Appendix Functional Analysis and Operators

Home , Bounded operator, Operator (mathematics), Operator theory, Volterra operator

A.1 Linear Space

Definition A.1 A real linear space Z is a set {z1, z2,...} in which operations addi- tion ( + ) and scalar multiplication by real numbers are defined. The following axioms are satisfied for all zi ∈ Z and all scalars α, β ∈ R:

1 z1 + z2 ∈ Z 2 αz ∈ Z 3 z1 + z2 = z2 + z1 4 (z1 + z2) + z3 = z1 + (z2 + z3) 5. There exists 0 ∈ Z with the property that z + 0 = z for all z ∈ Z 6. For every z ∈ Z there is z˜ ∈ Z such that z +˜z = 0 7 (αβ)z = (βα)z 8 (α + β)z = (β + α)z 9 α(z1 + z2) = αz1 + αz2 10 1z = z

The element 0 is known as the zero element. The scalars can also be the complex numbers, in which case it is a complex linear space. (Linear spaces are sometimes referred to as vector spaces.)

The following can all be veriﬁed to be real linear spaces: n – R , vectors (z1, z2,...,zn) of length n with real-valued entries, – C[a, b], the set of real-valued continuous functions on an interval [a, b]; while the following spaces are complex linear spaces n – C , vectors (z1, z2,...,zn) of length n with complex-valued entries,

© Springer Nature Switzerland AG 2020 253 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3 254 Appendix: Functional Analysis and Operators

– L2(a, b), the set of all scalar-valued functions f deﬁned on (a, b) with b | f (x)|2dx < ∞ a

and scalar multiplication by complex numbers. (This deﬁnition implies that f is well enough behaved that the above integral is well-deﬁned.)

A.2 Inner Products and Norms

Consider solving a problem such as 1 k(s, t) f (s)ds = g(t) (A.1) 0 for the function f. The kernel of the integral, k, and the right-hand side g are known continuous functions. This has a passing similarity to a matrix equation

Af = g, where matrix A ∈ Rn×n and vector g ∈ Rn are known while f ∈ Rn is to be calculated. We know how to solve matrix equations, so trying to reformulate (A.1)asa matrix problem is tempting. √ The following is a basic result in Fourier series. The symbol j = −1 while α indicates the complex conjugate of α. Theorem A.2 For integers n = 0, ±1, ±2,...deﬁne

j2πnt φn(t) = e = cos(2nπt) + j sin(2nπt). , = 1 φ ( )φ ( ) = 0 n m , 1. 0 n t m t dt , = 1 n m ∈ 2( , ) 1 ( )φ ( ) = 1 | ( )|2 = . 2. for any f L 0 1 , 0 f t n t dt 0 for all n implies 0 f t dt 0 This suggests writing ∞ f (t) = fi φi (t) (A.2) i=−∞ = 1 ( )φ ( ) . φ where fi 0 f t i t dt Multiplying each side of the above equation by m and integrating over [0, 1] yields, making use of property (1) above, 1 f (t)φm(t)dt = fm. (A.3) 0 Appendix: Functional Analysis and Operators 255

{ }∞ Thus, for every continuous function f there corresponds an inﬁnite vector fi i=−∞. Note that 1 1 ∞ ∞ 2 | f (t)| dt = fi φi (t) fm φm(t)dt 0 0 i=−∞ m=−∞ 1 ∞ = fi φi (t) fi φi (t)dt 0 i=−∞ ∞ 1 2 = | fi | φi (t)φi (t)dt i=−∞ 0 ∞ 2 = | fi | . i=−∞

The expansion (A.2) can be done for any function for which 1 | f (t)|2dt < ∞ . 0

In what sense is (A.2) valid? Deﬁne { fi } as in (A.3) and consider the error function

N h(t) = f (t) − lim fi φi (t). N→∞ i=−N

For any φm, 1 1 N h(t)φm(t)dt = fm − lim fi φi (t)φm(t)dt N→∞ 0 0 i=−N = fm − fm = 0.

