About Grupo the Mathematical de Investigación Foundation of Quantum Mechanics

About Grupo the Mathematical de Investigación Foundation of Quantum Mechanics M. Victoria Velasco Collado Departamento de Análisis Matemático Universidad de Granada (Spain) Operator Theory and The Principles of Quantum Mechanics CIMPA-MOROCCO research school, Meknès, September 8-17, 2014 Lecture nº 2 12-09-2014

About Grupo the Mathematical de Investigación Foundation of Quantum Mechanics Lecture 1: About the origins of the Quantum Mechanics Lecture 2: The mathematical foundations of Quantum Mechanics. Lecture 3: About the future of Quantum Mechanics. Some problems and challenges Lecture 2: The mathematical foundations of Quantum Mechanics Hilbert spaces: orthogonality, summable families, Fourier expansion, the prototype of Hilbert space, Hilbert basis, the Riesz-Fréchet theorem, the weak topology. Operators on Hilbert spaces: The adjoint operator, the spectral equation, the spectrum, compact operators on Hilbert spaces. CIMPA-MOROCCO MEKNÈS, September 2014 Operator Theory and The Principles of Quantum Mechanics

From Quantum Mechanics to Functional Analysis Definition: A normed space is a (real or complex) linear space X equipped with a norm, i. e. a function : X R satisfying i) x = 0 x = 0 (separates points) ii) αx = α x (absolute homogeneity) iii) x + y x + y triangle inequality (or subadditivity). Definition: A Banach space is a complete normed space. Definition: A Hilbert space is a inner product Banach space H. That is a Banach space H whose norm is given by x = x, x, for every x H, where, is an inner product, i. e. a mapping, : H H K such that: i) x, x = 0 x = 0 ii) x, x 0 iii) αx + βy, z = α x, z + β y, z iv) x, y = y, x Note: If H is a real space, then the conjugation is the identity map.

Hilbert spaces Examples of Banach spaces and Hilbert spaces: a) R, C (Hilbert) b) R n, C n (Hilbert) Espacios de Banach c) Sequences spaces l p (Hilbert si p = 2) Espacios de Hilbert d) Spaces of continuous functions C,a, be) Spaces of integrable functions L p,a, b- (Hilbert if p = 2) f) Matrices M n n (Hilbert) g) Spaces of bounded linear operators: L X, Y, L(H) Theorem (Cauchy-Schawarz inequality): If H is a linear space equipped with a inner product, then v u, v u, u v, v (u, v H) From the Cauchy-Schawarz inequality we obtain straightforwardly the Minkowsky inequality: u + v, u + v u, u + v, v u, v H Theorem: If H is a linear space equipped with a inner product normed space with the norm u = u, u (u H).,, then H is a

Hilbert spaces An inner product space is linear space H equipped with an inner product,. Let us rewrite Cauchy-Schwarz inequality in terms of the associated norm: Fact: If H is a inner product space, then u, v u v (u, v H) Cauchy-Schwarz Corollary: If H is a inner product space, then, is continuous. This means that if u = lim u n and v = lim v n, then u, v = lim u n, v n. I fact, in terms of the associated norm, the inner product is given by: Polarization identities: Let H be a inner product space over K If K = R, then u, v = 1 4 ( u + v 2 u v 2 ). If K = C, then u, v = 1 4 ( u + v 2 u v 2 )+ i 4 ( u + iv 2 u iv 2 ). Consequently, if H is a inner product space over K (= R or C) then Parallelogram identity: u + v 2 + u v 2 = 2( u 2 + v 2 ).

Hilbert spaces Question: Given a normed space H, how v to know if H is an inner product space? If this is the case, then such a norm needs to satisfy the paralelogram indetity Theorem (Parallelogram theorem): If (H, ) is a normed space then is given by an inner product if and only of if, for every v u, v H we have that u + v 2 + u v 2 = 2( u 2 + v 2 ) (Parallelogram identity) Therefore we know, for instance that: l p is Hilbert space p = 2. Theorem: Any inner product space may be completed to a Hilbert space. v u u v Proof (sketch): If u = lim u n and v = lim v n (for u, v H) then define u, v : = lim u n, v n. It happens that this not depend of the choice of the sequences. Moreover, u = lim u n = lim u n, u n = u, u If follows that H is Hilbert space from the parallelogram identity.

