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Home , Densely defined operator, Unbounded operator

II - LECTURE 2ND SEMESTER 2017-18

The informations will be added correspondingly to the lectures.

Recommended literature Reinhold Meise and Dietmar Vogt, Introduction to Functional Analysis Walter Rudin, Functional Analysis Nelson Dunford and Jacob T. Schwartz, Linear operators Dirk Werner, Funktionalanalysis

1. Unbounded operators on Hilbert spaces H over a eld of complex numbers, if we do not write otherwise. Motivation: Theory of bounded, possibly compact, operators can be applied to some integral operators. E.g. the operator , where R t is a compact f ∈ C[0, 1] 7→ T f T f(t) = 0 f operator. The for normal (in particular for self-adjoint operators) can be applied to self-adjoint operators on Hilbert spaces. As an example may serve 2 e.g. integral operators on L2[0, 1] with kernel k ∈ L2([0, 1] ). Typical dierential operators are not bounded. This is one of reasons for the study of unbounded operators.

0 Example. For f ∈ W := {f ∈ AC([0, 1]) : f ∈ L2([0, 1])} ⊂ H := L2([0, 1]) we put T f = f 0. It is a linear mapping of the vector space W ⊂ H to H; W is a proper subspace H which is dense in H; T is not continuous (so not bounded) on W ; the graph T is however closed (W cannot be closed in such situation - due to the Theorem). 1.1. The notion of an . The mapping T : D(T ) → H is an operator on H if D(T ) is a vector space H ane T : D(T ) → H is linear. The vector space D(T ) is a domain of T and its denition is part of the denition of the operator T . Range R(T ) of the operator T on H is the vector space T (D(T )) ⊂ H. The graph of the operator T is denoted by G(T ). The kernel of the operator T is denoted by N(T ). An operator T is an extension of the operator S if G(S) ⊂ G(T ) (we also write S ⊂ T ). H × H endowed with the operations of summing and multiplying by a complex number (coordinate-wise) is a vector space H ⊕ H. We dene a scalar product on it by h(x1, x2), (y1, y2)i = hx1, y1i+hx2, y2i. The space H ⊕H with this scalar product is a Hilbert space and we denote it H ⊕2 H. The operator T is closed if its graph G(T ) is closed in H ⊕2 H. 1 2 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

Porposition (bonded operators). (a) An operator T is in L(H) if and only if it is closed and D(T ) = H. (b) An operator S na H is bounded on D(S) if and only if it has the unique extension T ∈ L(D(S),H) (resp. it has an extension T ∈ L(H)). Lemma 1.1.1 (on the graph of the operator). (a) A linear space G ⊂ H ⊕ H is the graph of an operator if and only if {(x, y) ∈ G : x = 0} = {(0, 0)}.

(b) T has a closed extension if and only if xn → 0 v D(T ) a T (xn) → y implies y = 0.

For operators S, T on H and c ∈ C we dene S + T , ST , cT in the standard way, where the domains are dened by D(S +T ) = D(S)∩D(T ), D(ST ) = D(T )∩ T −1(D(S)), D(cT ) = D(T ) pro c 6= 0. As an exercise verify thatfor operators R, S, T on H we have • (R + S) + T = R + (S + T ); • (RS)T = R(ST ); • (R + S)T = RT + ST a T (R + S) ⊃ TR + TS. Find an example in which the inclusion cannot be replaced by equality. Proposition 1.1.2 (closeness of sum and composition). If S is a closed operator on H and T ∈ L(H), then (a) S + T is closed; (b) ST is closed. Problem. Find S, T closed say on H = `2 so that S + T is not closed, or even it cannot be extended to a closed operator. Find S closed on H and T ∈ L(H) such that TS is not closed.

Corollary 1.1.3. For any T closed on H and λ ∈ C we get that λI − T is closed. 1.2. The notion of an adjoint operator. An operator T on H is densely dened if D(T ) = H. The notion of an adjoint operator T ∗ for a densely dened operator T : Lemma 1.2.1 (denition of an adjoint operator). Let T be densely dened on H. Then (a) D := {y ∈ H : x 7→ hT x, yi is continuous} is a vector space and (b) there is the unique T ∗ : D(T ∗) = D → H such that hT x, yi = hx, T ∗yi for all y ∈ D and x ∈ D(T ). If T is a densely dened operator on H, the the operator T ∗ from the preceding lemma is called the adjoint operator of T . The end of the 1st lecture. The proof of the lemma on the existence of an adjoint operator. Proposition 1.2.2 (on kernel and range). For every densely dened operator T we have N(T ∗) = R(T )⊥. In particular, N(T ∗) is closed. Proposition 1.2.3 (on the operations + and ◦). (a) If S + T is densely dened, then S∗ + T ∗ ⊂ (S + T )∗. If moreover T ∈ L(H), then S∗ + T ∗ = (S + T )∗. FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 3

