Dixmier traces S. Richard Spring Semester 2017 2 Contents 1 Hilbert space and linear operators 5 1.1 Hilbert space . 5 1.2 Bounded linear operators . 9 1.3 Special classes of bounded linear operators . 11 1.4 Unbounded, closed, and self-adjoint operators . 15 1.5 Resolvent and spectrum . 19 1.6 Positivity and polar decomposition . 21 2 Normed ideals of K (H) 25 2.1 Compact operators and the canonical expansion . 25 2.2 Eigenvalues and singular values . 27 2.3 Technical interlude . 30 2.4 Normed ideals of B(H)........................... 35 2.5 The Schatten ideals Jp .......................... 39 2.6 Usual trace . 41 3 The Dixmier trace 47 3.1 Invariant states . 47 3.2 Additional sequence spaces . 49 3.3 Dixmier's construction . 50 3.4 Generalizations of the Dixmier trace . 53 3.4.1 Extended limits . 54 3.4.2 Additional spaces on R+ ...................... 56 3.4.3 Dixmier traces . 57 4 Heat kernel and ζ-function 61 4.1 ζ-function residue . 61 4.2 The heat kernel functional . 64 5 Traces of pseudo-differential operators 67 5.1 Pseudo-differential operators on Rd .................... 67 5.2 Noncommutative residue . 74 5.3 Modulated operators . 75 5.4 Connes' trace theorem . 83 3 4 CONTENTS Chapter 1 Hilbert space and linear operators The purpose of this first chapter is to introduce (or recall) many standard definitions related to the study of operators on a Hilbert space. Its content is mainly based on the first two chapters of the book [Amr]. 1.1 Hilbert space Definition 1.1.1. A (complex) Hilbert space H is a vector space on C with a strictly positive scalar product (or inner product) which is complete for the associated norm1 and which admits a countable orthonormal basis. The scalar product is denoted by ⟨·; ·⟩ and the corresponding norm by k · k. In particular, note that for any f; g; h 2 H and α 2 C the following properties hold: (i) hf; gi = hg; fi, (ii) hf; g + αhi = hf; gi + αhf; hi, (iii) kfk2 = hf; fi ≥ 0, and kfk = 0 if and only if f = 0. Note that hg; fi means the complex conjugate of hg; fi. Note also that the linearity in the second argument in (ii) is a matter of convention, many authors define the linearity in the first argument. In (iii) the norm of f is defined in terms of the scalar product hf; fi. We emphasize that the scalar product can also be defined in terms of the norm of H, this is the content of the polarisation identity: 4hf; gi = kf + gk2 − kf − gk2 − ikf + igk2 + ikf − igk2: (1.1) From now on, the symbol H will always denote a Hilbert space. 1Recall that H is said to be complete if any Cauchy sequence in H has a limit in H. More precisely, ffngn2N ⊂ H is a Cauchy sequence if for any " > 0 there exists N 2 N such that kfn − fmk < " for any n; m ≥ N. Then H is complete if for any such sequence there exists f1 2 H such that limn!1 kfn − f1k = 0. 5 6 CHAPTER 1. HILBERT SPACE AND LINEAR OPERATORS P H Cd h i d 2 Cd Examples 1.1.2. (i) = with α; β = j=1 αj βj for any α; β , P H Z h i 2 Z (ii) = `2( ) with a; b = j2Z aj bj for any a; b `2( ), R H 2 Rd h i 2 2 Rd (iii) = L ( ) with f; g = Rd f(x)g(x)dx for any f; g L ( ). Let us recall some useful inequalities: For any f; g 2 H one has jhf; gij ≤ kfkkgk Schwarz inequality; (1.2) kf + gk ≤ kfk + kgk triangle inequality; (1.3) kf + gk2 ≤ 2kfk2 + 2kgk2; (1.4) kfk − kgk ≤ kf − gk: (1.5) The proof of these inequalities is standard and is left as a free exercise, see also [Amr, p. 3-4]. Let us also recall that f; g 2 H are said to be orthogonal if hf; gi = 0. Definition 1.1.3. A sequence ffngn2N ⊂ H is strongly convergent to f1 2 H if limn!1 kfn − f1k = 0, or is weakly convergent to f1 2 H if for any g 2 H one has limn!1hg; fn − f1i = 0. One writes s− limn!1 fn = f1 if the sequence is strongly convergent, and w− limn!1 fn = f1 if the sequence is weakly convergent. Clearly, a strongly convergent sequence is also weakly convergent. The converse is not true. Exercise 1.1.4. In the Hilbert space L2(R), exhibit a sequence which is weakly conver- gent but not strongly convergent. Lemma 1.1.5. Consider a sequence ffngn2N ⊂ H. One has s− lim fn = f1 () w− lim fn = f1 and lim kfnk = kf1k: n!1 n!1 n!1 Proof. Assume first that s− limn!1 fn = f1. By the Schwarz inequality one infers that for any g 2 H: jhg; fn − f1ij ≤ kfn − f1kkgk ! 