Hilbert spaces: orthogonality Law of cosines (Euclides): a 2 = b 2 + c 2 2bc cosθ b a θ Consequently: c cosθ = u 2 + v 2 u v 2 2 u v = u,u + v,v u v,u v 2 u v = 2 u,v 2 u v = u u, v v v u v Fact : u v cosθ=0 u, v =0 θ v Definition: Let H be an inner space. It is said that u, v H are orthogonal vectors if u, v =0. If this is the case, then we write u v. Definition: Let H be an inner space. The orthogonal set of S H is defined by S u H: u s, s S. Proposition: If H is an inner space, and if S, W H then: i) 0 S iv) S W W S ii) 0 S S S = 0 v) S is a closed linear subspace of H iii) *0+ = H; H = 0 vi) S S

Hilbert spaces: orthogonality Definition: Let X be a linear space, and let Y and Z be linear subspaces. Then: X = Y Z X = Y + Z ; Y Z = 0. (Direct sum) Let X be a normed space such that X = Y Z. Let x = y + z and x n = y n + z n in Y Z. We hope that : x n x y n x y z n z. If this is the case, then we say that X = Y Z is a topological direct sum. Fact: If X is a Banach space then, X = Y Z is a topological direct sum the subspaces Y and Z are closed Theorem (best approximation): Let H be a Hilbert space, u H, and M a closed subspace of H. Then, there exists a unique m M such that u m = d u, M. Notation: P M u = m M: u m = d u, M = best approximation from u to M. Orthogonal projection theorem: Let H be Hilbert space and M a closed subspace. Then, H = M M. (Topological direct sum) Morever if π M : H M is the canonical projection then π M = P M. Also π M = 1 and u = π M (u) + u π M (u) = P M (u) + u P M (u). Remmark: Results also know as Hilbert projection theorem

Hilbert spaces: orthogonality Fact: Every closed subspace of a Hilbert space admits a topological complement. This is a very important property. Indeed it characterizes the Hilbert spaces Theorem (Lindestrauss-Tzafriri, 1971): Assume that every closed subspace of a Banach space X is complemented. Then X is isomophic to a Hilbert space. To be complemented means to admit a topological complement.

Hilbert spaces: orthogonality Definition: Let *e i + i I be a family of nonzero vectors of H. We say that *e i + i I is a orthogonal family if e i, e j = 0 i j. If in addition e i = 1, i I, then we say that *e i + i I is a ortonormal family. Proposición: Every orthogonal family of vectors in H is linearly independent. The Gram Schmidt process is a method for orthonormalising a set of vectors in an inner product space. It applies to a linearly independent countably infinite (or finite) family of vectors in a inner space. Proposition (Gram Schmidt process): Let *v n + n N be a linearly independent family of vectors in a inner space H. Then the family *u n + n N given by u 1 = v 1 n 1 u j, v n u n = v n u j u j, u j is a orthogonal family in H. Morever the family *e n + n N given by e n = u n u n j=1 is orthonormal.

Hilbert spaces : summable families Definition: A family *x i + i I in a normed space X is summable if there exists x X such that ε > 0 there exists a finite set J ε I, such that if J I is finite and J ε J then i J x i x < ε. If this is the case we write x = i I x i. Proposition: A family of positive real numbers *x i + i I is summable if and only if the set * : J I ; J finito+ is bounded. If this is the case then, i J x i x = i I x i = sup* x i : J I ; J finito+. i J Definition: A family *x i + i I in a normed space X satisfies the Cauchy condition if ε > 0 there exists a finite J ε I, such that if J I is finite, and if J J ε = then, < ε. i J x i Theorem: Let X be a Banach space and *x i + i I a family in X. Then, i) *x i + i I is summable *x i + i I satisfies the Cauchy condition. ii) *x i + i I summable i I: x i 0 is countable. Remark: Apply the Cauchy condition with ε = 1 n. Then we obtain J1 n such that i J x i < 1 if J J n ε =. Hence, I 0 J1 n then, x i < 1, n, so that x n i = 0. is a countable set. Moreover if i I I 0

Hilbert spaces : summable families Theorem: Let *e i + i I be a orthogonal family *e i + i I in a Hilbert space H. Then, *e i + i I summable (in H) * e 2 i + i I is summable en R, in whose case: e 2 2 i I i = i I e i. This result addresses the problem of the summability in H to the problem of the summability in R. On the other hand, it generalizes the Pythagoras's theorem. Remark: The canonical base of R 3 is orthogonal. Moreover if x R 3 is such that i=3 x = (x 1, x 2, x 3 ), that is x = i=1 x i e i, then The idea is to generalize this fact to an arbitrary H to have coordinates. Remark: Let *e i + i I be a othonormal family in a Hilbert space H. Then *f(e i )e i + i I summable in H * f(e i ) 2 + i I es summable in R. If this is the case, then: x = f(e i )e 2 i I i = i I f(e i ) 2. x, e i e i Particularly: * x, e i e i + i I summable in H * x, e i 2 + i I summable R. i=3 i=1