(b) If T and TS are densely dened, then T ∗S∗ ⊂ (ST )∗. If moreover T ∈ L(H), then T ∗S∗ ⊂ (ST )∗. The end of the 2nd lecture.

Example 1 (exercises). There are operators R, S, T on some H such that T (R+S) is a proper extension of TR + TS.

Example 2 (exercises). The T on `2 dened by equalities D(T ) = c00 a T x = P∞ P∞ is densely dened, it is not bounded, and it ( n=1 xn, 0, 0,... ) = ( n=1 xn)e1 does not have any closed extension.

Example 3 (exercises). Let us consider H = L2[0, 1] (and C = C[0, 1]) and linear mappings T : W = {f ∈ AC[0, 1] : f 0 ∈ H} ⊂ C ⊂ H → H dened by T f = f 0. Here AC[0, 1] is the set of all absolutely continuous functions (in fact of the equivalence classes in L2[0, 1] which contain an absolutely countinuous ). (a) Then the operator is densely dened and closed operator on the T1 = T C1[0,1] C (where the needed notions are dened appropriately to those for Hilbert spaces); (b) the operator is densely dened, but it is not closed on and T2 = T C1[0,1] H (c) the operator T is densely dened and closed. The end of the 1st exercises Let us dene an auxiliaty mapping V : H × H → H × H by V (x, y) = (−y, x).

Lemma 1.2.4 (on V ). ∗ −1 (a) V is a on H ⊕2 H with V = V = −V ; in particular, for any linear subspace G of H ⊕ H we have V (G⊥) = V (G)⊥. (b) G(T ∗) = (V (G(T )))⊥ = V (G(T )⊥) if T is a densely dened operator on H.

∗ Example. If T is closed and densely dened, then G(T ) ⊕⊥ V (G(T )). It follows that for every (a, b) ∈ H × H, there is the unique (x, y) ∈ H × H such that −T x + x = a a x + T ∗y = b.

Theorem 1.2.5 (adjoint and closed operators). Let T be densely dened on H. Then • The operator T ∗ is closed. • T has a closed extension if and only if T ∗ is densely dened. • If the operator T with G(T ) = G(T ) is an operator, then T = T ∗∗. • T is closed if and only if T = T ∗∗ (implicitely T ∗ is densely dened).

The notion of an inverse operator of an injective operator T : D(T ) ⊂ H → R(T ) ⊂ H.

Proposition 1.2.6 (adjoint operator and T −1). If T is injective on H, then D(T ) and R(T ) are dense in H if and only if (T ∗)−1 = (T −1)∗.

Proposition 1.2.7 (adjoint operators and inclusion). If S ⊂ T a D(S) = H, then T ∗ ⊂ S∗. The end of the 3rd lecture. 4 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

