Chapter 10 Linear Maps on Hilbert Spaces A special tool called the adjoint helps provide insight into the behavior of linear maps on Hilbert spaces. This chapter begins with a study of the adjoint and its connection to the null space and range of a linear map. Then we discuss various issues connected with the invertibility of operators on Hilbert spaces. These issues lead to the spectrum, which is a set of numbers that gives important information about an operator. This chapter then looks at special classes of operators on Hilbert spaces: self- adjoint operators, normal operators, isometries, unitary operators, integral operators, and compact operators. Even on infinite-dimensional Hilbert spaces, compact operators display many characteristics expected from finite-dimensional linear algebra. We will see that the powerful Spectral Theorem for compact operators greatly resembles the finite- dimensional version. Also, we develop the Singular Value Decomposition for an arbitrary compact operator, again quite similar to the finite-dimensional result. The Botanical Garden at Uppsala University (the oldest university in Sweden, founded in 1477), where Erik Fredholm (1866–1927) was a student. The theorem called the Fredholm Alternative, which we prove in this chapter, states that a compact operator minus a nonzero scalar multiple of the identity operator is injective if and only if it is surjective. CC-BY-SA Per Enström © Sheldon Axler 2020 S. Axler, Measure, Integration & Real Analysis, Graduate Texts 280 in Mathematics 282, https://doi.org/10.1007/978-3-030-33143-6_10 Section 10A Adjoints and Invertibility 281 10A Adjoints and Invertibility Adjoints of Linear Maps on Hilbert Spaces The next definition provides a key tool for studying linear maps on Hilbert spaces. 10.1 Definition adjoint; T∗ Suppose V and W are Hilbert spaces and T : V W is a bounded linear map. ! The adjoint of T is the function T : W V such that ∗ ! T f , g = f , T∗g h i h i for every f V and every g W. 2 2 To see why the definition above makes The word adjoint has two unrelated sense, fix g W. Consider the linear 2 meanings in linear algebra. We need functional on V defined by f T f , g . only the meaning defined above. This linear functional is bounded7! hbecausei T f , g T f g T g f jh ij ≤ k k k k ≤ k k k k k k for all f V; thus the linear functional f T f , g has norm at most T g . By 2 7! h i k k k k the Riesz Representation Theorem (8.47), there exists a unique element of V (with norm at most T g ) such that this linear functional is given by taking the inner k k k k product with it. We call this unique element T∗g. In other words, T∗g is the unique element of V such that 10.2 T f , g = f , T∗g h i h i for every f V. Furthermore, 2 10.3 T∗g T g . k k ≤ k kk k In 10.2, notice that the inner product on the left is the inner product in W and the inner product on the right is the inner product in V. 10.4 Example multiplication operators ¥ Suppose (X, , m) is a measure space and h (m). Define the multiplication S 2 L operator M : L2(m) L2(m) by h ! Mh f = f h. Then M is a bounded linear map and M h ¥. Because h k hk ≤ k k The complex conjugates that appear M f , g = f hg dm = f , M g h h i h h i in this example are unnecessary (but Z they do no harm) if F = R. for all f , g L2(m), we have M = M . 2 h∗ h 282 Chapter 10 Linear Maps on Hilbert Spaces 10.5 Example linear maps induced by integration Suppose (X, , m) and (Y, , n) are s-finite measure spaces and K 2(m n). S T 2 L × Define a linear map : L2(n) L2(m) by IK ! 10.6 ( K f )(x) = K(x, y) f (y) dn(y) I ZY for f L2(n) and x X. To see that this definition makes sense, first note that 2 2 there are no worrisome measurability issues because for each x X, the function 2 y K(x, y) is a -measurable function on Y (see 5.9). 