Spectral Theory of Compact Linear Operators and Applications

Spectral Theory of Compact Linear Operators and Applications

Chapter One

PREAMBLE OF THE STUDY

Basic notions and results from Functional Analysis

The purpose of this section is to refresh our minds on some basic fundamentals required to facilitate a smooth understanding of the study of compact linear operators and its applications.

Definition 1.1.1. A non-negative function on a vector space X is called a norm on X if and only if

||x||≥ 0 for every x ∈ X (Positivity).
||x||= 0 if and only if x = 0 (Nondegeneracy).
||λ(x)||= |λ|||x|| for every x ∈ X (Homogeneity).
||x+ y|| ≤ ||x|| + ||y|| for every x, y ∈ X (subadditivity).

A vector space X with a norm is denoted by (X, ) and is called a normed linear space (or just a normed space).

A sequence (x_n)_n_∈_N of elements in a normed linear space X is called Cauchy if ∀ϵ > 0, ∃N ∈ N, such that for m, n ∈ N, ||x_m − x_n|| ≤ ϵ, m, n ≥ N

A Banach space is a normed linear space (X, ) that is complete in the canonical metric defined by ρ(x, y) = x y for x, y X i.e every Cauchy sequence in X for the metric ρ converges to some point in X.

CHAPTER TWO

Linear Compact Operators on Banach Spaces

Definition 2.0.9.

Let X and Y be arbitrary Banach Spaces. A linear operator, T : X −→ Y is called compact if the image of the closed unit ball B_X(0, 1) = {x ∈ X : ǁxǁ_X ≤ 1} by T is a relatively compact subset of Y . In other words, T is compact if T (B_X) is compact.

This definition is equivalent to each of the following properties.

Foreach bounded B ⊂ X, the image T (B) is relatively compact in Y .
Forevery bounded sequence x_n n∈N X, Tx_n n∈N has a convergent subsequence in

We introduce the following notations

K(X, Y ) := {T : X −→ Y | T is linear and compact} K(X) = K(X, X)

lemma 2.0.10.

K(X, Y ) ⊂ B(X, Y ).

Proof. Suppose T ∈ K(X, Y ), we show that T ∈ B(X, Y ) i.e for all x ∈ X

there exist M > 0 such that ||T x|| ≤ M ||x||

Let x ∈ X. if x = 0, it holds trivially.

Assume x /= 0, ^x ∈ B_X

||x||

T (B_X) is compact since T ∈ K(X, Y ),so T (B_X) is bounded.

Therefore there exist M > 0 such that T ( ^x

||x||

) ≤ M , which implies tha

||T x|| ≤ M ||x||

Remark. Recall that Riesz theorem characterizes the compactness of the closed unit ball of a Banach space X by the finiteness of the dimension of X.

Thus for an infinite dimensional Banach space X, we have I ∈ B(X) \ K(X) and so the inclusion K(X) ⊂ B(X) is strict, that is,

K(X) Ç B(X) whenever dimX = +∞.

Properties of compact linear maps

Theorem 2.1.1.

Let X, Y and Z be Banach spaces, and let T : X −→ Y , S : Y −→ Z and

F : Z −→ X be bounded linear operators.

Ifthe R(T ) is finite dimensional (i.e dimR(T ) < +∞), then T is
IfT is compact, then T ◦ S and F ◦ T are

For every T₁,T₂ ∈ K(X, Y ) and every scalar α and β, we have αT₁+βT₂ ∈ K(X, Y ). That is K(X, Y ) is a linear subspace of B(X, Y ).

Proof.

Supposethat R(T ) is finite dimensional and let B be any bounded subset of X. We show that T (B) is compact in Y .

T (B) is a bounded subset of Y , since T is bounded. So T (B) is compact as a closed and bounded subset of a finite dimensional space. Hence T is compact.

Let B be any bounded set in X. We need to show that T (S(B)) and

F (T (B)) are compact.

S(B) is bounded since S is bounded. Therefore T (S(B)) is compact since T

is compact. This shows that T ◦ S is compact.

Now (F ◦ T )(B) = F (T (B)) ⊂ F (T (B)), since T (B) ⊂ T (B).

(F ◦ T )(B) ⊂ F (T (B))

And since T is compact, T (B) is compact, and so F (T (B)) is compact as a continuous image of compact set. Therefore F (T (B)) ⊂ F (T (B)) is compact as a closed subset of a compact set. Thus F ◦ T is compact.

ClearlyαT₁ + βT₂ ∈ B(X, Y ). Let B_X be the closed unit ball of X, we show that (αT₁ + βT₂)(B_X) is compact in Y .

(αT₁ + βT₂)(B_X) ⊆ αT₁(B_X) + βT₂(B_X) by linearity of T₁ and T₂.

Thus (αT₁ + βT₂)(B_X) ⊂ αT₁(B_X) + βT₂(B_X) since αT₁(B_X) and βT₂(B_X) are subsets of αT₁(B_X) and βT₂(B_X) respectively. αT₁(B_X) + βT₂(B_X) is

CHAPTER THREE

Application to Linear Elliptic Boundary value Problems

Notations and definitions

All funtions and Vector fields used are of class atleast C²

Definition 3.1.1. We define the gradient of the scalar function f ∈ Rn as the vector field of the partial derivatives of f denoted by ∇f i.e

Definition 3.1.2. The divergence of the vector field F = (f₁, f₂, …, f_n) in the coordinates (x₁, x₂, …, x_n), is given by

∂f

divF = ∇ · F =

+ ∂f2 + … + ∂fn .

Definition 3.1.3. Let φ C^k(Ω), k 2, where Ω is open in Rn. We define the Laplacian operator of φ by

Δφ = div(∇φ)

Proposition 3.1.4. By taking φ, ψ ∈ C^k(Ω), k ≥ 2, we get,

Δ(φ+ ψ) = Δφ + Δψ
div(φ∇ψ)= φ(Δψ) + ⟨∇φ, ∇ψ⟩

Bibliography

Gohberg an S. Seymour; Basic Operator theory, 1980
EChidume; Applicable Functional Analysis, International Centre for Theoretical Physics Trieste, Italy, July 2006
Fabian, P. Habala, P. Hajek, V. Montesinos and V. Zizler; Banach space theory for linear and non-linear Analysis. , Springer 2006
AMunoz, Y. Sarantopolous, A. Tonge; Complexification of real Banach Spaces, Studia Mathematica 1999.
Djitte; Lecture note on Sobolev spaces and linear elliptic partial differ- ential equations, African University of Science and Technology, Abuja,Nigeria. 2011
Albert; Lecture note on Functional Analysis, 2011

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