1.3. Self-adjoint and symmetric operators. The notions of self-adjoint and symmetric operators. Remark. Self-adjoint operators are densely dened and closed. Proposition 1.3.1 (characterizations of self-adjoint and symmetric operators). Let T be densely dened on H. Then (a) T is symmetric if and only if T ⊂ T ∗. (b) T is self-adjoint, if it is symmetric and D(T ) = D(T ∗). (c) If T is self-adjoint, then it is a maximal symmetric operator (it does not have a proper symmetric extension). Remark. (c) cannot be reversed. We will show it later using the and the indeces of defect. Theorem 1.3.2 (adjoint and symmetric operators - further properties). Let T be symmetric and densely dened on H. (a) Then T is symmetric. (b) If T is dened everywhere, then it is bounded and self-adjoint. (c) If R(T ) is dense, then T is injective. (d) If T is self-adjoint and injective, then T −1 is self-adjoint. (e) If R(T ) = H, then T is self-adjoint, injective, and T −1 ∈ L(H). The end of the 4th lecture. Example 4 (exercises). Let and P , where D(T ) = D(R) T (ϕ) = n∈ ϕ(n)en {en : is an orthonormal basis of , is an operator on N . Then ∗ n ∈ N} H H = L2(R) D(T ) = {0} (and T ∗(0) = 0). P 2 2 Example 5 (exercises). Let H = `2, D(T ) = {x ∈ `2 : n |xn| < ∞}, and P n∈N P T (x) = T1(x) + T2(x), where T1(x) = xne1 and T2(x) = nxnen+1, is n∈N n∈N an operator on H. Here en(n) = 1 and en(k) = 0 if k 6= n. Then T and T2 are closed densely dened and T − T2 = T1 is not closed. Also ST is not closed, where S is the Sx = x1e1 dened on all H. The end of the 2nd exercises Proof of Theorem 1.3.2 1.4. Spectrum of unbounded operators. Notions the ρ(T ), spectrum σ(T ) and resolvent mapping RT : ρ(T ) → L(H) for an operator T on H. Proposition 1.4.1. (a) If T is closed on H, then (λI − T )−1 ∈ L(H) if and only if λI − T is bijective from D(T ) onto H. (b) If ρ(T ) 6= ∅, then T is closed. (c) σ(T ) is a closed subset of C. 0 0 Example. Show that T f = −if on L2(0, 1) with D(T ) = {f ∈ AC([0, 1]) : f ∈ is closed, densely dened, symmetric, and . L2(0, 1), f(0) = 0} σ(T ) = σp(T ) = C −1 −1 Show that T ∈ L(L2[0, 1]) and σ(T ) = {0}. Lemma 1.4.2 (empty spectrum and T −1). If a closed operator on H has empty spectrum, then T −1 ∈ L(H) and σ(T −1) = {0}. Lemma 1.4.3 (symmetric operators and λ∈ / R). Let S be symmetric on H and λ ∈ C \ R. Then FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 5

(a) λI − S is injective and open mapping of D(λI − S) = D(S) onto R(λI − S). (b) S is closed if and only if R(λI − S) is closed. The end of 5th lecture. Proof of Lemma 1.4.3. Theorem 1.4.4 (spectrum of self-adjoint operator). If S is a self-adjoint operator on H 6= {0}, then ∅= 6 σ(S) ⊂ R. The end of 6th lecture. Example 6 (exercises). The continuation of the example of closed operators T and T2 on `2 and a bounded operator S on `2 such that T + (−T2) = T1 and ST are not closable. Example 7 (exercises). Prove that the operator T f = f 0 on W = {f ∈ AC[0, 1] : 0 is closed (see a former exercises), R 1 is a bounded operator f ∈ L2[0, 1]} Sg = 0 g on L2[0, 1], and ST is not closable. 0 Example 8 (exercises). Tkf = if , k = 1, 2, 3, where D(T1) = {f ∈ AC([0, 1]) : 0 2 0 2 f ∈ L ([0, 1])}, D(T2) = {f ∈ AC([0, 1]) : f ∈ L ([0, 1]), f(0) = f(1)} and 0 2 . Show that ∗ , D(T3) = {f ∈ AC([0, 1]) : f ∈ L ([0, 1]), f(0) = f(1) = 0} T2 = T2 ∗ , ∗ . T1 = T3 T3 = T1 The end of 3rd exercise. Corollary 1.4.5 (characterization of self-adjoint operators). Let S be densely de- ned on H. The the following are equivalent: (a) S is self-adjoint; (b) S is symmetric and σ(S) ⊂ R; (c) S is symmetric and there is a λ ∈ C \ R such that λ, λ ∈ ρ(S). 1.5. Cayley's transformation. The idea of transformation of a self-adjoint ope- rator to a unitary one inspired by our knowledge of the function calcuus for bounded normal operators. Let us consider the function i+z which transforms boun- r(z) = −i−z ded with a real spectrum to a unitary operator with its spectrum in the unit circle of complex units. We extend this transformation to self-adjoint operators and use our knowledge of the spectral theory of normal, in particular unitary, operators. Notice that for and 1+z for and . |r(t)| = 1 t ∈ R t = i 1−z ∈ R z 6= 1 |z| = 1 Denition 1.5.1. Given a symmetric operator S on H we dene its Cayley trans- form U = C(S) = (iI − S)(−iI − S)−1. Remark. Due to Lemma 1.4.3 S + iI is injective for symmetric operator S. This ensures that the denition of the Caley's transform is correct. Lemma 1.5.2 (on with the graph of a symmetric operator). Let S be symmetric on H. Then for j ∈ {i, −i} (a) 2 2 2 , (i.e. is an kjx − Sx}H = kxkH + kSxkH x ∈ D(S) (x, y) ∈ G(S) 7→ jx − y isometry of G(S) ⊂ H ⊕2 H onto R(jI − S). (b) If R(jI − S) = H, the operator S is maximal symmetric. Theorem 1.5.3 (properties of C(S)). Let S be symmetric on H and U = C(S). Then 6 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

