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Spectral Theory of Elliptic Differential Operators

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Spectral Theory of Elliptic Differential Operators Spectral theory of elliptic differential operators 802628S Lecture Notes 1st Edition Second printing Valeriy Serov University of Oulu 2009 Edited by Markus Harju Contents 1 Inner product spaces and Hilbert spaces 1 2 Symmetric operators in the Hilbert space 11 3 J. von Neumann’s spectral theorem 20 4 Spectrum of self-adjoint operators 33 5 Quadratic forms. Friedrichs extension. 48 6 Elliptic differential operators 52 7 Spectral function 61 8 Fundamental solution 64 9 Fractional powers of self-adjoint operators 85 Index 106 i 1 Inner product spaces and Hilbert spaces A collection of elements is called a complex (real) vector space (linear space) H if the following axioms are satisfied: 1) To every pair x, y H there corresponds a vector x + y, called the sum, with the properties: ∈ a) x + y = y + x b) x +(y + z)=(x + y)+ z x + y + z ≡ c) there exists unique 0 H such that x +0= x ∈ d) for every x H there exists unique y1 H such that x+y1 = 0. We denote y := x. ∈ ∈ 1 − 2) For every x H and every λ,µ C there corresponds a vector λ x such that ∈ ∈ · a) λ(µx)=(λµ)x λµx ≡ b) (λ + µ)x = λx + µx c) λ(x + y)= λx + λy d) 1 x = x. · Definition. For a linear space H every mapping ( , ) : H H C is called an inner product or a scalar product if · · × → 1) (x, x) 0 and (x, x) = 0 if and only if x = 0 ≥ 2) (x, y + z)=(x, y)+(x, z) 3) (λx, y)= λ(x, y) 4) (x, y)= (y, x) for every x, y, z H and λ C. A linear space equipped with an inner product is called an inner product∈ space∈. An immediate consequence of this definition is that (λx + µy,z) = λ(x, z)+ µ(y, z), (x, λy) = λ(x, y) for every x, y, z H and λ,µ C. ∈ ∈ Example 1.1. On the complex Euclidean space H = Cn the standard inner product is n (x, y)= xjyj, jX=1 where x =(x ,...,x ) Cn and y =(y ,...,y ) Cn. 1 n ∈ 1 n ∈ 1 Example 1.2. On the linear space C[a, b] of continuous complex-valued functions, the formula b (f,g)= f(x)g(x)dx Za defines an inner product. Definition. Suppose H is an inner product space. One calls 1) x H orthogonal to y H if (x, y) = 0. ∈ ∈ 1, α = β 2) a system xα α∈A H orthonormal if (xα, xβ) = δα,β = , where A { } ⊂ 0, α = β 6 is some index set. 3) x := (x, x) is called the length of x H. k k ∈ k N Exercise 1. Prove the Theorem of Pythagoras: If xj j=1, k is an orthonormal system in an» inner product space H, then { } ∈ 2 k k x 2 = (x, x ) 2 + x (x, x )x j j j k k j=1 | | − j=1 X X for every x H. ∈ k Exercise 2. Prove Bessel’s inequality: If xj j=1, k is an orthonormal system then { } ≤ ∞ k (x, x ) 2 x 2 , | j | ≤ k k jX=1 for every x H. ∈ Exercise 3. Prove the Cauchy-Schwarz-Bunjakovskii inequality: (x, y) x y , x, y H. | | ≤ k k k k ∈ Prove also that ( , ) is continuous as a map from H H to C. · · × If H is an inner product space, then x := (x, x) k k has the following properties: » 1) x 0 for every x H and x = 0 if and only if x = 0. k k ≥ ∈ k k 2) λx = λ x for every x H and λ C. k k | | k k ∈ ∈ 3) x + y x + y for every x, y H. This is the triangle inequality. k k ≤ k k k k ∈ 2 The function = ( , ) is thus a norm on H. It is called the norm induced by the inner product.k·k · · Every inner product» space H is a normed space under the induced norm. The neighborhood of x H is the open ball Br(x)= y H : x y < r . This system of neighborhoods defines∈ the norm topology on H{such∈ that:k − k } 1) The addition x + y is a continuous map H H H. × → 2) The scalar multiplication λ x is a continuous map C H H. · × → ∞ Definition. 1) A sequence xj j=1 H is called a Cauchy sequence if for every ε> 0 there exists n N {such} that⊂ x x <ε for k, j n . 