Hilbert spaces : summable families When * x, e 2 i + i I is summable in R? When is x = i I x, e i e i? Let J I, such that J is finite. Then 0 x x, e i e i i J 2 = x x, e i. e i, x x, e i e i i J i J = = x 2 x, e i 2 i J Therefore, x, e 2 i J i Consequently x 2 for every J I, with J finite. u, e 2 i I i = sup* u, e 2 i J i : J I ; J finite+ x 2 This means that * x, e i 2 + i I is summable in R and hence we have the following: Theorem (Bessel's inequality): If *e i + i I is a orthormal family in a Hilbert space H, then for every u H, the family i I u, e i e i is summable. Moreover u, e i e i u i I

Hilbert spaces : summable families Corollary (characterization of the maximal ortonormal families): Let *e i + i I be an orthonormal system in a Hilbert space H. The following conditions are equivalent: i) u = u, e i e i i I ii) u, v = u, e i e i, v i I iii) u = u, e i 2 i I (Parseval s identy) iv) *e i + i I is a maximal orthonormal maximal v) u e i i u = 0 vi) H = lin e i : i I.. Proof: TPO+Bessel: u H = lin e i : i I u = The other assertions are easy to prove. i I u, e i e i. Fact: By Zorn s lemma there exist maximal ortonormal basis. Definition: A Hilbert basis (or an orthonormal basis) in H is a maximal orthonormal family of vectors in H. Example: *e n + n N is an orthonormal family in l 2 = *u = α n n N α n 2 < + such that u e n n u = 0. Therefore B *e n : n N+ is a Hilbert basis of l 2. Note that B is not a Hamel basis. Inndeed the linear span of B is c 00 (sequences with finite support). Similarly with l 2 (I) where l 2 = l 2 (N).

Hilbert spaces : Fourier expansion Theorem: Every Hilbert space H has an orthonormal basis. Moreover all the orthonormal bases of H have have the same cardinal. Definition: The Hilbert space dimension of a Hilbert space H is the cardinality of an orthonormal basis of H. Remark: Denote by dimh the algebraic dimension of H. If dimh <, then B = e 1,, e n is a Hilbert basis B is a Hamel basis. Indeed, H = lin*e 1,, e n + = lin*e 1,, e n +. If dimh =, and if B = *e i + i I is a Hilbert basis then we have H = lin*e i : i I+. But *e i + i I do not need to be a spanning set of H. Therefore Hilbert basis Hamel basis Definition: Let H be a Hilbert space and *e i + i I a Hilbert basis. For every u H, u = i I u, e i e i (Fourier expansion) Fact: The Fourier coefficients * u, e i + i I are the unique family of scalars *α i + i I such that u = i I α i e i vector coordinates in -dim!!

Hilbert spaces : Fourier expansion Example: Let be the Lebesgue σ-algebra of π, π and let m be the Lebesgue measure. Let μ = m if K = R, and μ = m if K = C. Then π 2π The family e int : n Z is a Hilbert basis of L 2 ( π, π,, μ) C. It is known as the trigonometric system. Therefore, for f L 2 π, π we have f t = n Z f (n) e int (Fourier expansion) convergence in 2, where f (n):= 1 π f(t)e int dt 2π π Similarly, the family 1, cos nt, sin(nt) : n N is a Hilbert basis of 2 L 2 ( π, π,, μ) R. Therefore, for f L 2 π, π we have f(t) = a 0 + (a 2 n=1 ncos nt + b n sen nt ) (Fourier expansion) Convergence in 2, where: a 0 = 1 π π π f(t) dt ; a n = 1 π π f(t) cos nt dt π ; b n = 1 π f(t) sen nt dt π π

Hilbert spaces : The prototype Fact: If H is a Hilbert space and B *e i : i S+ is a Hilbert basis then, every u H has a unique expansion given by u = i I u, e i e i and u = u, e 2 i I i. Consequently, the mapping u u, e i i I defines an isometric isomorphism from H onto l 2 (I). Therefore, l 2 (I) becomes the Hilbert space prototype. Example: The mapping L 2 π, π C l 2 (Z), given by f f (n) n Z isometric isomorphism. defines an For this reason the Hilbert spaces have coordinates, as well as the euclidean space (which is also a Hilbert space) has its own coordinates. Theorem: Two Hilbert spaces are isometrically isomorphic, if and only if, they have the same Hilbert dimension. (that is that an orthonormal basis of one of this spaces has the same cardinality of an orthonormal basis of the other one).