(a) U is a linear isometry of D(U) = R(−iI − S) onto R(U) = R(iI − S). (b) U is closed if and only if S is closed. The end of 7th lecture. Proof of Theorem 1.5.3.

Lemma 1.5.4 (on isometric operators). Let U be an operator on H which is an isometry of D(U) onto R(U). Then (a) hUx, Uyi = hx, yi for all x, y ∈ D(U). In particular, U is unitary, i.e. UU ∗ = U ∗U = I, if and only if D(U) = R(U) = H. (b) N(I − U) ⊂ (R(I − U))⊥. In particular, I − U is injective if R(I − U) is dense in H. (c) The spaces D(U) ⊂ H, R(U) ⊂ H and G(U) ⊂ H × H are closed if one of them is closed.

Theorem 1.5.5 (range of Cayley's transform). If U is a linear isometry of D(U) ⊂ H onto R(U) ⊂ H and I − U is injective, then U is the Caley transform of the symmetric operator S = i(I + U)(I − U)−1. The end of the 8th lecture. Solution of Example 8. The end of the 4th exercises.

Theorem 1.5.6 (self-adjoint operators and Caley's transform). If U is the Cayley transform of a symmetric operator S on H, then U is unitary if and only it S is self-adjoint. Corollary 1.5.7. Symmetric densely dened operator has a closed symmetric ex- tension. Denition of deciency indexes of a closed symmetric operator. The end of the 9th lecture.

Theorem 1.5.8 (deciency indexes and self-adjoint extensions). Let S a closed symmetric and densely dened operator on H. Then ∗ (a) S = S if and only if n+(S) = n−(S) = 0. (b) S is maximal symmetric if and only if n+(S) = 0 or n−(S) = 0. (c) S has a self-adjoint extension if and only if n+(S) = n−(S). Example (maximální symetrický nesamoadjungovaný operátor). Ukaºte, ºe exis- tuje symetrický operátor S, jehoº Cayleyova transformace je operátor pravého po- sunu U :(x1, x2,... ) 7→ (0, x1,... ) na `2. Jeho indexy defektu jsou 0 a 1. S je maximální symetrický hust¥ denovaný, uzav°ený, ale není samoadjungovaný. 1.6. The integral of unbounded function with respect to a spectral mea- sure. Let B(C) be the σ-algebra of all Borel subsets of C. B(C) denotes the set of all complex-valued Borel measurable functions of to ; denotes the set of C C Bb(C) bounded elements of B(C). The mapping P : B(C) → L(H) is a spectral measure (or "decomposition of the unity") if it maps Borel sets in C to orthogonal projections and satises P (C) = I and P S in for every at most countable family of j=1,... P (Aj)x = P ( j=1,... Aj)x H pairwise disjoint Borel subsets of C and x ∈ H. (It follows that P (∅) = 0 and P (A ∩ B) = P (A)P (B) = P (B)P (A).) FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 7

Remark. We needed just spectral measure with a bounded support for the spectral decomposition of bounded normal operators. Now we shall need spectral measures with suport in R. We denote again Pxy(B) = hP (B)x, yi and Px(B) = Px,x(B) = hP (B)x, xi for . is a non-negative measure with 2. B ∈ B(C) Px Px(B) ≤ kxk Lemma 1.6.1 (density of and approximation of in for by Dh,P h B(C) x ∈ Dh,P elements of in ). If is a spectral measure on with Bb(C) L2(Px) P : B(C) → L(H) C values in L(H) and h ∈ B(C), then (a) Z 2 Dh,P = {x ∈ H : |h| dPx < ∞} is a dense vector subspace of H and L2(Px) (b) h ∈ Bb(Ω) for every x ∈ Dh,P . The end of the 10th lecture.

