Spectral Theory Notes for discussion on Spring 2020 31.3.2020 The next Zoom session of the course will start on Tuesday March 31 at 14:15, in the same Zoom room as the previous one. (The link can be found in Moodle if you have lost the previous invitation.) Before the session, please study the following sections from the textbook Mathematical Methods in Quantum Mechanics With Applications to Schrödinger Operators by G. Teschl: Section 1.4, half page, up to and including the rst Example: The main topic is the denition of the (external) direct sum of Hilbert spaces, L , also called the orthog- i Hi onal sum of Hilbert spaces. Section 2.2, 4 pages, until but excluding Example (Dierential operator): The main topic is introduction of Multiplication operators in the rst Example. This also serves as an introduction to unbounded operators on a Hilbert space, as linear maps A : Dom(A) !H, and to the denition of the adjoint of a densely dened unbounded operator (denition of A∗ if Dom(A) = H) and, consequently, the concept of self-adjoint unbounded operators (A∗ = A and Dom(A) = H). We also need the denition of S + T and ST on their natural domains for unbounded operators S, T . Section 2.4, two new items: Essential range of a measurable function and its relation to the corresponding multiplication operator (Example (multiplication operator)). Spectrum of an inverse (Lemma 2.17). Section 2.5, excluding statements about closed operators and closures: For the spectral decomposition theorem, we will need direct (orthogonal) sums of op- erators dened on a direct sum of Hilbert spaces in Section 1.4 above. Motivations for the selection above: • The spectral decomposition theorem of self-adjoint operators (bounded or unbounded) A on H can be loosely summarized as follows: There exists a collection of Hilbert spaces , , and a unitary map L which transforms the operator into a Hi i 2 I U : H! i2I Hi A direct sum of multiplication operators on , i.e., L . This version Ai Hi UA = i2I Ai U gives the appropriate generalization of the matrix result that real symmetric matrices can be diagonalized by orthogonal matrices (and complex Hermitian matrices can be diagonalized by unitary matrices). Compare this also to the statement in Theorem 5.4.1 in the textbook by Davies. • Why include unbounded operators but not their whole theory? There is very little dierence in the proof of the spectral decomposition theorem between bounded and unbounded operators. In addition, most unbounded operators in applications arise in this manner from multiplication operators, via a suitable unitary map. This holds also for dierential operators (using Fourier transforms and leading to Sobolev space theory) but these topics would need their own course to be done properly. Hence, we are skipping all examples related to closed operators, closures and dierential operators in the textbook by Teschl. Below is a summary of denitions for bounded and unbounded operators on a Hilbert space H. Let us rst x some related terminology for this course: there are variations on the related notations and terms, so please keep an open mind when reading other sources. In particular, below you can nd a comparison between our notations and those used in Teschl's textbook. Scalar product: We use hf; gi which is conjugate linear in the second argument (in g). In Teschl's textbook, the scalar product is conjugate linear in the rst argument (in f). ∗ Therefore, hf; giTeschl = hg; fi = hf; gi . 1 Subspace operator: A is a subspace operator on H if its domain Dom(A) and image Image(A) are linear subspaces of H, and A : Dom(A) ! Image(A) is a linear map. In Teschl's textbook, one denotes D(A) = Dom(A). Range: The range of a subspace operator A is Ran(A) := fAf j f 2 Dom(A)g, and it is a linear subspace of Image(A), hence also of H. Operator: A subspace operator A is called an operator if Image(A) = H. (Sometimes operator is used as a shorthand for bounded operator, as dened below.) In Teschl one assumes that an operator is always densely dened (see below). Boundedness: A subspace operator A is called bounded if kAk := sup fkAfk j f 2 Dom(A) ; kfk ≤ 1g < 1 : A subspace operator is a continuous map (in the norm topologies inherited from H) if and only if it is bounded. Unbounded