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An introduction to some aspects of functional analysis, 2: Bounded linear operators Stephen Semmes Rice University Abstract These notes are largely concerned with the strong and weak operator topologies on spaces of bounded linear operators, especially on Hilbert spaces, and related matters. Contents I Basic notions 7 1 Norms and seminorms 7 2 ℓp spaces 7 3 Bounded linear mappings 8 4 Dual spaces 9 5 Shift operators 10 6 Arbitrary sequences 11 7 Metrizability 11 8 Completeness 12 9 Continuous extensions 12 10 Uniform boundedness 13 11 Bounded linear mappings, 2 13 12 Separability 14 13 Inner product spaces 14 1 14 Hilbert spaces 16 15 Orthogonal projections 17 16 Orthogonal complements 18 17 Dual spaces, 2 18 18 The Hahn–Banach theorem 18 19 The weak topology 19 20 The weak∗ topology 20 21 Convergence of sequences 21 22 The strong operator topology 22 23 Convergence of sequences, 2 22 24 Shift operators, 2 23 25 Multiplication operators 24 26 The weak operator topology 24 27 Convergence of sequences, 3 25 28 The weak∗ operator topology 26 29 Fourier series 27 30 Adjoints 28 31 Unitary transformations 29 32 Self-adjoint operators 30 33 Nonnegative self-adjoint operators 31 34 Cauchy–Schwarz, revisited 31 35 Continuity of compositions 32 36 Multiplication operators, 2 33 37 Complex Hilbert spaces 34 38 The C∗ identity 34 2 39 Finite-dimensional spaces 35 40 Continuous linear functionals 36 41 Operator topologies, revisited 37 42 Continuous functions 38 43 von Neumann algebras 38 44 Commutants 39 45 Second commutants 39 46 Invertibility 40 47 Positivity 41 48 Invertibility, 2 42 49 Invertibility, 3 44 50 The Hardy space 44 51 Fourier series, 2 46 52 Multiplication on H2 46 53 The Poisson kernel 47 54 Fourier series, 3 49 55 Hardy spaces, continued 49 56 Multiplication, continued 50 57 Convolution on Z 52 58 Convolution on Z, 2 53 59 Convolution on T 54 II Topological groups 56 60 Topological groups 56 61 Locally compact groups 57 3 62 Uniform continuity 57 63 Haar measure 58 64 Convolution, revisited 60 65 Convolution of measures 61 66 Locally compact Hausdorff spaces 63 67 Simple estimates 63 68 Translation operators 65 69 Uniform continuity, 2 66 70 Continuity and convolution 67 71 Convolution operators on C0 67 72 Compactness and σ-compactness 68 73 Equicontinuity 69 74 Locally compact spaces, 2 70 75 Convolution operators on L1 71 76 Simple estimates, 2 72 77 Convolution operators into C0 73 78 L1 and L∞ 74 79 Conjugation by multiplication 75 80 L1 into Lp 76 81 Integral pairings 78 82 Operators on L2 79 83 Dual operators 79 84 Operators on Lp 80 85 Group representations 81 86 Local uniform boundedness 82 4 87 Uniform convexity 83 88 Bounded representations 85 89 Dual representations 86 90 Invertibility, 4 87 91 Uniform continuity, 3 88 92 Invariant subspaces 89 93 Dual subspaces 90 94 Irreducibility 91 95 Vector-valued integration 92 96 Strongly continuous representations 95 97 Dual representations, 2 96 98 Compositions 96 99 Unbounded continuous functions 98 100 Bounded continuous functions 100 101 Discrete sets and groups 102 102 Group algebras 104 III Self-adjoint linear operators 105 103 Self-adjoint operators, 2 105 104 Invertibility and compositions 107 105 A few simple applications 108 106 Polynomials 109 107Nonnegativerealpolynomials 111 108 Polynomials and operators 112 109 Functional calculus 113 5 110 Step functions 114 111Composition of functions 115 112 Step functions, continued 116 113 Algebras of operators 118 114 Irreducibility, 2 118 115 Restriction of scalars 119 116 Quaternions 120 117Real -algebras 121 ∗ 118 Another approach 123 119Finite dimensions 123 120 Multiplication operators, 3 124 121 Measures 125 122Measures, 2 127 123 Complex-valued functions 127 124 Pointwise convergence 128 125 Borel functions 128 126 Approximations 129 127 Densities 131 128 Eigenvalues 131 129 Another limit 132 130 Kernels and projections 134 131 Projections and subspaces 134 132 Sequences of subspaces 135 133Families of subspaces 136 134 The positive square root 137 6 135 Positive self-adjoint part 139 136Polar decompositions 140 References 142 Part I Basic notions 1 Norms and seminorms Let V be a vector space over the real numbers R or the complex numbers C. A nonnegative real-valued function N on V is a seminorm