0 1 Example 9. Consider the operator T4f = if on D(T4) = {f ∈ C [0, 1] : f(0) = . Show that is , ∗ ∗ , and make conclusions on the f(1) = 0} T4 T3 T4 = T3 = T1 properties of T4 from it. Example 10. Let T be a densely dened operator in H. Then (a) If T is symmetric, then T ⊂ T ∗∗ ⊂ T ∗ = T ∗∗∗. In particular, T ∗∗ = T is symmetric. (b) T is closed and symmetric if and only if T = T ∗∗ ⊂ T ∗. (c) T = T ∗ if and only if T = T ∗∗ = T ∗.

Example 11. Consider the operator Mha = ha, where (Ω, Σ, µ) is a measure space (e.g. (R, λ) or a set with the arithmetic measure on it), h :Ω → R be measurable and D(Mh) = {a ∈ L2(µ): ha ∈ L2(µ)}. Show that Mh is self-adjoint. The end of 5th exercises. Proof of Lemma 1.6.1.

Theorem 1.6.2 (integral R h dP of unbounded functions). For an L(H)-valued R spectral measure P and a Borel measurable h : C → C there is unique h dP such that R (a) D( h dP ) = Dh,P , R R (b) h( h dP )x, xi = h dPx pro x ∈ Dh,P , and (c) R for . k( h dP )xk = khkL2(Px) x ∈ Dh,P Theorem 1.6.3 (R f dE as a "∗-homomorphism", normality and closeness). If P : B(Ω) → L(H) is a spectral measure, then for f, g ∈ B(C) and the notation P hi(h) = R h dP we have (a) Φ(f) + Φ(g) ⊂ Φ(f + g) (equality holds if one of f, g is bounded). (b) Φ(f)Φ(g) ⊂ Φ(fg) and D(Φ(f)Φ(f)) = D(Φ(g) ∩ D(Φ(fg)) (equality holds if g is bounded). (c) Φ(f)∗ = Φ(f) and Φ(f)Φ(f)∗ = Φ(|f|2) = Φ(f)∗Φ(f). (d) Φ(f) is a closed operator. The end of the 11th lecture. The proof of Theorem 1.6.3. 8 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

The end of the 12th lecture.

Proof of the statement on the multiplication operator Mh formulated at the end of the previous exercises.

Example 12. Consider the operator Mha = ha, where (Ω, Σ, µ) is a measure space (e.g. (R, λ) or a set with the arithmetic measure on it), h :Ω → R be measurable and D(Mh) = {a ∈ L2(µ): ha ∈ L2(µ)}. Show that Mh is self-adjoint. Example 13. There is a (not densely dened) operator on `(Z) with nonsymmetric closure. Hints for its denition and ideas of proof.

The end of 6th exercises.

Theorem 1.6.4 (spectrum of R f dP ). If P : B(Ω) → L(H) is a spectral measure and h ∈ B(C), then Z [ open −1 σ( h dP ) = RP (h) := C \ {G ⊂ C : G ,P (h (G)) = 0}.

Moreover: (a) R −1 (b) λ ∈ σp( h dP ) ⇔ P (h (λ)) 6= 0. R R R −1 (c) λ ∈ σ( h dP )(= RP (h)) \ σp( h dP ) ⇒ (λI − h dP ) is densely dened, closed, and discontinuous.

1.7. Spectral decomposition of self-adjoint operators.

Lemma 1.7.1 (on image of a spectral measure). Let Q be an L(H)-valued spectral measure and ρ ∈ B(C). Then P (B) = Q(δ−1(B)) is an L(H)-valued spectral measure such that Z Z h dP = h ◦ ρ dQ for every h ∈ B(C). The end of 13th lecture. Proof of Lemma.

Theorem 1.7.2 (spectral decomposition of self-adjoint operators). Let S be a self- adjoint operator on a Hilbert space H. Then there is unique spectral measure P with values in L(H) such that S = R r dP (r). R Moreover, σ(S) = RP (id ) is the support of P . First part of the proof. The end of the 14th lecture.

Example 14. More detailed hints to example of a symmetric operator which does not have a closed symmetric extension.

Example 15. Two proofs of the fact that for a unitary operator. σ(U) ⊂ SC Proof of - conclusion. FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 9

2. Localy convex spaces - continuation 2.1. Krein-Milman theorem.

Denition 2.1.1. Let C ⊂ X be a convex subset of a vector space. Then x ∈ C is and (x ∈ ext C) if C \{x} is convex. Examples of balls in NLS's and their extreme points.

