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Operator Theory on Hilbert Spaces

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Operator Theory on Hilbert Spaces Operator theory on Hilbert spaces S. Richard Spring Semester 2019 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space . .5 1.2 Vector-valued functions . .9 1.3 Bounded linear operators . 11 1.4 Special classes of bounded linear operators . 13 1.5 Operator-valued maps . 17 2 Unbounded operators 19 2.1 Unbounded, closed, and self-adjoint operators . 19 2.2 Resolvent and spectrum . 23 2.3 Perturbation theory for self-adjoint operators . 25 3 Examples 31 3.1 Multiplication and convolution operators . 31 3.1.1 The harmonic oscillator . 35 3.2 Schr¨odingeroperators . 35 3.2.1 The hydrogen atom . 36 3.3 The Weyl calculus . 37 1 3.4 Schr¨odingeroperators with x2 -potential . 39 3.4.1 Two families of Schr¨odingeroperators . 40 4 Spectral theory for self-adjoint operators 43 4.1 Stieltjes measures . 43 4.2 Spectral measures . 45 4.3 Spectral parts of a self-adjoint operator . 51 4.4 The resolvent near the spectrum . 55 5 Scattering theory 63 5.1 Evolution groups . 63 5.2 Wave operators . 67 5.3 Scattering operator and completeness . 73 3 4 CONTENTS 6 Commutator methods 77 6.1 Main result . 77 6.2 Regularity classes . 78 6.3 Affiliation . 82 6.4 Locally smooth operators . 86 6.5 Limiting absorption principle . 88 6.6 The method of differential inequalities . 90 Chapter 1 Hilbert space and bounded linear operators This chapter is mainly based on the first two chapters of the book [Amr]. Its content is quite standard and this theory can be seen as a special instance of bounded linear operators on more general Banach spaces. 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 h·; ·i 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) 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 BOUNDED LINEAR OPERATORS From now on, the symbol H will always denote a Hilbert space. d Pd d Examples 1.1.2. (i) H = C with hα; βi = j=1 αj βj for any α; β 2 C , (ii) H = l2( ) with ha; bi = P a b for any a; b 2 l2( ), Z j2Z j j Z (iii) H = L2( d) with hf; gi = R f(x)g(x)dx for any f; g 2 L2( d). R Rd R 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, then supn2N kfnk < 1. 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 is uniformly bounded, i.e. there exists c > 0 such that supλ 'λ(f) ≤ c for any f 2 H with kfk = 1. In the next definition, we introduce the notion of a linear manifold and of a subspace of a Hilbert space. Definition 1.1.9. A linear manifold 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. If M is closed (, any Cauchy sequence in M converges strongly in M), then M is called a subspace of H. Note that if M is closed, then M is a Hilbert space in itself, with the scalar product and norm inherited from H. Be aware that some authors call subspace what we have defined as a linear manifold, and call closed subspace what we simply call a subspace. Examples 1.1.10. (i) If f1; : : : ; fn 2 H, then Vect(f1; : : : ; fn) is the closed vector space generated by the linear combinations of f1; : : : fn. Vect(f1; : : : ; fn) is a sub- space. (ii) If M is a subset of H, then M? := ff 2 H j hf; gi = 0; 8g 2 Mg (1.7) is a subspace of H. 2 'λ is continuous if 'λ(fn) ! 'λ(f1) whenever s− limn!1 fn = f1. 8 CHAPTER 1. HILBERT SPACE AND BOUNDED LINEAR OPERATORS Exercise 1.1.11. Check that in the above example the set M? is a subspace of H. Exercise 1.1.12. Check that a linear manifold M ⊂ H is dense in H if and only if M? = f0g. If M is a subset of H the 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 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 : 06=f2H kfk ∗ Clearly, if g 2 H, then g defines an element 'g of H by setting 'g(f) := hg; fi.
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