Hilbert spaces: the Riesz-Fréchet theorem Let H be a Hilbert space over K. Who is H? This is to determine the set of continuous linear functionals f: H K. For general Banach spaces, this was open problem along more of 20 years. The answer was the Hahn-Banach theorem (a cornerstone theorem of the Functional Analysis). Fréchet, Maurice 1878-1973 If dim H = n then H = K n. Fix an orthonormal basis B = *e 1,, e n +. If f: H K is linear then, for u = α i e i, we have n n i=1 n f u = i=1 α i f(e i ) = u, v where v = i=1 f(e i )e i. Therefore f =, v. Conversely, f =, v is a continuous linear functional, obviously. Thus, H =, v : v H. If dimh = then,, v : v H H trivially. What F. Riesz and M. Fréchet showed (in independent papers published in Comptes Redus in 1907) is that, in fact,, v : v H = H. Note also that if f v : =, v, then f v = v.

Espacios de Hilbert: El teorema de Riesz-Fréchet Riesz-Fréchet theorem (1907): Let H be a Hilbert space and let f H. Then, there exists a unique v H such that f = f v =, v. Moreover, f = v. Proof: Let f H. If f = 0, then obviously f = f 0 =, 0. If f 0, then kerf is a proper closed subspace of H, so that kerf is nonzero. Therefore, by the OPT there exists ω kerf such that f(ω) 0. Now it is easy to check that v = f(ω) ω ω 2 is the vector that we look for, and the result follows. Corollary: If H is a Hilbert space, then there exists an isometric conjugate-linear bijection between H and H. Proof: Consider the map H H given by f v =, v v. Corollary: If H is a Hilbert space, then H is also a Hilbert space. Proof: Define f u, f v u, v and check it. Corollary: Every Hilbert space is reflexive (i. e. H = H ). Corollary: Any inner product space may be completed to a Hilbert space.

Hilbert spaces: The weak topology Corollary: If H is a Hilbert space and M is a subspace, then every bounded linear functional f: M K admits a bounded linear extension f: H K with f = f. Bounded linear functionals in a Hilbert space do exist in abundance, as we have shown. Consider the smallest topology that makes that all of them are continuous. This is the so-called weak topology (the strong topology is the norm topology). Unless that dim H <, the weak topology is not normable. Definition: A sequence u n, in a Hilbert H, converges weakly to u H (and we write u n ω u ) whenever un, v u, v, for every v H. From the continuity of the inner product we deduce the following result. Corollary: If in a Hilbert space, u n u u n ω u. The converse does not hold. For instance, in l 2 we have that e n, v 0 = 0, v, so that e n ω 0, meanwhile en e m = 2. Corollary: The (norm)-closed unit ball in a Hilbert space is weakly compact.

Operators in Hilbert spaces: The adjoint operator Let H 1 and H 2 be Hilbert spaces and let T L H 1, H 2. As a consequence of the Riesz-Fréchet theorem, there exists a unique bounded linear operator T : H 2 H 1 such that Tu, ω = u, T ω, (u H 1, ω H 2 ). Note that Therefore T ω 2 = T ω, T ω = TT ω, ω T T ω ω, T ω T ω. Consequently, T is continuous and T T. Definition: Let H 1 and H 2 be Hilbert spaces and let T: H 1 H 2 be a bounded linear operator. The adjoint operador of T is defined as the unique bounded linear operator T : H 2 H 1 such that Tu, ω = u, T ω, (u H 1, ω H 2 ). Fact: T = T. Indeed, if u H 2, and ω H 1, then: T u, ω = ω, T u = Tω, u = u, Tω = u, T ω. Since T ω is unique, we obtain that T ω = Tω.

Operators in Hilbert spaces: The adjoint operator Proposition: Let H 1 and H 2 be Hilbert spaces and let T, S L(H 1, H 2 ). Then: i) T = T ii) αt = αt (α K) iii) T + S = T + S Moreover, if H 3 is a Hilbert space and R L(H 2, H 3 ), then (RT) = T R. Since T = T and T T, we obtain that T = T. Indeed: Proposition: Let H 1 and H 2 be Hilbert spaces and let T L(H 1, H 2 ). Then, T = T = T T = TT (Gelfand-Naimark). Fact: The completeness is essential for the existence of the adjoint. Example: Let T: c 00 c 00 be the operador Tx = x = x 1, x 2.. x n, Tx = * n=1 n x n x n n e 1. That is n=1, 0, 0, +. Let y = *y n + with y 1 0. Let T y = z n. Then T y c 00.