Theorem 2.1.2 (Krein-Milman). If K is a nonempty compact convex subset of a LCS X, then K = conv ext K. (In particular, nonempty convex compact set contains an extreme point.) Example of unit balls in reexive spaces and their extreme points. Denition of a face of K. Remark. • K is a face of K. • Nonempty intersection of faces is a face. •{x} ⊂ K is a face if and only if x ∈ ext K. Lemma 2.1.3 (Bauer's maximal principle). Let X be a LCS, K ⊂ X be no- nempty, compact, and convex. Then each x∗ ∈ X∗ (or even each convex lower semi-continuous x∗) attains its maximum on K at some x ∈ ext K. Proof of the Krein-Milman Theorem. The end of the 15th lecture.

2.2. Locally convex with the same dual. Example: (X, w)∗ = (X, k k)∗ and the weak w is the weakest locally convex topology with this property.

Denition 2.2.1. A locally convex topology τ on X is (X,X∗)-admissible if (X, τ)∗ = X∗. Theorem 2.2.2 (Mackey). Let be a LCS and be any ∗ - X = (X, τ) ν (X,Xτ ) admissible topology. Then A ⊂ X is τ-bounded if and only if A is ν-bounded. The end of the 16th lecture.

Example 16. The proof of the existence of an isometry U with I −U injective and I − U not injective. Example 17. Formulation of the statement about nonegative self-adjoint operators, their spectrum and square root of them. Part of the proof about the spectrum. Remark about the as the of the unit ball. the unit ball as the polar of the dual unit ball.

Denition 2.2.3. Let X be a LCS. Then µ = µ(X,X∗) is the if it is generated by the family of pseudonorms , being all absolutely convex pM0 M w∗-compact sets in X∗. Theorem 2.2.4 (Mackey-Arens). Let X be a LCS. Then µ = µ(X,X∗) is the strongest (X,X∗)-admissible topology. 10 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

2.3. bw∗-topology on NLS - completeness, theorems of Banach-Dieudonné and Krein-’mulyan.

∗ ∗ Denition 2.3.1. Let X be a NLS. Then F ⊂ X is bw -closed if F ∩ rBX∗ is w∗-closed for every r > 0. Remark on bw∗-open sets, and on sets A ∩ B for all bounded sets B and bw∗- closed or open sets A. Lemma 2.3.2. Let X be NLS. The sets A< = {x∗ ∈ X∗ : |x∗(a)| < 1 for all a ∈ for , , form a local base of the locally convex topoogy A} A = {an : n ∈ N} an → 0 bw∗. Beginning of the proof. The end of the 17th lecture. The conclusion of the proof of Lemma 1.

Theorem 2.3.3 (Banach-Dieudonné). Let X be a NLS. Then ε(X) = (X∗, bw∗)∗. (Since ∗ ∗ ∗, is complete if and only if ∗ is ∗ ∗ ∗ - ε(X) = (X , w ) X bw (X , (X )w∗ ) admissible.) The end of the 18th lecture. Example 18. The conclusion of the proof of the characterization of nonnegative self-adjoint operators in terms of their spectrum. The proof of the existence of a square root of a self-adjoint nonnegative operator. Example 19. Reminder of unitary equivalence of any bounded normal operator to a bounded multiplication operator on some L2(µ). Similar statement for unbounded operators will be deduced next time. The end of 9th exercises. Proof of the Banach-Dieudonné theorem.

Corollary 2.3.4 (Krein-’mulyan theorem). Let X be a Banach space and A ⊂ X∗ be convex. Then A is bw∗-closed if and only if A is w∗-closed. Corollary 2.3.5 (Banach-Dieudonné theorem). Let X be a Banach space, f ∈ ∗ #, and be ∗-continuous. Then for some . (X ) f BX∗ w f = ε(x) x ∈ X The end of the 19th lecture.

Corollary 2.3.6 (embedding BS to C(BX∗ )). Let X be a Banach space and J : ∗ ∗ , where . Then is an isometry of onto X → C(BX , w ) Jx = ε(x) BX∗ J X J(X) (known fact), J is a w∗-homeomorphism of X onto J(X) (easy follows from ∗ denitions), and J(X) is τp-closed in C(BX∗ , w ). Informal complements about completeness of TLS, equivalent formulations for LCS, Fréchet spaces, and connection to the Krein-’mulyan theorem.

