝑇_𝑙−𝐴⁻¹_𝐿𝐴⁻¹𝑇1−𝐴⁻¹𝑇1−𝐴⁻¹𝑇_𝑙=2𝐴⁻¹_𝐿−𝐴⁻¹_𝐿𝐴⁻¹𝑇_𝐿1−𝐴⁻¹𝑇1−𝐴⁻¹𝑇_𝑙,⟹(𝐴−𝑇)⁻¹_𝐿≤𝐴⁻¹_𝐿2−𝐴⁻¹𝑇_𝐿1−𝐴⁻¹𝑇1−𝐴⁻¹𝑇_𝑙. (3.15) To prove the second part, consider the sequence of approximating operators 𝐵₀=𝐼,𝐵_𝑛=𝐼+𝐴⁻¹𝑇𝐵_𝑛−1,for𝑛=1,2,3,…. (3.16)
We claim that𝐵_𝑛+1𝑥−𝐵_𝑛𝑥≤𝐴⁻¹𝑇^𝑛_𝑙𝐴⁻¹𝑇𝑥. (3.17)
For 𝑛=0,‖𝐵₁𝑥−𝐵₀𝑥‖=‖(𝐼+𝐴⁻¹𝑇)𝐼𝑥−𝐼𝑥‖=‖𝐴⁻¹𝑇𝑥‖.
For 𝑛=1,‖𝐵₂𝑥−𝐵₁𝑥‖=‖(𝐼+(𝐴⁻¹𝑇)𝐵₁)𝑥−(𝐼+(𝐴⁻¹𝑇)𝐵₀)𝑥‖≤‖𝐴⁻¹𝑇‖¹_𝑙‖𝐴⁻¹𝑇𝑥‖.
Assume that (3.17) is true for 𝑛=𝑘−1, that is ‖𝐵_𝑘𝑥−𝐵_𝑘−1𝑥‖≤‖𝐴⁻¹𝑇‖_𝑙^𝑘−1‖𝐴⁻¹𝑇𝑥‖.
Now for 𝑛=𝑘,𝐵_𝑘+1𝑥−𝐵_𝑘𝑥=𝐼+𝐴⁻¹𝑇𝐵_𝑘𝑥−𝐼+𝐴⁻¹𝑇𝐵_𝑘−1𝑥=𝐴⁻¹𝑇𝐵_𝑘𝑥−𝐴⁻¹𝑇𝐵_𝑘−1𝑥≤𝐴⁻¹𝑇_𝑙𝐵_𝑘𝑥−𝐵_𝑘−1𝑥≤𝐴⁻¹𝑇^𝑘_𝑙𝐴⁻¹𝑇𝑥. (3.18)Now for a positive integer 𝑝, 𝐵_𝑛+𝑝𝑥−𝐵_𝑛𝑥=^𝑝−1_𝑘=0𝐵_𝑛+𝑘+1𝑥−𝐵_𝑛+𝑘𝑥≤^𝑝−1_𝑘=0𝐵_𝑛+𝑘+1𝑥−𝐵_𝑛+𝑘𝑥≤^𝑝−1_𝑘=0𝐴⁻¹𝑇_𝑙^𝑛+𝑘𝐴⁻¹=𝐴𝑇𝑥₋₁𝑇^𝑛_𝑙𝐴⁻¹𝑇𝑥1+𝐴⁻¹𝑇_𝑙+𝐴⁻¹𝑇²_𝑙+⋯+𝐴⁻¹𝑇_𝑙^𝑝−1≤𝐴⁻¹𝑇^𝑛_𝑙𝐴⁻¹𝑇𝑥1−𝐴⁻¹𝑇_𝑙⁻¹. (3.19)Since ‖𝐴⁻¹𝑇‖_𝑙<1, the sequence {𝐵_𝑛𝑥} is a Cauchy sequence. Again since 𝑋 is complete, we have lim_𝑚→∞𝐵_𝑚(𝑥)=𝐸𝑥 exists for all 𝑥∈𝑋.
For 𝑚=𝑛+𝑝,𝐸𝑥−𝐵_𝑛𝑥=lim_𝑝→∞𝐵_𝑛+𝑝𝑥−𝐵_𝑛𝑥≤𝐴⁻¹𝑇^𝑛_𝑙𝐴⁻¹𝑇𝑥1−𝐴⁻¹𝑇_𝑙⁻¹. (3.20)As 𝐴⁻¹𝑇 is continuous, we have 𝐸𝑥=lim_𝑛→∞𝐵_𝑛(𝑥)=lim_𝑛→∞𝐼+𝐴⁻¹𝑇𝐵_𝑛−1𝑥=𝐼+𝐴⁻¹𝑇𝐸𝑥⟹𝐸𝑥−𝐴⁻¹𝑇𝐸𝑥=𝐼𝑥⟹𝐸=𝐼−𝐴⁻¹𝑇⁻¹. (3.21)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.

Acknowledgments

The authors are thankful to the referees and editors for their valuable suggestions which improved the presentation of the paper. The authors are also thankful to Professor R. N. Mohapatra, University of Central Florida, USA, for his valuable discussions and suggestions. The first author is very much thankful to the Council of Scientific and Industrial Research (CSIR), India, for its financial support in executing this study.