Property (2) in Theorem A.2 then implies statement (A.2) in the sense that the error 1 | ( )|2 = . h satisﬁes 0 h t dt 0 For notational convenience, renumber the indices: 0, −1, 1, −2, −2,...Deﬁning f ={f0, f−1, f1, f−2, f2 ...},

∞ 2 1 f =( | fi | ) 2 . (A.4) i=1 ∞ | |2 < ∞ . The linear space of vectors of inﬁnite length for which i=1 fi is called 2 256 Appendix: Functional Analysis and Operators

Returning to (A.1), assume g, f ∈ L2(0, 1). Replacing f, g by their Fourier series (A.2), 1 ∞ ∞ k(s, t) f j φ j (s)ds = g j φ j (t) 0 j=1 j=1 ∞ 1 ∞ k(s, t)φ j (s)dsfj = g j φ j (t). j=1 0 j=1

Multiply each side by φi (t) and integrate with respect to t over [0, 1], 1 ∞ 1 1 ∞ k(s, t)φ j (s)dsφi (t)dtfj = g j φ j (t)φi (t)dt 0 j=1 0 0 j=1 ∞ 1 1 k(s, t)φ j (s)dsφi (t)dt f j = gi . j=1 0 0

[A]ij

Thus, deﬁning ⎡ ⎤ ⎡ ⎤ g1 f1 ⎢ ⎥ ⎢ ⎥ 1 1 g = ⎣g2⎦ , f = ⎣ f2⎦ , [A] = k(s, t)φ (s)dsφ (t)dt, . . ij j i . . 0 0

∞ [A]ij f j = gi j=1 or Af= g.

An approximation to f can be calculated by solving the ﬁrst N × N sub-block of the { }N inﬁnite-matrix A for fi i=1, if this system of linear equations has a solution. Then,

N fN (t) = fi φi (t) i=1 is an approximate solution to the integral equation (A.1). It needs to be shown that the error in approximating f by fN is small in some sense. This approach to solving an integral Eq. A.1 motivates the definition of an infinite “dot” product for infinite products for vectors in 2: Appendix: Functional Analysis and Operators 257

∞ f · g = fi gi . (A.5) i=1

Also, ∞ f · g = fi gi i=1 ∞ 1 ∞ = fi φi (t) g j φ j (t)dt i=1 0 j=1 1 ∞ ∞ = fi φi (t) g j φ j (t)dt 0 i=1 j=1 1 = f (t)g(t)dt. 0

This suggests deﬁning a “dot product” on L2(0, 1) by 1 f, g = f (t)g(t)dt. (A.6) 0

These scalar products are special cases of what is generally known as an inner product.

Deﬁnition A.3 Let X be a complex linear space. A function ·, · : X × X → C is an inner product if, for all x, y, z ∈ X , α ∈ C, 1. x + y, z =x, z +y, z 2. αx, y =αx, y 3. x, y =y, x 4. x, x > 0ifx = 0

The deﬁnition for a real linear space is identical, except that the scalars are real numbers and property (3) reduces to x, y =y, x . An inner product space is a linear space together with an inner product. Examples of inner product spaces include 2 with the inner product (A.5), L2(0, 1) with the inner product (A.6). Some other inner product spaces are – continuously differentiable functions on [a, b] with the inner product 1 1 f, g = f (t)g(t)dt + f (t)g (t)dt 0 0

– continuous functions deﬁned on some closed region ⊂ R3 with 258 Appendix: Functional Analysis and Operators f, g = f (x)g(x)dx.

For vectors in Rn, the Euclidean inner product deﬁnes the length of the vector,

n 2 1 x=( |xi | ) 2 . i=1

This concept extends to general inner product spaces. Deﬁnition A.4 A real-valued function ·:Z → R on a linear space Z is a norm if for all y, z ∈ Z, α ∈ R, 1. z≥0 (non-negative) 2. z=0 if and only z = 0 (strictly positive) 3. αz=|α|z(homogeneous) 4. y + z≤y+z (triangle inequality) For any inner product space, deﬁne

1 f =f, f 2 . (A.7)

It is clear that the first three properties of a norm are satisfied. The last property, the triangle inequality, can be verified using the following result, known as the Cauchy– Schwarz Inequality. Theorem A.5 (Cauchy–Schwarz Inequality) For any x, y in an inner product space X, |x, y | ≤ xy where ·is defined in (A.7). Theorem A.6 Every inner product defines a norm.

