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Hilbert Space Operators and Functional Integrals

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Hilbert Space Operators and Functional Integrals APPENDIX TO PART I Hilbert Space Operators and Functional Integrals Mathematical background material assumed throughout Part I is presented here. A basic knowledge of Hilbert space and self-adjoint operators through the spectral theorem is assumed; this material is summarized in the opening sections. The notions of trace class operators and nuclear spaces are devel­ oped, leading to Gaussian measures over the Schwartz space fl" of tempered distributions. Basic properties of measures on Y' are presented, including the Bochner theorem and the Feynman-Kac formula. A.I Bounded and Unbounded Operators on Hilbert Space A Hilbert space Yf is a vector space over the complex numbers having a positive definite inner product < , ), and which is complete in the corre­ sponding norm 11811 = <8,8)1/2. The inner product < , ) is a map from ff- x Yf to the scalars which is linear in its second argument and antilinear in its first argument. Positive definiteness is the statement 0::; <8,8) with equality only for 8 = o. Some useful elementary formulas relating the norm and inner product are: 118 + xl12 + 118 - xl12 = 211811 2 + 211xl12 (Parallelogram law), (A.U) <8, X) = H 118 + xl12 - 118 - xl12 (Polarization), (A. 1.2) + illx + Wll 2 - illx - i8112} 122 A.l Bounded and Unbounded Operators on Hilbert Space 123 <0, X) ~ 11011 Ilxll (Schwarz inequality), (A. 1.3) 110 + xii ~ 11011 + Ilxll (Triangle inequality), (A. 1.4) Completeness of Yf means that Cauchy sequences have limits. Thus if then there is a 0 E Yf for which On ~ O. To give examples and fix notation we list as Hilbert spaces 12 (square summable sequences), L2 (square summable functions), and en (complex n-space). Vectors 0 and X are perpendicular (orthogonal) if <0, X) = 0, also written as 0.1 X. A set of vectors {O;}7=1 is orthonormal if <0. 0.) = J.. = {O for i -# j, " J 'J 1 for i = j. Proposition A.I.}. Let Xl' ... , Xn be orthonormal and let X E Yf. Then (Bessel's inequality). A Hilbert space Yf is separable if it has a countable dense subset. An orthonormal sequence (or net) {eJ7=1' n ~ 00 is a basis provided the finite linear combinations Lf=1 Aiei> N < 00 are dense in Yf. The number of basis elements n is the dimension of Yf, and is independent of the basis {e i }7=1. Also Yf is separable if and only if it has a finite or denumerable basis. Most Hilbert spaces encountered in mathematics and physics are separable. The notion of a net generalizes that of a sequence, and is necessary only for limiting processes in nonseparable Hilbert spaces. Proposition A.1.2. Let {ei} be an orthonormal basis. If Ai is a sequence (or net) of scalars with L IAil2 < 00, then (J = LAieiEYf and every 8 E Yf is uniquely representable by such a series. Proposition A.l.3. Let 0 E Yf and let % be a closed linear subspace of Yf. There is a unique vector XO E % which minimizes the distance from % to Yf. For g £; Yf, let g.1 = {8: <X, 0) = °for all XE g}. The fundamental theorem of orthogonal decomposition of a Hilbert space is: Proposition A.l.4. Let % be a closed linear subspace of a Hilbert space Yf. Then each 8 E Yf can be uniquely written as 8 = 81 + 82 124 Appendix to Part I Hilbert Space Operators and Functional Integrals with 81 E %,82 E %~. Let [I' be any subset of Yf. Then [l'H is the smallest closed linear subspace of Yf containing [1'. In particular, % = %H. Let EE> denote a direct sum of Hilbert spaces. With the obvious identifica­ tions, the proposition states that Yf = % EE> %~. A linear operator T from a Hilbert space Yf to a Hilbert space % is bounded if for some constant c, IIT811 :::;; c11811, all 8EYf. Then II T II is defined to be the smallest such constant, so that IIT811 :::;; II TIl 11811. The property that Tis bounded is equivalent to Tbeing continuous in the topology defined by the norm on Yf. Elementary properties of the norm are II T + 2SI1 :::;; II Til + 12111SII, IITSII:::;; IITIIIISII· Now use the notation of Proposition A. 1.4 and define an operator E = E,AI' by the formula E8 = 81, From the uniqueness part of Proposition A1.4, E is linear. Because 81 J.. 82, 1181 II :::;; 11811 and so II E II :::;; 1. However, also by uniqueness of the decomposi­ tion, E81 = 81 so IIEII = 