operator: Consistently with the terminology so far: A is an unbounded op- erator on H if it is a subspace operator for which Image(A) = H and kAk = 1. Densely dened: A subspace operator A is densely dened if Dom(A) = H. (Closure taken in the norm topology of H.) Bounded operator: A is a a bounded operator on H if it is a subspace operator for which Dom(A) = H = Image(A) and kAk < 1. (Note that for a bounded subspace operator Dom(A) or Image(A) could both be proper subspaces of H.) We denote the collection of all bounded operators by L (H), in Teschl's textbook this is denoted by L(H). Extension of operators: Suppose A and B are both operators on H, i.e., both are subspace operators with Image(A) = H = Image(B). We say that A is extended by B, or equivalently that B extends A, if Dom(A) ⊂ Dom(B) and Bf = Af for all f 2 Dom(A). This is denoted by A ⊂ B, and in Teschl's textbook by A ⊆ B. Natural domain of a sum of operators: Suppose A and B are both operators on H. The natural domain of A + B is Dom(A + B) := Dom(A) \ Dom(B) since this is the largest collection of f 2 H for which both Af and Bf are dened. Note that Dom(A + B) is a linear subspace and setting (A + B)(f) = Af + Bf denes then an operator A + B. Natural domain of a product of operators: Suppose A and B are both operators on H. The natural domain of AB is Dom(AB) := ff 2 Dom(B) j Bf 2 Dom(A)g ⊂ Dom(B) since this is the largest collection of f 2 H for which both Bf and A(Bf) are dened. Note that Dom(AB) is a linear subspace and setting (AB)(f) = A(Bf) denes an operator AB. Note the following consequences of the denitions: • An unbounded operator cannot have bounded operator extensions. • A bounded subspace operator A always has a unique bounded extension into a subspace operator B with Dom(B) = Dom(A) and Image(B) = Image(A). However, if A is not densely dened, A might have operator extensions which are unbounded. • If A; B 2 L (H), then the above natural domain denitions yield the same bounded operators A + B and AB 2 L (H) as we have used up to now. The following properties are not needed (yet), but let me include them here if you wish to read also some of the skipped parts of the textbook. 2 Graph of an operator: If A is an operator, then its graph is the following subset G(A) := f(f; Af) j f 2 Dom(A)g of the Hilbert space H ⊕ H which is equal to the product set with a scalar product . In particular, H×H h(f1; g1); (f2; g2)iH⊕H = hf1; f2iH+hg1; g2iH is here endowed with the norm p 2 2 . In Teschl's H ⊕ H k(f; g)kH⊕H = kfkH + kgkH textbook the graph is denoted by Γ(A). Closed operator: An operator A is closed if its graph G(A) is a closed subset of H ⊕ H. Note the following consequences of the denitions: • The graph of an operator is always a linear subspace of H ⊕ H. • If A 2 L (H), then A is closed. (General topological property, using the fact that then A is continuous, H × H = Dom(A) × Image(A), and checking that the topology of H ⊕ H agrees with the product topology.) • Conversely, by the closed graph theorem, if A is a closed subspace operator with Dom(A) = H = Image(A), then A is bounded, and thus A 2 L (H). Section 1.4 Explanation of the Example: Take and for each set which is a one- J = N j 2 N Hj = C dimensional Hilbert space such that each g 2 Hj has the norm kgkj := jgj. Dene further L . Then if , we have where for all , and H := Hj f 2 H f = (fj)j2N fj 2 C j j2N X X kfk2 = kf k2 = jf j2 = kfk2 : H j j j `2 j2N j2N n P 2 o 2 Thus H := f 2 N kfjk < 1 is isometrically isomorphic with ` ( ; ) (the scalar C j2N j N C products agree by the polarization identity). Section 2.2 Some comments to connect the results to the textbook by Davies: • The Axioms refer to an axiomatic version of quantum mechanics and can be ignored. • The quadratic form of an operator A, qA(f) := hAf; fi, f 2 Dom(A), appeared already in the Lemmata in Section 5.1 of Davies when A was a self-adjoint bounded operator. • To dene the adjoint A∗ of an unbounded operator A, it is crucial that it is densely dened (this is the main reason why Teschl includes the assumption in the denition of an operator), see the discussion below. • It should be stressed that, in general, (A∗)∗ is no longer automatically well dened: for this one needs to require that also Dom(A∗) is dense.