Although every inner product deﬁnes a norm, not every norm is derived from an inner product. For example, for a vector x in Rn,

x∞ = max |xi | 1≤i≤n is a norm, but there is no corresponding inner product. The following result is now straightforward. Theorem A.7 (Pythogoreas) Let x, y be elements of any inner product space.

x + y2 =x2 +y2

if and only if x, y =0. Appendix: Functional Analysis and Operators 259

Definition A.8 Elements f, g in an inner product space are orthogonal if f, g =0. Definition A.9 A linear space Z with a mapping ·:Z → R that satisfies the definition of a norm (A.4) is called a normed linear space. As noted above, every inner product defines a norm, so all inner product spaces are normed linear spaces. n The linear space R with the norm ·∞ is an example of a normed linear space. Other examples of normed linear spaces are – real-valued continuous functions defined on a closed bounded set ⊂ Rn with norm f =max | f (x)|, x∈

– L1(), the linear space of all functions integrable on ⊂ Rn, with norm

f 1 = | f (x)|dx.

A.3 Linear Independence and Bases

Deﬁnition A.10 A set of elements zi , i = 1 ...n in a linear space Z is linearly independent if a1z1 + a2z2 ···+an zn = 0 implies a1 = a2 ···=an = 0.

If a set is not linearly independent, then it is linearly dependent. Deﬁnition A.11 A linear space Z is ﬁnite-dimensional if there is an integer n such that Z contains a linearly independent set of n elements, while any set of n + 1or more elements is linearly dependent. The number n is the dimension of Z.Ifdim Z = {φ }n Z n a linearly independent set of n elements i i=1 is called a basis of .

Theorem A.12 Let {φi } be a basis for an n-dimensional linear space Z. For every z ∈ Z there is a unique set of n scalars {ai } such that

z = a1φ1 +···anφn.

Thus, each element of a finite-dimensional space corresponds to a vector in Rn through its coefficients in some basis. If a space is not finite-dimensional, then by definition it has an infinite set of linearly independent elements. Such spaces are said to be infinite-dimensional. Con- 2 sider and the set of elements φn where all the coefficients are 0 except the nth. 260 Appendix: Functional Analysis and Operators

This is an infinite set, and no φn can be written as a linear combination of other ele- 2 {φ }∞ ments from this set. Thus, is infinite-dimensional. Similarly, considering n n=−∞ { ( π )}∞ 2( , ) defined in Theorem A.2, or alternatively the set of functions sin n x n=1, L 0 1 is infinite-dimensional. Bases can also be defined for infinite-dimensional spaces.

Deﬁnition A.13 Aset{φα} in a inner product space Z is a basis if z, φα =0for all φn implies z = 0, the zero element.

Deﬁnition A.14 An inner product linear space with a countable basis is said to be separable.

Deﬁnition A.15 If for a set {φn}, φn, φm =0forn = m and φn=1, the set is orthonormal.

Any set of vectors in Rn can be replaced by a set of orthonormal vectors with the same span using Gram–Schmidt orthogonalization. Similarly, given any countable {w }∞ linearly independent set n n=1 in an inner product space, Gram–Schmidt orthogonalization can be used to construct an orthonormal set with the same span:

1 φ1 = w1, w1 φ2 = α2(w2 −w2, φ1 φ1), α2 chosen so φ2=1, . . n φn+1 = αn+1(wn+1 − wn+1, φk φk ), αn+1 chosen so φn+1=1 k=1 . .

Although this is straightforward in theory, this process may have numerical problems in practice, particularly if some of the basis elements are close to collinear; that is wm,wn is not small for m = n. j2πnt Theorem A.2 states that φn(t) = e = cos(2nπx) + j sin(2nπx), n = 0, ±1, ±2,...is a (countable) orthonormal basis for L2(0, 1). Any function f ∈ L2(0, 1) can be expanded in terms of {φn} as ∞ f (t) = fnφi (x) (A.8) n=−∞ = 1 ( )φ ( ) . where fn 0 f t n t dt (See A.2 above.) Here equality is understood to hold in the sense that N lim f − fnφn2 = 0 N→∞ n=−N Appendix: Functional Analysis and Operators 261