if (1.1) N(t v)= t N(v) | | for every t R or C, as appropriate, and v V , and ∈ ∈ (1.2) N(v + w) N(v)+ N(w) ≤ for every v, w V . Here t denotes the absolute value of t R or the modulus of t C. If also∈ N(v) > 0| | when v = 0, then N is a norm on∈ V . If∈N is a norm on V , then 6 (1.3) d(v, w)= N(v w) − is a metric on V . A collection of seminorms on V will be called nice if for every v V with v = 0 there isN an N such that N(v) > 0. In this case, the topology∈ on V associated6 to is defined∈ N by saying that U V is an open set if for every u U there are finitelyN many seminorms N ,...,N⊆ and ∈ 1 l ∈ N finitely many positive real numbers r1,...,rl such that (1.4) v V : N (u v) < r for j =1,...,l U. { ∈ j − j }⊆ It is easy to see that V is Hausdorff with respect to this topology, and that the vector space operations of addition and scalar multiplication are continuous. 2 ℓp spaces p ∞ Let p be a real number, p 1, and let ℓ be the space of sequences a = aj j=1 of real or complex numbers≥ such that the infinite series { } ∞ (2.1) a p | j | Xj=1 7 converges. It is well known that this is a vector space with respect to termwise addition and scalar multiplication, and that ∞ 1/p (2.2) a = a p k kp | j | Xj=1 defines a norm on this space. Similarly, the space ℓ∞ of bounded sequences of real or complex numbers is a vector space, and (2.3) a = sup a : j 1 k k∞ {| j | ≥ } is a norm on ℓ∞. p If aj = 0 for all but finitely many j, then a is contained in ℓ for each p. Moreover, the space of these sequences is a dense linear subspace of ℓp when ∞ p< . The closure of this space of sequences in ℓ is the space c0 of sequences that∞ converge to 0. p These spaces may also be denoted ℓ (Z+), c0(Z+), where Z+ is the set of positive integers. There are versions of these spaces for doubly-infinite sequences ∞ p a = aj j=−∞ as well, denoted ℓ (Z), c0(Z), where Z is the set of all integers. More{ precisely,} if p < , then ℓp(Z) is the space of doubly-infinite sequences such that ∞ ∞ (2.4) a p | j | j=X−∞ converges, which is the same as saying that ∞ ∞ (2.5) a p, a p | j | | −j | Xj=1 Xj=0 both converge, equipped with the norm ∞ 1/p (2.6) a = a p . k kp | j | j=X−∞ The space ℓ∞(Z) consists of bounded doubly-infinite sequences, with the norm (2.7) a = sup a : j Z . k k∞ {| j| ∈ } As before, the space of doubly-infinite sequences a such that aj 0 as j ∞ → → ±∞ defines a closed linear subspace c0(Z) of ℓ (Z). 3 Bounded linear mappings Let V , W be vector spaces, both real or both complex, and equipped with norms v V , w W , respectively. A linear mapping T : V W is said to be bounded ifk therek k isk an L 0 such that → ≥ (3.1) T (v) L v k kW ≤ k kV 8 for every v V . It is easy to see that bounded linear mappings are continuous and even uniformly∈ continuous with respect to the metrics on V , W associated to their norms. Conversely, a linear mapping is bounded if it is continuous at 0. The operator norm of a bounded linear mapping T : V W is defined by → (3.2) T = sup T (v) : v V, v 1 . k kop {k kW ∈ k kV ≤ } Equivalently, L = T op is the smallest nonnegative real number that satis- fies the previous condition.k k The space (V, W ) of bounded linear mappings from V into W is a vector space with respectBL to pointwise addition and scalar multiplication, and it is easy to see that T defines a norm on this space. k kop Suppose that V1, V2, V3 are vector spaces, all real or all complex, and equipped with norms , , . If T : V V and T : V V are k·k1 k·k2 k·k3 1 1 → 2 2 2 → 3 bounded linear mappings, then their composition T2 T1 : V1 V3 is also a bounded linear mapping, with ◦ → (3.3) T T T T , k 2 ◦ 1kop,13 ≤k 1kop,12 k 2kop,23 where the subscripts indicate the spaces and norms involved. In particular, the space (V )= (V, V ) of bounded linear operators on a real or complex BL BL vector space V with a norm v V is an algebra with respect to composition. Of course, the identity operatorkIkon V has operator norm 1. 4 Dual spaces Let V be a real or complex vector space, equipped with a norm v V . A bounded linear functional on V is a bounded linear mapping from V intok kR or C, using the standard absolute value or modulus as the norm on the latter. The vector space of bounded linear functionals on V is the same as (V, R) or (V, C), and will be denoted V ′.
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