Proposition 2.3.7 (characterization of dual operators). Let X,Y be Banach spaces and S ∈ L(X∗,Y ∗). The the following statements are equivalent: (a) S = T 0 for some T ∈ L(X,Y ). (b) is continuous from ∗ ∗ to ∗ ∗ . S (Y , wY ∗ ) (X , wX∗ ) (c) S0(ε(X)) ⊂ ε(Y ). FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 11

The end of the 20th lecture. Example 19. Essential part of the proof of unitary equivalence of (unbounded) self-adjoint operator to a multiplication operator on some L2(µ) space. We are deducing it from the knowledge of the counterpart concerning bounded normal operators. The end of 10th exercises.

∗ 2.4. Weak compactness. Known facts about weak compactness of BX and w - compactness of BX∗ for a Banach space X. Denition 2.4.1. Let K be a subset of a Hausdor topological space T . (1) The topological space K is compact if every open cover has a nite subcover. (2) The topological space K is countably compact if every open countable cover has a nite subcover. (3) The topological space K is sequentially compact if every has a subsequence converging in K. (4) K ⊂ T is relatively compact if S{F : F ∈ F} 6= ∅ in T for every centered familyF of subsets of K. (5) is relatively countably compact if every sequence ∞ in has K ⊂ T (xn)n=1 K a cluster point in , i.e. T in . x T n∈ {xn,... }= 6 ∅ T (6) is relatively countably compactN if every sequence ∞ in has K ⊂ T (xn)n=1 K a convergent subsequence in T . Some equivalent formulations. Relation of relative compactness dened as above and compactness of K in T (equivalence in regular spaces). The implications between these notions which hold in general. The equivalences which hold in metrizable spaces. Lemma 2.4.2 (Eberlein's theorem). Let K be a Hausdor compact space and A ⊂ (C(K), τp) = Cp(K) be relatively countably compact. Then A is relatively compact. The end of the 21st lecture. Lemma 2.4.3 (’mulyan's theorem). Let K be a Hausdor compact space, A ⊂ τp (C(K), τp) = Cp(K) be relatively countably compact, and f ∈ A in C(K). Then there is a sequence (fn) ⊂ A which converges to f in Cp(K). The end of the 22nd lecture. Example 19. The conclusion of the proof of the unitary equivalence of a self- adjoint operator to the operator of multiplication of L2(µ)-functions by measurable functions on L2(µ). The end of 11th exercises.

Theorem 2.4.4 (Eberlein-’mulyan for τp). Let K be a Hausdor compact space and A ⊂ Cp(K). Then the following statements are equivalent.

(a) A is relatively compact in Cp(K). (b) A is relatively countably compact in Cp(K). (c) A is relatively sequentially compact in Cp(K).

The following statements about (A, τp) are also equivelent. (a') A is compact. (b') A is countably compact in Cp(K). 12 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

(c') A is sequentially compact in Cp(K). Theorem 2.4.5 (Eberlein-’mulyan for w). Let X be a Banach space and A ⊂ X. Then the following statements are equivalent. (a) A is relatively weakly compact (A is compact in (X, w)). (b) A is relatively weakly countably compact (in (X, w)). (c) A is relatively weakly sequentially compact (in (X, w)).

The following statements about (A, τp) are also equivelent. (a') A is weakly compact. (b') A is weakly countably compact in (X, w). (c') A is weakly sequentially compact in (X, w).

Theorem 2.4.6 (Grothendieck on τp and w in C(K)). Let K be a Hausdor compact space and A ⊂ Cp(K) be norm-bounded in the Banach space C(K). Then (a) A is relatively weakly compact (A is relatively compact in (C(K), w)) if and only if A is relatively τp-compact. (b) A is weakly compact (A is compact in (C(K), w)) if and only if A is τp-compact. 2.5. Preservation of weak compactness.

Denition 2.5.1. Let X,Y be Banach spaces. A linear operator T : X → Y is weakly compact if T (BX ) is weakly compact. Remark 2.5.2. (a) The closure in the denition can be equivalently understood in the norm or in the . (b) If T is weakly compact, it is bounded. (c) If T is compact, it is weakly compact. (d) If T is bounded and X or Y is reexive, then T is weakly compact. Examples of weakly compact non-compact operators. Remark on the Schauder theorem on compactness of dual operators.