2 where ·2 is the usual norm on L (0, 1). The expansion (A.8) generalizes to any countable orthonormal basis for a inner product space in a straightforward way. Let f be an element of a inner product space ·, · · {φ }∞ . with inner product , corresponding norm and an orthonormal basis n n=1 Then ∞ f = f, φn φn n=1 in the sense that N lim f − φn=0. N→∞ n=−N

By deﬁnition, any inner product space with a countable basis is separable. Most inner product spaces of interest in applications are separable. Consider the following examples. {φ }∞ – Deﬁne the set of elements n n=1 in the space 2 by the nth entry is 1 and all other entries are zero; that is,

(φn)n = 1,(φn) j = 0, j = n.

Clearly this set is countable and also an orthonormal basis for 2 and so 2 is separable. φ ( ) = j2πnt, {φ }∞ 2( , ). – Deﬁning n t e consider the set n n=−∞ on L 0 1 Since for any square integrable function f , 1 f (t)φn(t)dt = 0 0 1 | ( )|2 = 2( , ) for all n implies 0 f t dt 0 (Theorem A.2) this set is a basis for L 0 1 and L2(0, 1) is a separable space. – For any domain , L2() is separable. The following theorem makes the above discussion of generalizing Fourier series precise. {φ }∞ Z. Theorem A.16 Let n n=1 be an orthonormal set in an inner product space The following properties are equivalent. {φ }∞ Z. 1. n n=1 is an orthonormal basis for 2. z, φn =0 for all φn implies that z = 0. 3. For any z ∈ Z, ∞ z = z, φn φn. (A.9) n=1 262 Appendix: Functional Analysis and Operators

4. For any y, z ∈ Z, ∞ y, z = y, φn z, φn . (A.10) n=1

The series (A.9) is sometimes called a generalized Fourier series, or more simply, a Fourier series, for z. The equality (A.10)isParseval’s Equality. Setting z = y in Parseval’s Equality implies that for any z ∈ Z,

∞ 2 2 z = |z, φn | . n=1

A.4 Convergence and Completeness

Consider a sequence {zn}⊂Z where Z is a linear space. If Z is the real numbers, this sequence is said to converge to some z0 if for every > 0, there is N so that |zn − z0| < for all n > N. That is, by going far enough in the sequence it is possible n to get arbitrarily close to z0. This idea is extended to vectors in R by defining distance between 2 vectors, using the Euclidean norm, or any other norm on Rn. In the previous section, a norm was defined on a general linear space (Definition A.4). This generalizes the concept of “closeness” and so allows limits of sequences in any normed space to be defined.

Deﬁnition A.17 Let {zn}⊂Z where Z is a normed space. The statement

lim zn = z0 (A.11) n→∞ means that z0 ∈ Z and for every > 0 there is N so that

zn − z0 < , for all n > N.

By definition, for a convergent sequence, the limit z0 can be approximated arbitrarily closely by terms in the sequence {zn}, by taking n large enough. R { 1 }∞ . Example A.18 On , the familiar sequence n n=1 converges to 0 Example A.19 Define the normed space consisting of continuously differentiable functions defined on [0, 1] along with the norm 1 z2 = z(x)2dx. 0

Consider the sequence Appendix: Functional Analysis and Operators 263

1 z (x) = sin(nπx). n n Since 1 z 2 =z , z = , n n n 2n2 the sequence converges to the zero function 0. Now consider the same linear space, continuously differentiable functions deﬁned on [0, 1], but now create a different normed space using the norm 1 1 2 2 z1 = z(x) dx + z (x) dx. 0 0

With this norm, 1 π2 z 2 = + . n 1 2n2 2

The sequence {zn} does not converge to the zero function 0 in the norm ·1.

Deﬁnition A.20 A norm ·a on a linear space Z is equivalent to another norm ·b on Z if there are real numbers m > 0, M > 0 such that for all z ∈ Z,

mza ≤zb ≤ Mza.

If two norms are equivalent, then convergence in one norm implies convergence in the other and vice versa.

Theorem A.21 Every norm on a ﬁnite-dimensional space Z is equivalent to any other norm on Z.