Operators in Hilbert spaces: The adjoint operator From the above result we obtain straightforwardly the following proposition. Proposition: Let H 1 and H 2 be Hilbert spaces and let T L(H 1, H 2 ). Then, i) T is injective T has dense range. (Ker T = (Im T) ) ii) T is injective T has dense range. (Ker T = (ImT ) ) iii) T is biyective T is bijective, in whose case (T ) 1 = (T 1 ). Notation: L(H) = L(H, H). Examples: Let H be a Hilbert space and let B = *e i + i I be a Hilbert basis. i) If T L H is diagonal (i.e. Te i = λ i e i ), then T e i = λ i e i. ii) If T L H is the orthogonal projection on a closed subspace, then T = T. Examples: i) If, repect to the basis B, the matrix associated to T L l 2 is a ij, then the matrix associated to T is b ij = a ij. ii) If T L L 2 a, b is an integral operator with kernel k L 2 ( a, b a, b ), then T is an integral operator with kernel k t, s = k s, t.

Operators in Hilbert spaces: The adjoint operator Fact: From now on, we will consider operators from a Hilbert space into itself, that is the space L H : = L H, H of bounded linear operators T: H H. Recall that if T L H, then T is the unique operator L(H) such that Tu, ω = u, T ω, (u, v H). Definition: T L(H) is a self-adjoint operator if T = T. Proposición: Let S, T L H be selft-adjoint operators. Then, i) S + T is selft-adjoint. ii) ST selft-adjoint ST = TS. Proposition: Let T L H. If T = T, then H = ker T ImT. Proposition: Let H be a complex Hilbert space and let T L H. Then: T = T Tu, u R, for every u H. Moreover, if T is selft-adjoint, then T = sup Tu, u : u 1 = sup* Tu, u : u = 1+.

Operators in Hilbert spaces: The adjoint operator Proposition: Let H be a complex Hilbert space and let T L(H). Then, there exist self-adjoint operators R, S L H (which are unique) such that T = R + is. Proof: T = T+T 2 + i T T. 2i The unique self-adjoint operators R, S L H, such that T = R + is, are called the real part and the imaginary part of T, respectivelty. Definition: T L H is called normal if TT = T T. T L H self-adjoint T normal T L(H) diagonal T normal Proposition: Let T L H. Then T normal Tu = T u u H. Proposition: Let H be a complex Hilbert space. Then T L H is normal if, and only if, its real and imaginary parts commute. Proposition: If T L(H) is such that T 2 = T (i.e. T is a projection), then, T normal T is a orthogonal projection T = 1.

Operators in Hilbert spaces: The spectral equation Goal: To solve the spectral equation T λi x = y. Let T: C n C n be a linear operator. If A is the matrix associated to T respect to the canonical basis, then we like to solve the equation A λi x = y, where A is an n n matrix, I is the identity matrix, and x and y are the coordinates of the corresponding vectors (where y is known and x is not). If det A λi 0, then A λi is an invertible matrix and the given equation has a unique solution given by x = A λi 1 y. If det A λi = 0, then T λi is not injective, nor surjective. Therefore the system has a solution (not unique) if, and only if, y Im T λi. Thus, determining the set of all λ C such that T λi is not invertible is relevant: *λ C det A λi = 0+. Definition: T L H is invertible if there exists R L H such that TR = RT = I. If turns out that R is unique (whenever it exists). We denote R = T 1.

Operators in Hilbert spaces: The spectrum Definition: Let H be a complex Hilbert space. The spectrum of T L H is the set σ T = *λ C T λi is not invertible+ If T: C n C n is a linear operator, and if A is its matrix respecto to the canonical basis, then σ T = *λ C T λi is not invertible+ = σ p T = *λ C det A λi = 0+. In fact: T invertible T injective T surjective Thus, σ p T = *λ C det A λi = 0+ = *λ C T λi is not injective+. Definition: Let H be a complex Hilbert space. The pointwise spectrum of T L(H) is the set given by σ p T = *λ C T λi is not injective+. The elements of σ p T are called eigenvalues of T. If x ker T λi then x is an eigenvector of T associated to the eigenvalue λ. Fact: Let H be a finite-dimensional complex Hilbert space. Then, σ T = σ p T, for every T L H (i. e. the spectrum and the pointwise spectrum coincide).