1, and E2 = E. E is called the orthogonal projection onto %. A linear operator mapping from a Hilbert space Yf to the scalars, C, is called a linear form. For any 8 E Yf, the map x ---+ (e, X> is a linear form, and it is continuous by the Schwarz inequality. The converse to this example is the Riesz representation theorem. Proposition A.I.5. Let f be a continuous linear form on a Hilbert space Yf. Then there is a unique 8 E Yf such that f(x) = (8, X>, The adjoint T* of a bounded operator T: Yf ---+ %, is defined by the formula (e, TX> = (T*8, X>· Note that T*: % ---+ Yf. Elementary properties of the adjoint of bounded operators are (ST)* = T*S*, A.1 Bounded and Unbounded Operators on Hilbert Space 125 (S + AT)* = S* + IT*, II TIl = II T*II = II T* Til 1/2, T** = T. An operator T is self-adjoint if T* = T. It is normal if T* T= TT*. It is isometric if II T811 = 11811 for all 8 E.Yt', which is equivalent to the relation T* T = I. Tis unitary if it is isometric and its range is the full Hilbert space, which is equivalent to T* T = TT* = I. Tis an orthogonal projection if and only if T= T* = T2. The spectral theorem states that a commuting family of normal operators can be simultaneously diagonalized. Theorem A.1.6. Let r be a commuting family of bounded normal operators on.Yt'. There is a measure space (X, dv) and a unitary map U: L 2(X, dv) -+.Yt' such that U* TU is a multiplication operator by a complex valued measurable function for each T E To In the theorem, eachf E Loo(X, dv) defines a bounded operator on L 2(X, dv) by the formula 8 -+ f8, where (f8)(x) = f(x)(J(x). The spectral theorem is often formulated in terms of spectral projections. Consider a single self-adjoint operator T and the corresponding real valued f ELoo(X, dv). Let e). be the characteristic function of the set {x:f(x) ~ A} and let E). = Ue).U*. Theorem A.l.7. In the above notation, T= SA dE). where the integral is a Riemann-Stieljes integral. Other types of convergence besides norm convergence are useful in the study of operators. Let 8n -+ 8 be a sequence of vectors in .Yt' and let An be a sequence of bounded operators. Then 8n -+ 8, weak convergence, if Also An -+ A, weak operator convergence, if <x, An 8) -+ <X, A8), all X, 8 E.Yt' and An -+ A, strong operator convergence, if For nonseparable Hilbert spaces, sequential convergence as above should be replaced by convergence through nets. An elementary but very useful fact is Proposition A.1.S. The unit sphere in .Yt' is compact in the weak topology. 126 Appendix to Part I Hilbert Space Operators and Functional Integrals A main tool in using the spectral theorem is the functional calculus. Theorem A.1.9. Let F be a bounded Borel measurable function of a real variable and let T be self-adjoint. We can define an operator F(T) so that T = f A dE). => F(T) = fF(A) dE)., U* TU = fELoo => U* F(T)U = F(f)ELoo. The mapping T --+ F(T) respects polynomial operations, composite functions and uniform limits of the function F. The most interesting operators are unbounded. However, bounded oper­ ators are used to study unbounded ones. Let an operator A be defined on a linear subspace £»(A) (called the domain of A) of a Hilbert space Yf, and let A map £»(A) into Yf. In general, £»(A) is not closed and A is not bounded, but we suppose that £»(A) is dense in Yf, as this occurs in most examples. The graph ~(A) is the linear subspace of Yf E8 Yf defined by ~(A) = {(e, Ae): eE£»(A)} s; £» E8 Yf. A is closed if ~(A) is closed. A is closable if ~(A)- is the graph of an operator A -. In this case and A - is called the closure of A. A is an extension of B, written B s; A, if ~(B) s; ~(A). Thus A-is an extension of A. A criterion for A to be closable is that if I/In E .0J(A). and I/In --+ 0 and also Al/ln --+ e E Yf, then e = o. The adjoint A * of an unbounded operator A is defined by the formula <A*I/I, () = <1/1, A(), (A. 1.5) provided (A. 1.6) for some constant c. By the Riesz representation theorem, (A. 1.5) defines A *1/1 uniquely, provided .0J(A) is dense. Then .0J(A *) is the set of all t/I E Yf for which (A. 1.6) holds. Proposition A.t.tO. A * is closed. A is closable if and only if A * is densely defined. In this case, A - = A **. Definition. A is symmetric if As; A*, A is essentially self-adjoint if A- = A*, or equivalently A* = A**, and A is self-adjoint if A = A*.
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