Thus, in Rn, or any other finite-dimensional normed linear space, if a sequence converges in one norm, then it converges in any other norm. The fact that in a general normed space different norms may not be equivalent is illustrated by Example A.19. Theorem A.12 can be used to show even further, that by mapping a basis in a given finite-dimensional space of dimension n to a basis of vectors for Rn, every finite-dimensional space is essentially the same as Rn. It is useful to have a test for convergence that does not require already knowing the limit L.

Deﬁnition A.22 A sequence {zk } is Cauchy if for all > 0 there is N so that for all k > j > N, zk − z j < .

Example A.23 Letting ln indicate the natural logarithm function, consider the sequence xk = ln k, k = 1, 2,....For any k ≥ 1, 264 Appendix: Functional Analysis and Operators

xk+1 − xk = ln(k + 1) − ln k k + 1 = ln( ). k

k+1 = − = . Since limk→∞ k 1 and ln is a continuous function, limk→∞ xk+1 xk 0 But ln k →∞as k →∞and the sequence diverges. Note that for any k,

x2k − xk = ln (2k) − ln k = ln 2.

For any < ln 2 there is no N so that |xk − x j | < for all k, j > N. The sequence is not Cauchy.

The previous example illustrates that a sequence is not necessarily Cauchy if zk is near zk+1 for large k. A Cauchy sequence has its elements essentially “clumping” as n →∞. A sequence must be Cauchy in order to converge.

Theorem A.24 If a sequence is convergent then it is Cauchy.

Let Q indicate the normed linear space of rational numbers with absolute value as the norm. There are many Cauchy sequences in Q that converge to an irrational number. For example, –3, 3.1, 3.14, 3.141, 3.1415, 3.14159 ...converges to π; = k 1 . – xk j=0 j! converges to e Thus, not every Cauchy sequence in Q converges. To get convergence of Cauchy sequences of rational numbers, the “holes” need to be ﬁlled, which yields the real numbers R. Real numbers as limits of sequence of rational numbers is the funda- mental deﬁnition of the real numbers. This gives us a different way to think about R. Any real number can be regarded as the limit of a Cauchy sequence of rational numbers. That is, any real number can be approximated by a rational number to arbitrary accuracy. The number e can be regarded as the limit of any of the following sequences: 1 (1 + )n , n

2., 2.7, 2.71, 2.718,...,

n 1/i! . i=1

Deﬁnition A.25 Let (Z, ·) be a normed space. If every Cauchy sequence converges to an element of Z, the space is complete. A complete normed space is called a Banach space and a complete inner product space is called a Hilbert space.

As mentioned above, R is complete: every Cauchy sequence of real numbers converges to a real number. The space of rational numbers is not complete. Appendix: Functional Analysis and Operators 265

Example A.26 Consider the inner product space of continuous functions on [−1, 1] with the inner product 1 y, z = y(x)z(x)dx. −1

Deﬁne the sequence of continuous functions ⎧ ⎨ 0 −1 ≤ x < 0 z (x) = nx 0 ≤ x < 1 n ⎩ n 1 ≤ ≤ . 1 n x 1

Consider any 2 terms in the sequence, zm and zn where n = m + p, p > 0.

1 1 n m 2 2 2 zn − zm = (nx − mx) dx + (1 − mx) dx 1 0 n 2 = p 3m(m + p)2 ≤ 1 . 3m

1 > + − < > < . Thus, for any 0, zm p zm for all m N where N is chosen so 3N 2 By deﬁnition, the sequence is Cauchy. However, {zn} does not converge to any continuous function. It is straightforward to show, deﬁning 0 −1 ≤ x < 0 z (x) = , 0 10≤ x ≤ 1 that limn→∞ zn − z0=0.