Operators in Hilbert spaces: The spectrum Fact: If H is an infinite-dimensional complex Hilbert space, then σ p T and these sets do not need to coincide. σ T Example: If π: H M is the projection of H over a closed subspace M, then σ p T = 0,1. Moreover M is the invariant subspace associated to the eigenvalue 0 and M the one associated to the eigenvalue 1. Example: The Volterra operator T: L 2,0,1- L 2 0,1 is such that σ p T =. Fact: Let T L H. By the Banach isomorphism theorem we have that there exists T 1 L(H) such that TT 1 = T 1 T = I if, and only if, T is bijective. Definition: Let H be a complex Hilbert space. The surjective spectrum of T L H is defined by σ su T = *λ C T λi is not surjective+. Proposition: Let H be a complex Hilbert space. For every T L H we have σ T = σ p T σ su T.

Operators in Hilbert spaces: The spectrum Theorem: Let H be a complex Hilbert space, and let T L H. Then, σ T is a non-empty compact subset of C. Remark: If H is a real Hilbert space, then the set *λ R T λi is not bijective+ may be empty (in whose case, no information is provided). Example: T L C R given by T x = ix. In fact, (T λi)x = i λ x always is bijective because if λ R, then i λ 0. Definition: Let H be a real Hilbert space. The complexification of H is defined as the complex Hilbert space H C H ih. Moreover, the spectrum of T L(H) is defined as the spectrum of the operator T C L(H C ) given by T C u + iv = T u + it v. Theorem: Let H be a Hilbert space (either real or complex). If T L(H) then σ T is a non-empty compact subset of C. Agreement: From now on, only complex Hilbert spaces will be considered.

Operators in Hilbert spaces: The spectrum Proposition : If T L H then σ T = *λ λ σ T + (the conjugate set of σ T ). Proof: T λi invertible T λi R = R T λi = I R T λi = T λi R = I R T λ I = R T λ I = I T λ I invertible. Corollary: If T L H is self-adjoint then σ T R. Proposition: If T L H is normal then, λ σ(t) there exists x n with x n = 1 such that (T λi)x n 0. Note: This is equivalent to the following fact: λ C\σ(T) there exists c > 0 such that T λi x c x, x H. Corollary: If T L H is normal, then T = max λ : λ σ T. Proof: This is a consequence of the fact that if T L H is normal, then T = sup x =1 Tx, x (the numerical radius).

Compact operators on Hilbert spaces Let (Ω, Σ, μ) be a measure space and let k(s, t) L 2 (Ω Ω, Σ Σ, μ μ). Then, the integral operator with kernel k(s, t) is the operator T: L 2 (μ) L 2 (μ) given by Tx s = k s, t x t dμ(t) Ω (s Ω, x L 2 (μ)) In Quantum Mechanics the integral equation T λi x s integral operator with kernel k, is essential. = y s, where T is the Definition: Let X and Y be normed spaces. An operator T L X, Y is compact if, for every bounded sequence x n in X, the sequence Tx n has a convergent subsequence in Y. Example: The integral operator T: L 2 (μ) L 2 (μ) with kernel k is compact. Proposition: Let X and Y be normed spaces and let T L X, Y. Then, the following assertions are equivalent: i) T is compact, ii) For every sequence x n in X with x n = 1, the sequence Tx n has a convergent subsequence in Y, iii) T(B X ) is compact in Y (where B X denotes the closed unit ball of X).

Compact operators on Hilbert spaces Example: If T L X, Y is a finite rank opertor, then T is compact. If x n = 1, then Tx n is a bounded sequence in a finite dimensional normed space Y, and hence it has a convergent subsequence. Example: If H is an infinite dimensional Hilbert space, then the identity operator I: H H is not compact. In fact, if e n is an orthonormal sequence, then Ie n does not have a convergent subsequence. Notation: Let X and Y be normed spaces. Let denote K(X, Y)= *T L X, Y : T is compact+. Proposition: K X, Y is closed whenever Y is a Banach space. Proposition: Let X and Y be normed spaces and let T K X, Y. Then, i) T(X) is a separable subspace of Y. ii) If Y is a Hilbert space, and if B = *e n : n N+ is a Hilbert basis, then T = limπ n T where π n is the orthogonal projection on Lin e 1,, e n.

