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Operator Theory and Integral Equations 802660S Lecture Notes

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Operator Theory and Integral Equations 802660S Lecture Notes Operator theory and integral equations 802660S Lecture Notes Second printing Valery Serov University of Oulu 2012 Edited by Markus Harju Contents 1 Inner product spaces and Hilbert spaces 1 2 Symmetric operators in the Hilbert space 12 3 J. von Neumann’s spectral theorem 25 4 Spectrum of self-adjoint operators 38 5 Quadratic forms. Friedrichs extension. 54 6 Elliptic differential operators 58 7 Spectral function 67 8 Integral operators with weak singularities. Integral equations of the first and second kind. 71 9 Volterra and singular integral equations 81 10 Approximate methods 88 Index 98 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 a 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, j=1 X 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. We say that 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 { } ⊂ 0, α = β A ( 6 is some index set. 3) x := (x,x) is called the length of x H. k k ∈ k Exercise 1. pProve the Theorem of Pythagoras: If xj j=1, k N is an orthonormal system in an inner product space H, then { } ∈ k k 2 2 2 x = (x,xj) + x (x,xj)xj 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 j=1 X 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: p 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 · · p 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. · × → 3) The inner product (x,y): H H C is continuous. × → ∞ 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 that on C[a,b] the norm f = max f(x) k k x∈[a,b] | | is not induced by an inner product. 3 Exercise 7. Give an example of an inner product space which is not complete. 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| ∞ ( j=1 ) X The estimates x + y 2 2 x 2 + y 2 , λx 2 = λ 2 x 2 | j j| ≤ | j| | j| | j| | | | j| and 1 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 ∞ (x,y) := xjyj j=1 X 2 (k) ∞ 2 and prove that l (C) is complete. Suppose that x k=1 l (C) 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→∞ 4 This fact and l x(k) x(m) 2 <ε2, l N | j − j | ∈ j=1 X imply that l l (k) (m) 2 (k) 2 2 lim xj xj = xj xj ε m→∞ | − | | − | ≤ j=1 j=1 X X for all k n and l N. Therefore the sequence ≥ 0 ∈ l s := x(k) x 2, k n l | j − j| ≥ 0 j=1 X 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→∞ | − | ≤ j=1 j=1 X X 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 |Xα|≤k Z 5 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, for all x M . { ∈ ∈ } Remark. It is clear that M ⊥ is a linear subspace of H. Moreover, M M ⊥ = 0 since 0 M always. ∩ { } ∈ Definition.
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