0 as n ! 1; which means that w− limn!1 fn = f1. In addition, by (1.5) one also gets kfnk − kf1k ≤ kfn − f1k ! 0 as n ! 1; and thus limn!1 kfnk = kf1k. For the reverse implication, observe first that 2 2 2 kfn − f1k = kfnk + kf1k − hfn; f1i − hf1; fni: (1.6) If w− limn!1 fn = f1 and limn!1 kfnk = kf1k, then the right-hand side of (1.6) 2 2 2 2 converges to kf1k + kf1k − kf1k − kf1k = 0, so that s− limn!1 fn = f1. 1.1. HILBERT SPACE 7 Let us also note that if s− limn!1 fn = f1 and s− limn!1 gn = g1 then one has lim hfn; gni = hf1; g1i n!1 by a simple application of the Schwarz inequality. Exercise 1.1.6. Let fengn2N be an orthonormal basis of an infinite dimensional Hilbert space. Show that w− limn!1 en = 0, but that s− limn!1 en does not exist. Exercise 1.1.7. Show that the limit of a strong or a weak Cauchy sequence is unique. Show also that such a sequence is bounded, i.e. if ffngn2N denotes this Cauchy sequence, k k 1 then supn2N fn < . For the weak Cauchy sequence, the boundedness can be obtained from the follow- ing quite general result which will be useful later on. Its proof can be found in [Kat, Thm. III.1.29]. In the statement, Λ is simply a set. Theorem 1.1.8 (Uniform boundedness principle). Let f'λgλ2Λ be a family of contin- 2 uous maps 'λ : H! [0; 1) satisfying 'λ(f + g) ≤ 'λ(f) + 'λ(g) 8f; g 2 H: If the set f'λ(f)gλ2Λ ⊂ [0; 1) is bounded for any fixed f 2 H, then the family f'λgλ2Λ ≤ 2 H is uniformly bounded, i.e. there exists c > 0 such that supλ 'λ(f) c for any f with kfk = 1. In the next definition, we introduce the notion of subspace of a Hilbert space. Definition 1.1.9. A subspace M of a Hilbert space H is a linear subset of H, or more precisely 8f; g 2 M and α 2 C one has f + αg 2 M. The subspace M is closed if any Cauchy sequence in M converges strongly in M. Note that if M is closed, then M is a Hilbert space in itself, with the scalar product and norm inherited from H. Examples 1.1.10. (i) If f1; : : : ; fn 2 H, then Span(f1; : : : ; fn) is the closed vector space generated by the linear combinations of f1; : : : fn. Span(f1; : : : ; fn) is a closed subspace. (ii) If M is a subset of H, then M? := ff 2 H j hf; gi = 0; 8g 2 Mg (1.7) is a closed subspace of H. Exercise 1.1.11. Check that in the above example the set M? is a closed subspace of H. 2 'λ is continuous if 'λ(fn) ! 'λ(f1) whenever s− limn!1 fn = f1. 8 CHAPTER 1. HILBERT SPACE AND LINEAR OPERATORS Exercise 1.1.12. Check that a subspace M ⊂ H is dense in H if and only if M? = f0g. If M is a subset of H the closed subspace M? is called the orthocomplement of M in H. The following result is important in the setting of Hilbert spaces. Its proof is not complicated but a little bit lengthy, we thus refer to [Amr, Prop. 1.7]. Proposition 1.1.13 (Projection Theorem). Let M be a closed subspace of a Hilbert ? space H. Then, for any f 2 H there exist a unique f1 2 M and a unique f2 2 M such that f = f1 + f2. Let us close this section with the so-called Riesz Lemma. For that purpose, recall first that the dual H∗ of the Hilbert space H consists in the set of all bounded linear functionals on H, i.e. H∗ consists in all mappings ' : H! C satisfying for any f; g 2 H and α 2 C (i) '(f + αg) = '(f) + α'(g), (linearity) (ii) j'(f)j ≤ ckfk, (boundedness) where c is a constant independent of f. One then sets j'(f)j k'kH∗ := sup : 0=6 f2H kfk ∗ Clearly, if g 2 H, then g defines an element 'g of H by setting 'g(f) := hg; fi. Indeed 'g is linear and one has 1 1 k'gkH∗ := sup jhg; fij ≤ sup kgkkfk = kgk: 0=6 f2H kfk 0=6 f2H kfk k k ∗ k k 1 1 k k2 k k In fact, note that 'g H = g since kgk 'g(g) = kgk g = g .