Compact operators on Hilbert spaces Notation: Let H and K be Hilbert spaces. Let denote F H, K = *T L H, K : T has finite rank+. Corollary (Jordan): Let H and K be Hilbert spaces. Then, K H, K = F(H, K). Note that the integral operator T with kernel k is the limit of the sequence of finiterank operators T n, where T n is the integral operators with kernel k n for a sequence k n of simple functions with k n k. Corollary (Theorem of the adjoint): Let H and K Hilbert spaces. Then, T K H, K T K H, K. Theorem: Let T L(H) be a compact operator and let λ 0. Then, i) dim ker T λ I < (Theorem of the kernel) (Riesz) ii) (T λi)(h) is closed (Theorem of the rank). Corollary: Let T L(H) a compact operators and let λ 0. Then, (T λi)(h) = H ker T λ I = *0+. Consequently, σ su (T)\*0+ = σ p (T)\*0+ = σ(t)\*0+. In finite dimension σ su (T) = σ p (T) = σ(t).

Compact operators on Hilbert spaces Theorem (Fredholm alternative): Let T L(H) be a compact operator and let λ 0. Consider the following equations: (a) T λ I x = y (b) T λ I x = y (c) T λ I x = 0 (d) T λ I x = 0 Then either i) the equations (a) and (b) has a solution x and x, for every y, y H, resp. ii) or the homogeneous system of equations (c) and (d) have a non-trivial solution. In the case (i), the solutions x and x are unique and depend continously on y and y respectively. In the case (ii), the equation (a) has a unique solution x if, and only if, y is orthogonal to all the solutions of (d). Similarly (b) has a unique solution x if, and only if, y is orthogonal to all the solutions of (a). Proof: If λ σ(t) then we have (i) obviously, because σ(t) = σ(t ). In fact, x = (T λi) 1 y meanwhile x = (T λi) 1 y. Otherwise, λ σ(t) *0+ = σ p (T) *0+, and hence (c) and (d) have a non-trivial solution. Moreover, (a) has a solution y T λ I H = ker T λ I. Similarly (b) has a solution y T λ I H = ker T λ I.

Compact operators on Hilbert spaces Corollary: If T L(H) is a self-adjoint compact operator, then T or T is an eigenvalue of T. Indeed it is known that if T is normal, then T = max λ : λ σ T. Remarks: i) Recall that if T is compact, then σ p (T)\*0+ = σ(t)\*0+. ii) The restriction of a self-adjoint compact operator to an invariant subspace is a self-adjoint compact operator. Diagonalization of a selft-adjoint compact operator Let T L H a self-adjoint compact operator (T 0). Let λ 1 σ p (T) such that λ 1 = T. Let u 1 with u 1 = 1 such that T λ 1 I u 1 = 0. Let H 1 = H and H 2 = u H 1 : u u 1 = u 1. If u H 2, then u, u 1 = 0 and hence, 0 = λ 1 u, u 1 = u, λ 1 u 1 = u, Tu 1 = T u, u 1 = λ 1 u, u 1 = Tu, u 1, Thus, T(H 2 )= H 2, so that T H2 is self-adjoint and compact.

Compact operators on Hilbert spaces Repeat the process with T H2. Let λ 2 σ p (T H2 ) σ p (T) such that λ 2 = T H2. Note that λ 2 λ 1 because T H2 T. Let H 3 = u H 2 : u u 2. Then T H2 is self-adjoint and compact and T H2 (H 3 ) = H 3. Note that H 2 = u H 1 : u u 1 = u 1 H 3 = u H 2 : u u 2 = u 1 u 2 = u 1, u 2 Reiterating the process we obtain nonzero eigenvalues λ 1, λ 2,, λ n such that λ n λ 2 λ 1, with unital eigenvectors u 1, u 2,, u n, and closed invariant subspaces H 1, H 2,, H n, where H j+1 = u H j : u u j is such that H n H 2 H 1 and λ j = T Hj. The process stops only when T Hn+1 = 0. Since H n+1 = u 1, u 2,, u n and H = u 1, u 2,, u n u 1, u 2,, u n (OPT) we n have: if u H, then u = i=1 u, u i u i + v with v H n+1. Thus, if T Hn+1 = 0 then (this is the case, particularly, if dimh< ). Tu = n i=1 λ i u, u i u i

Compact operators on Hilbert spaces If the process does not stop, then we obtain a sequence of eigenvalues λ n 0. 1 Indeed, if λ n λ > 0 then u λ n is bounded so that T( 1 u n λ n )= u n has a n covergent subsequence which contradicts that u n u m = 2. Hence, in this case, for every n N we have that n H = u 1, u 2,, u n u 1, u 2,, u n. If u = i=1 u, u i u i + v n with v n u 1, u 2,, u n = H n+1, then we obtain that Tv n = T Hn+1 v n T Hn+1 v n = λ n+1 v n λ n+1 u 0, And hence, Tu = i=1 λ i u, u i u i Note that if λ is a nonzero eigenvalue of T, then λ λ n : n N. Otherwise, if u is an associated eigenvalue then, λu = Tu = 0, which is impossible.