7! T Suppose f L2(n). Use the Cauchy–Schwarz inequality (8.11) or Hölder’s inequality (7.9)2 to show that 1/2 2 10.7 K(x, y) f (y) dn(y) K(x, y) dn(y) f L2(n). Yj j j j ≤ Yj j k k Z Z for every x X. Squaring both sides of the inequality above and then integrating on 2 X with respect to m gives 2 ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) 2 K x, y f y dn y dm x K x, y dn y dm x f L2(n) X Yj j j j ≤ X Yj j k k Z Z Z Z = 2 2 K L2(m n) f L2(n), k k × k k where the last line holds by Tonelli’s Theorem (5.28). The inequality above implies that the integral on the left side of 10.7 is finite for m-almost every x X. Thus 2 the integral in 10.6 makes sense for m-almost every x X. Now the last inequality above shows that 2 2 = ( )( ) 2 ( ) 2 2 K f L2(m) K f x dm x K L2(m n) f L2(n). kI k ZXj I j ≤ k k × k k Thus is a bounded linear map from L2(n) to L2(m) and IK 10.8 K K L2(m n). kI k ≤ k k × Define K : Y X F by ∗ × ! K∗(y, x) = K(x, y), 2( ) 2( ) 2( ) and note that K∗ L n m . Thus K∗ : L m L n is a bounded linear map. Using Tonelli’s Theorem2 × (5.28) and Fubini’sI Theorem! (5.32), we have K f , g = K(x, y) f (y) dn(y)g(x) dm(x) hI i ZX ZY = f (y) K(x, y)g(x) dm(x) dn(y) ZY ZX = ( ) ( ) ( ) = f y K∗ g y dn y f , K∗ g ZY I h I i for all f L2(n) and all g L2(m). Thus 2 2 10.9 ( )∗ = . IK IK∗ Section 10A Adjoints and Invertibility 283 10.10 Example linear maps induced by matrices As a special case of the previous example, suppose m, n Z+, m is counting 2 measure on 1, . , m , n is counting measure on 1, . , n , and K is an m-by-n f g f g matrix with entry K(i, j) F in row i, column j. In this case, the linear map 2 : L2(n) L2(m) induced by integration is given by the equation IK ! n ( K f )(i) = ∑ K(i, j) f (j) I j=1 for f L2(n). If we identify L2(n) and L2(m) with Fn and Fm and then think of elements2 of Fn and Fm as column vectors, then the equation above shows that the linear map : Fn Fm is simply matrix multiplication by K. IK ! In this setting, K∗ is called the conjugate transpose of K because the n-by-m matrix K∗ is obtained by interchanging the rows and the columns of K and then taking the complex conjugate of each entry. The previous example now shows that m n 1/2 2 K ∑ ∑ K(i, j) . kI k ≤ = = j j i 1 j 1 Furthermore, the previous example shows that the adjoint of the linear map of multiplication by the matrix K is the linear map of multiplication by the conjugate transpose matrix K∗, a result that may be familiar to you from linear algebra. If T is a bounded linear map from a Hilbert space V to a Hilbert space W, then the adjoint T∗ has been defined as a function from W to V. We now show that the adjoint T is linear and bounded. Recall that (V, W) denotes the Banach space of ∗ B bounded linear maps from V to W. 10.11 T∗ is a bounded linear map Suppose V and W are Hilbert spaces and T (V, W). Then 2 B T∗ (W, V), (T∗)∗ = T, and T∗ = T . 2 B k k k k Proof Suppose g , g W. Then 1 2 2 f , T∗(g + g ) = T f , g + g = T f , g + T f , g h 1 2 i h 1 2i h 1i h 2i = f , T∗g + f , T∗g h 1i h 2i = f , T∗g + T∗g h 1 2i for all f V. Thus T (g + g ) = T g + T g . 2 ∗ 1 2 ∗ 1 ∗ 2 Suppose a F and g W. Then 2 2 f , T∗(ag) = T f , ag = a T f , g = a f , T∗g = f , aT∗g h i h i h i h i h i for all f V. Thus T (ag) = aT g. 2 ∗ ∗ We have now shown that T : W V is a linear map. From 10.3, we see that T ∗ ! ∗ is bounded. In other words, T (W, V). ∗ 2 B 284 Chapter 10 Linear Maps on Hilbert Spaces Because T∗ (W, V), its adjoint (T∗)∗ : V W is defined. Suppose f V. Then 2 B ! 2 (T∗)∗ f , g = g, (T ) f = T g, f = f , T∗g = T f , g h i h ∗ ∗ i h ∗ i h i h i for all g W.