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Math212a1403 the Spectral Theorem for Compact Self-Adjoint Operators

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Math212a1403 the Spectral Theorem for Compact Self-Adjoint Operators Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Math212a1403 The spectral theorem for compact self-adjoint operators. Shlomo Sternberg Sept. 9, 2014 Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. 1 Review of last lecture. 2 The Riesz representation theorem. 3 Bessel's inequality. 4 Self-adjoint transformations. Non-negative self-adjoint transformations. 5 Compact self-adjoint transformations. 6 The spectral theorem for compact self-adjoint operators. 7 Fourier's Fourier series. Proof by integration by parts. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. This w is characterized as being the unique element of M which minimizes kv − xk; x 2 M. The idea of the proof was to use the parallelogram law to conclude that if fxng is a sequence of elements of M such that kv − xnk approaches the greatest lower bound of kv − xk; x 2 M then fxng converges to the desired w, The map v 7! w is called orthogonal projection of H onto M and is denoted by πM . Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Orthogonal projection onto a complete subspace. If M is a complete subspace of a pre-Hilbert space H then for any v 2 H there is a unique w 2 M such that (v − w) ? M. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. The idea of the proof was to use the parallelogram law to conclude that if fxng is a sequence of elements of M such that kv − xnk approaches the greatest lower bound of kv − xk; x 2 M then fxng converges to the desired w, The map v 7! w is called orthogonal projection of H onto M and is denoted by πM . Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Orthogonal projection onto a complete subspace. If M is a complete subspace of a pre-Hilbert space H then for any v 2 H there is a unique w 2 M such that (v − w) ? M. This w is characterized as being the unique element of M which minimizes kv − xk; x 2 M. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. The map v 7! w is called orthogonal projection of H onto M and is denoted by πM . Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Orthogonal projection onto a complete subspace. If M is a complete subspace of a pre-Hilbert space H then for any v 2 H there is a unique w 2 M such that (v − w) ? M. This w is characterized as being the unique element of M which minimizes kv − xk; x 2 M. The idea of the proof was to use the parallelogram law to conclude that if fxng is a sequence of elements of M such that kv − xnk approaches the greatest lower bound of kv − xk; x 2 M then fxng converges to the desired w, Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Orthogonal projection onto a complete subspace. If M is a complete subspace of a pre-Hilbert space H then for any v 2 H there is a unique w 2 M such that (v − w) ? M. This w is characterized as being the unique element of M which minimizes kv − xk; x 2 M. The idea of the proof was to use the parallelogram law to conclude that if fxng is a sequence of elements of M such that kv − xnk approaches the greatest lower bound of kv − xk; x 2 M then fxng converges to the desired w, The map v 7! w is called orthogonal projection of H onto M and is denoted by πM . Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Linear and antilinear maps. Let V and W be two complex vector spaces. A map T : V ! W is called linear if T (λx + µy) = λT (x) + µT (y) 8 x; y 2 V ; λ, µ 2 C and is called anti-linear if T (λx + µy) = λT (x) + µT (y) 8 x; y 2 V λ, µ 2 C: Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Linear functions. If ` : V ! C is a linear map, (also known as a linear function) then ker ` := fx 2 V j `(x) = 0g has codimension one (unless ` ≡ 0). Indeed, if `(y) 6= 0 then 1 `(x) = 1 where x = y `(y) and for any z 2 V , z − `(z)x 2 ker `: Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. If V is a normed space and ` is continuous, then ker (`) is a closed subspace. The space of continuous linear functions is denoted by V ∗. It has its own norm defined by k`k := sup j`(x)j=kxk: x2V ;kxk6=0 Suppose that H is a pre-Hilbert space. We have an antilinear map φ : H ! H∗; (φ(g))(f ) := (f ; g): The Cauchy-Schwarz inequality implies that kφ(g)k ≤ kgk and (g; g) = kgk2 shows that kφ(g)k = kgk: In particular the map φ is injective. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. The Riesz representation theorem. Let H be a pre-hilbert space. Then we have an antilinear map φ : H ! H∗; (φ(g))(f ) := (f ; g): The Cauchy-Schwarz inequality implies that kφ(g)k ≤ kgk In fact (g; g) = kgk2: This shows that kφ(g)k = kgk: In particular the map φ is injective. The Riesz representation theorem says that if H is a Hilbert space, then this map is surjective: Theorem Every continuous linear function on a Hilbert space H is given by scalar product by some element of H. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Proof of the Riesz representation theorem. The proof follows from theorem about projections applied to N := ker `: If ` = 0 there is nothing to prove. If ` 6= 0 then N is a closed subspace of codimension one. Choose v 62 N. Then there is an x 2 N with (v − x) ? N: Let 1 y := (v − x): kv − xk Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Then y ? N and kyk = 1: For any f 2 H, [f − (f ; y)y] ? y so f − (f ; y)y 2 N or `(f ) = (f ; y)`(y); so if we set g := `(y)y then (f ; g) = `(f ) for all f 2 H: Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Claim: πM exists and X πM (v) = πMi (v): (1) Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Projection onto a finite direct sum. Suppose that the closed subspace M of a pre-Hilbert space H is the orthogonal direct sum of a finite number of subspaces M M = Mi i meaning that the Mi are mutually perpendicular and every element x of M can be written as X x = xi ; xi 2 Mi : (The orthogonality guarantees that such a decomposition is unique.) Suppose further that each Mi is such that the projection πMi exists. Shlomo Sternberg Math212a1403 The spectral theorem for compact self-adjoint operators. Outline Review of last lecture. The Riesz representation theorem. Bessel's inequality. Self-adjoint transformations. Compact self-adjoint transformations. The spectral theorem for compact self-adjoint operators. Fourier's Fourier series. Projection onto a finite direct sum. Suppose that the closed subspace M of a pre-Hilbert space H is the orthogonal direct sum of a finite number of subspaces M M = Mi i meaning that the Mi are mutually perpendicular and every element x of M can be written as X x = xi ; xi 2 Mi : (The orthogonality guarantees that such a decomposition is unique.) Suppose further that each Mi is such that the projection πMi exists.
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