Compact operators on Hilbert spaces: spectral theorem Theorem (spectral theorem for compact self-adjoint operators): Let T L(H) be a compact self-adjoint operator. Then T is diagonalizable. Indeed, one of the following assertions holds: i) There exist eignevalues λ 1, λ 2,, λ n and a system of associated orthonormal eigenvectors u 1, u 2,, u n such that, for every u H, Tu = n i=1 λ i u, u i u i (uniform convergence over the compact subsets of H). ii) There exists a sequence λ n of eigenvalues such that λ n 0, and a sequence of associated orthonormal eigenvectors u n such that, u H, Tu = λ i u, u i u i i=1 (uniform convergence over the compact subsets of H). In (i) as well as in (ii), if λ is a non-zero eigenvalue, then λ λ 1, λ 2. Moreover, the dimension of the invariant subspace associated to λ coincides with the number of times that λ appears in λ 1, λ 2.

Compact operators on Hilbert spaces: spectral theorem Corollary: T L H is a compact self-adjoint operator T is diagonalizable, i.e. T = λ i, u i u i i for a countable family of real numbers λ 1, λ 2 and a orthonormal system u 1, u 2. Rearranging the above sum, fix k and denote P λk = λi =λ k, u i u i. Since the linear subspace generated by u i, with λ i = λ k, is precisely ker T λ k I, we obtain that P k is nothing but the projection of H over ker T λ k I. Theorem (spectral resolution of a compact self-adjoint operator): Let T L(H) be a compact self-adjoint operator. For every eignevalue λ let P λ be the orthogonal projection on ker T λi. Then the family λp λ λ σp (T) is summable in the Banach space L H, and T = λp λ. Remark: Now each λ appears only once. λ σ p (T) Moreover, for every λ σ p (T)\*0+, the corresponding projection P λ has finite rank and these projections are mutually orthogonal, i.e. if λ and μ are non equal eigenvalues, then P λ P μ = P μ P λ = 0.

Compact operators on Hilbert spaces: spectral theorem Recall that T L(H) is normal T = R + is where R and S are self-adjoint operators such that RS = SR. Theorem (spectral resolution of a compact normal operator): Let T L H be a compact normal operator. For every eigenvalue λ, let P λ be the orthogonal projection on the invariant subspace ker T λi. Then, the family λp λ λ σp (T) is summable in the Banach space L H, and T = λp λ. λ σ p (T) Moreover, for every λ σ p (T)\*0+, the corresponding projection P λ has finite rank, and for every non-equal eigenvalues λ, and μ, we have that P λ P μ =P μ P λ =0. Corollary: If T L(H) is a compact normal operator, then T is diagonalizable. In fact, T = λ i, u i u i where λ 1, λ 2 is the set of non-zero eigenvalues of T and orthonormal system of associated eigenvectors. i u 1, u 2 is an

Compact operators on Hilbert spaces: spectral theorem Corollary: Let T L(H) be a compact normal operator. If T = i λ i, u i u i then, for every y H and λ 0 we have that: i) If λ λ 1, λ 2, then the equation T λi x = y has a unique solution for every y H. This solution is given by x = 1 λ ( λ λ λ k k y,u k λ k λ u k y). ii) Otherwise, the equation T λi x = y has a solution y ker T λi. In this is the case, the general solution is given by x = 1 λ ( λ λ λ k k y,u k λ k λ u k y) + z (z ker T λi ). Proof: If T λi x = y, then Tx = λx + y, so that Tx, u k = λx + y, u k = λ x, u k + y, u k Since Tx, u k = i λ i x, u i u i, u k = λ k x, u k we obtain that λ x, u k + y, u k = λ k x, u k, and hence (λ k λ) x, u k = y, u k. Therefore if (λ k λ) 0, then x, u k = 1 λ k λ x = 1 λ (Tx y) = 1 λ ( λ i x, u i u i y) = 1 λ ( y, u k. Consequenly λ λ λ i i y,u i λ i λ u i y).