Theorem 2.5.3 (Gantmacher). Let X,Y be Banach spaces and T : X → Y be a bounded linear operator. Then the following statements are equivalent. (a) T is weakly compact. (b) T 0 is weakly compact. (c) T 00(X∗∗) ⊂ ε(Y ). The end of the 23rd lecture. Conclusion of the proof of Gantmacher's theorem. Remark on closed convex hulls of compact sets in nite-dimensional spaces and in Banach spaces.

Theorem 2.5.4 (Krein). Let X be a Banach space and K ⊂ X be weakly compact. Then the closed absolutely convex hull of K is weakly compact. The end of the 24th lecture. Example 20. (a) Unbounded of on [0, 1] which converge pointwise to zero give examples of τp relatively compact sets which are not weakly compact. (b) The set A = {f ∈ [0, 1]Γ : spt f nite} for Γ uncountable is dense and not closed in [0, 1]Γ but it is sequentially compact. FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18 13

Example 21 (a criterion for self-adjoint extendability). Let T be a densely dened operator on a Hilbert space H and hT x, xi ≥ Ckxk2 on D(T ) or hT x, xi ≤ Ckxk2 on D(T ). Then T has a self-adjoint extension. The end of 12th exercises.

2.6. Krein-Milman theorem and generalized convex combinations. We consider locally convex spaces over R only.

Denition 2.6.1. Let K 6= ∅ be a compact subset of a LCS X and µ ∈ M1(K) be a Radon probability measure on K. We say that x ∈ X is represented by µ if ∗ R ∗ ∗ for every ∗ ∗. We write and say that x (x) = K x dµ(=: µ(x )) x ∈ X x = b(µ) x is a barycenter of µ in such a case.

Remark. x = b(µ) for some µ ∈ M1(K) is a generalization of x is a convex combination of some nitely many elements of K. Proposition 2.6.2 (the mapping b). Let C be a compact convex subset of a LCS X, C is the closed convex hull of a compact K in X, and µ ∈ M1(K). Then (a) there is unique b(µ) ∈ C; (b) the mapping b : µ 7→ b(µ) is ane (it maps convex combinations of elements of M1(K) to convex combinations of their images); ∗ ∗ (c) the mapping b : M1(K) ⊂ (C(K) , w ) → C ⊂ (X, τX ) is continuous. Proposition 2.6.3 (closed convex hull of compact sets and generalized convex combinations). Let K be a compact subset of a LCS X. Then x is an element of the closed convex hull of K if and only if there is a µ ∈ M1(K) such that x = b(µ). The end of the 25th lecture.

Theorem 2.6.4 (corollary of Krein-Milman). Let C be a compact convex subset of a LCS X. Then C = {b(µ): µ ∈ M1(ext C)}. Lemma 2.6.5 (Bauer). Let C be a compact convex subset of a LCS X and x ∈ C. Then x ∈ ext C if and only if δx is the unique element of M1(C) such that x = b(µ). Theorem 2.6.6 (Milman). Let C be a compact convex subset of a LCS X. If C is the closed convex hull of a F ⊂ X, then ext C ⊂ F . (In other words, ext C is the smallest closed set such that C is its closed convex hull.)

2.7. Countable tightness of Cp(K).

Theorem 2.7.1 (Kaplansky). Let K be a compact Hausdor space,A ⊂ Cp(K), and f ∈ A. Then there is a countable subset D of A such that f ∈ D. Compare this theorem with ’mulyan's lemma above. The end of the 26th lecture.

Example 22 (criteria for existence of the self-adjointness closure). Let S be a symmmetric densely dened operator on a Hilbert space H. Then the following statements are equivalent. (a) The closure of S is self-adjoint. (b) N(S∗ + iI) = {0} and N(S∗ − iI) = {0} (c) R(S + iI) and R(S − iI) are dense in H. 14 FUNCTIONAL ANALYSIS II - LECTURE 2ND SEMESTER 2017-18

Example 23 (Laplace operator). Let on n n . T = −∆ D(T ) = S(R ) ⊂ L2(R ) Then T is symmetric (by the Green formulas), the closure of T is self-adjoint, D(T ∗∗) = W 2(Rn), and T ∗∗ = −∆ with the partial dened as the genralized derivatives in n (by the unitary equivalence of and with the L2(R ) T T multiplication operator on and its closure on 2 n , where is the Mx7→x2 S F(W (R )) F Fourier-Plancherel transform on n ). L2(R ) The end of the 13th exercises.