0 ∈ k k − jk ≥ 0 ∞ 2) A sequence xj j=1 H is said to be convergent if there exists x H such that for every ε>{ 0} there⊂ exists n N such that x x <ε whenever∈ j n . 0 ∈ k − jk ≥ 0 3) The inner product space H is complete space if every Cauchy sequence in H converges. Corollary. 1) Every convergent sequence is a Cauchy sequence. 2) If x ∞ converges to x H then { j}j=1 ∈ lim xj = x . j→∞ k k k k Definition. (J. von Neumann, 1925) A Hilbert space is an inner product space which is complete (with respect to its norm topology). Exercise 4. Prove that in an inner product space the norm induced by this inner product satisfies the parallelogram law x + y 2 + x y 2 = 2 x 2 + 2 y 2 . k k k − k k k k k Exercise 5. Prove that if in a normed space H the parallelogram law holds, then there is an inner product on H such that x 2 =(x, x) and that this inner product is defined by the polarization identity k k 1 (x, y) := x + y 2 x y 2 + i x + iy 2 i x iy 2 . 4 k k − k − k k k − k − k Exercise 6. Prove thatÄ on C[a, b] the norm ä f = max f(x) k k x∈[a,b] | | is not induced by an inner product. Exercise 7. Give an example of an inner product space which is not complete. 3 Next we list some examples of Hilbert spaces. 1) The Euclidean spaces Rn and Cn. 2) The matrix space Mn(C) consisting of n n -matrices whose elements are complex numbers. For A, B M (C) the inner product× is given by ∈ n n ∗ (A, B)= akjbkj = Tr(AB ), k,jX=1 T where B∗ = B . 3) The sequence space l2(C) defined by ∞ l2(C) := x ∞ , x C : x 2 < . { j}j=1 j ∈ | j| ∞ jX=1 The estimates x + y 2 2 x 2 + y 2 , λx 2 = λ 2 x 2 | j j| ≤ | j| | j| | j| | | | j| and Ä 1 ä x y x 2 + y 2 | j j| ≤ 2 | j| | j| imply that l2(C) is a linear space. Let us define the inner product by Ä ä ∞ (x, y) := xjyj jX=1 2 C (k) ∞ 2 C and prove that l ( ) is complete. Suppose that x k=1 l ( ) is a Cauchy sequence. Then for every ε> 0 there exists n N{ such} that∈ 0 ∈ ∞ 2 x(k) x(m) = x(k) x(m) 2 <ε2 − | j − j | j=1 X for k,m n . It implies that ≥ 0 x(k) x(m) <ε, j = 1, 2,... | j − j | or that x(k) ∞ is a Cauchy sequence in C for every j = 1, 2,.... Since C is { j }k=1 a complete space then x(k) ∞ converges for every fixed j = 1, 2,... i.e. there { j }k=1 exists xj C such that ∈ (k) xj = lim xj . k→∞ This fact and l x(k) x(m) 2 <ε2, l N | j − j | ∈ jX=1 4 imply that l l (k) (m) 2 (k) 2 2 lim x x = x xj ε m→∞ | j − j | | j − | ≤ jX=1 jX=1 for all k n and l N. Therefore the sequence ≥ 0 ∈ l s := x(k) x 2, k n l | j − j| ≥ 0 jX=1 is a monotone increasing sequence which is bounded from above by ε2. Hence this sequence has a limit with the same upper bound i.e. ∞ l (k) 2 (k) 2 2 xj xj = lim xj xj ε . | − | l→∞ | − | ≤ jX=1 jX=1 That’s why we may conclude that x x(k) + x(k) x x(k) + ε k k ≤ − ≤ and x l2(C). ∈ 4) The Lebesgue space L2(Ω), where Ω Rn is an open set. The space L2(Ω) consists of all Lebesgue measurable functions⊂ f which are square integrable i.e. f(x) 2dx < . ZΩ | | ∞ It is a linear space with the inner product (f,g)= f(x)g(x)dx ZΩ and the Riesz-Fisher theorem reads as: L2(Ω) is a Hilbert space. k 2 5) The Sobolev spaces W2 (Ω) consisting of functions f L (Ω) whose weak or distributional derivatives Dαf also belong to L2(Ω) up∈ to order α k, k = k | | ≤ 1, 2,.... On the space W2 (Ω) the natural inner product is (f,g)= Dαf(x)Dαg(x)dx. Ω |αX|≤k Z Definition. Let H be an inner product space. For any subspace M H the orthogonal complement of M is defined as ⊂ M ⊥ := y H :(y, x) = 0, x M . { ∈ ∈ } Remark. It is clear that M ⊥ is a linear subspace of H. Moreover, M M ⊥ = 0 since 0 M always. ∩ { } ∈ 5 Definition. A closed subspace of a Hilbert space H is a linear subspace of H which is closed (i.e. M = M) with respect to the induced norm.
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