Operators on Hilbert spaces Jan Derezi´nski Department of Mathematical Methods in Physics Faculty of Physics University of Warsaw ul. Pasteura 5, 02-093, Warszawa, Poland October 17, 2019 Contents 1 Introduction 9 2 Banach spaces 11 2.1 Vector spaces . 11 2.2 Norms and seminorms . 11 2.3 Banach spaces . 12 2.4 Bounded operators . 13 2.5 Continuous embedding . 15 2.6 Direct sum of Banach spaces . 16 2.7 Vector valued functions . 16 3 Partial operators on Banach spaces 19 3.1 Relations . 19 3.2 Linear partial operators . 20 3.3 Closed operators . 21 3.4 Bounded operators as closed operators . 22 3.5 Closable operators . 22 3.6 Essential domains . 23 3.7 Perturbations of closed operators . 23 3.8 Invertible operators . 25 3.9 Product of operators . 27 4 Spectral theory of operators on Banach spaces 29 4.1 Spectrum . 29 4.2 Spectral radius . 31 4.3 Examples . 32 4.4 Functional calculus . 34 4.5 Idempotents . 36 4.6 Spectral idempotents . 37 4.7 Isolated eigenvalues . 38 4.8 Spectral theory in finite dimension . 39 4.9 Functional calculus for several commuting operators . 39 4.10 Examples of unbounded operators . 40 4.11 Pseudoresolvents . 41 3 5 One-parameter semigroups on Banach spaces 43 5.1 (M; β)-type semigroups . 43 5.2 Generator of a semigroup . 44 5.3 One-parameter groups . 46 5.4 Norm continuous semigroups . 47 5.5 Essential domains of generators . 48 5.6 Operators of (M; β)-type . 48 5.7 The Hille-Philips-Yosida theorem . 49 5.8 Semigroups of contractions and their generators . 51 6 Hilbert spaces 53 6.1 Scalar product spaces . 53 6.2 The definition and examples of Hilbert spaces . 54 6.3 Complementary subspaces . 55 6.4 Orthonormal basis . 56 6.5 The Riesz Lemma . 59 6.6 Quadratic forms . 59 6.7 Adjoint operators . 60 6.8 Numerical range . 61 6.9 Self-adjoint operators . 62 6.10 Projectors . 65 6.11 Isometries and partial isometries . 65 6.12 Unitary operators . 66 6.13 Normal operators . 67 6.14 Normal operators as multiplication operators . 68 6.15 Convergence . 69 6.16 Monotone convergence of selfadjoint operators . 71 7 Spectral theorems 73 7.1 Continuous functional calculus for self-adjoint and unitary oper- ators . 73 7.2 Projector valued measures . 75 7.3 Continuous and singular PVM's . 77 7.4 Projector valued Riesz-Markov theorem . 78 7.5 Alternative approaches to the projector valued Riesz-Markov the- orem . 79 7.6 Spectral theorem for bounded Borel functions . 80 7.7 Spectral theorem in terms of L2 spaces . 81 7.8 Ideals in commutative C∗-algebras . 81 7.9 Spectrum of a ∗-homomorphisms of C(X)............. 82 7.10 Commuting self-adjoint operators . 83 7.11 Functional calculus for a single normal operator . 84 7.12 Functional calculus for a family of commuting normal operators . 85 8 Compact operators 87 8.1 Finite rank operators . 87 8.2 Compact operators on Banach spaces . 87 8.3 Compact operators on a Hilbert space . 89 8.4 The Fredholm alternative . 89 8.5 Positive trace class operators . 90 8.6 Hilbert-Schmidt operators . 92 8.7 Trace class operators . 94 9 Unbounded operators on Hilbert spaces 99 9.1 Graph scalar product . 99 9.2 The adjoint of an operator . 99 9.3 Inverse of the adjoint operator . 101 9.4 The adjoint of a product of operators . 102 9.5 Numerical range and maximal operators . 103 9.6 Dissipative operators . 104 9.7 Hermitian operators . 105 9.8 Self-adjoint operators . 106 9.9 Spectral theorem . 107 9.10 Essentially self-adjoint operators . 109 9.11 Rigged Hilbert space . 110 9.12 Polar decomposition . 111 9.13 Scale of Hilbert spaces I . 113 9.14 Scale of Hilbert spaces II . 114 9.15 Complex interpolation . 114 9.16 Relative operator boundedness . 115 9.17 Relative form boundedness . 116 9.18 Discrete and essential spectrum . 117 9.19 The mini-max and max-min principle . 118 9.20 Singular values of an operator . 119 9.21 Convergence of unbounded operators . 119 10 Positive forms 121 10.1 Quadratic forms . 121 10.2 Sesquilinear quasiforms . 122 10.3 Closed positive forms . 122 10.4 Closable positive forms . 123 10.5 Operators associated with positive forms . 124 10.6 Perturbations of positive forms . 124 10.7 Friedrichs extensions . 125 11 Non-maximal operators 127 11.1 Defect indices . 127 11.2 Extensions of hermitian operators . 128 11.3 Extension of positive operators . 130 12 Aronszajn-Donoghue Hamiltonians and their renormalization 133 12.1 Construction . 133 12.2 Cut-off method . 135 12.3 Extensions of hermitian operators . 136 12.4 Positive H0 .............................. 136 13 Friedrichs Hamiltonians and their renormalization 139 13.1 Construction . 139 13.2 The cut-off method . 141 13.3 Eigenvectors and resonances . 141 13.4 Dissipative semigroup from a Friedrichs Hamiltonian . 142 14 Convolutions and Fourier transformation 145 14.1 Introduction to convolutions . 145 14.2 Modulus of continuity . 145 1 1 14.3 The special case of the Young inequality with p + q = 1 . 146 14.4 Convolution by an L1 function . 146 14.5 The Young inequality . 147 14.6 Fourier transformation on L1 [ L2(Rd)............... 148 14.7 Tempered distributions on Rd .................... 151 14.8 Spaces of sequences . 153 14.9 The oscillator representation of S(X) and S0(X)......... 154 14.10Convolution of distributions . 156 14.11The Hardy-Littlewood-Sobolev inequality . 158 14.12Self-adjointness of Schr¨odingeroperators . 160 15 Momentum in one dimension 163 15.1 Distributions on R .......................... 163 15.2 Momentum on the line . 163 15.3 Momentum on the half-line . 166 15.4 Momentum on an interval I . 167 15.5 Momentum on an interval II . 168 15.6 Momentum on an interval III . 168 16 Laplacian 169 16.1 Sobolev spaces in one dimension . 169 16.2 Laplacian on the line . 169 16.3 Laplacian on the halfline I . 171 16.4 Laplacian on the halfline II . 171 16.5 Neumann Laplacian on a halfline with the delta potential . 173 16.6 Dirichlet Laplacian on a halfline with the δ0 potential . 173 16.7.