In order to guarantee convergence of Cauchy sequences of continuous functions in the above norm, the linear space needs to be extended to include the limits of Cauchy sequences, just as the space of rational functions is extended to form the real numbers. The notation S for a set S in a normed linear space X indicates the closure of the set in the norm on X . For any closed bounded set of Rn with piecewise continuously differentiable boundary indicate continuous functions on by C() and deﬁne the inner product y, z = y(x)z(x) dx (A.12) with induced norm ·. The complete space formed by the limits of all Cauchy sequences of continuous functions is called L2(). Any function in L2() can be approximated to arbitrary accuracy by a continuous function, with the error measured in the norm (A.12). 266 Appendix: Functional Analysis and Operators

Two functions y, z ∈ L2() are regarded as the same function if y − z=0. For instance, the function that is everywhere 0, the limit of the sequence ⎧ ⎨ + − 1 ≤ ≤ 1 nx n x 0 z (x) = − nx < x ≤ 1 n ⎩ 1 0 n 0else and 1 x = 0.5 g(x) = 0 x = 0.5 are all regarded as the zero function in L2(0, 1). The space L2() can also be deﬁned to be the set of all measurable functions 2 1 for which f =( | f (x)| dx) 2 < ∞ where the integral here is is the Lebesgue integral. The Lebesgue integral is a generalization of the Riemann integral and is needed because the space of Riemann integrable functions is not complete in the norm ·2.

Deﬁnition A.27 A bounded region ⊂ Rn is a domain if it is a connected open set and its boundary ∂ can be locally represented by Lipshitz continuous functions. That, is, for any x ∈ ∂, there exists a neighborhood of x, G, such that G ∩ ∂ is the graph of a Lipschitz continuous function. Furthermore, is locally on one side of ∂.

Any polygon in Rn is a domain, and furthermore any bounded connected open set in Rn with a piecewise continuously differentiable boundary is a domain.

Deﬁnition A.28 For any domain the Sobolev space Hm() is the set of all functions z deﬁned on such that for every multi-index α with |α|≤m, the mixed partial derivative |α| α ∂ z D z = α α ∂ 1 ...∂ n x1 xn is in L2(). In other words, Hm() = z ∈ L2() : Dαz ∈ L2() for all |α|≤m .

The corresponding inner product is w, z = (Dαw)(x)(Dαz)(x) dx. |α|≤m

Deﬁnition A.29 Let Z be a Hilbert space. The subspace W ⊂ Z is dense in Z, written W = Z, or W → Z, if for every z ∈ Z and > 0, there is w ∈ W such that w − z < . Appendix: Functional Analysis and Operators 267

The statement that W is dense in Z means that every element of Z can be approximated arbitrarily closely by an element of W. Let ⊂ Rn be a domain and consider functions defined on the closure of , , that are m-times differentiable, with continuous derivatives. Indicate the space of such functions by Cm (). The notation C∞() indicates functions for which all derivatives exist and are continuous. Theorem A.30 The spaces Cm() and C∞() are dense in Hm(). Definition A.31 AsetB ⊂ Z where Z is a normed linear space is compact if every sequence in B has a sub-sequence that is convergent in the norm on Z. Theorem A.32 A finite-dimensional set B ⊂ Z is compact if and only if it is closed and bounded. Although it is always true that every compact set is closed and bounded, the converse statement is false for infinite-dimensional sets, as the following example illustrates. Example A.33 Consider the Hilbert space L2(0, 1) and the closed and bounded set

B ={z ∈ L2(0, 1)|z≤1}.

This set contains zn(t) = sin(nπt).

For any n, m, n = m, since {sin(nπt)} is an orthogonal set on L2(0, 1), 1 2 2 zn − zm = | sin(nπt) − sin(mπt)| dt 0 1 1 = | sin(nπt)|2dt + | sin(mπt)|2 dt 0 0 = 1 and so {zn} doesn’t have a convergent subsequence. Deﬁnition A.34 Consider normed linear spaces W ⊂ Z. The space W is compact in Z if the unit ball in W,

BW ={w ∈ W|wW ≤ 1} is a compact set in the Z-norm. Equivalently, every set that is bounded in the W-norm contains a subsequence that is convergent in the Z-norm. Theorem A.35 (Rellich’s Theorem) Let be a domain in RN . For any integer ≥ , Hm() 2(). N < Hm() m 1 is compact in L Also, if 2 m then is compact in C(). 268 Appendix: Functional Analysis and Operators

A.5 Operators

An operator is a mapping from one linear space to another (possibly the same) linear space. There are many commonly used operators. For instance, – The function x2 + y2 is an operator from R2 to R. [ , ] – Denote the linear space of integrable functions deﬁned on 0 1 with norm 1 | ( )| 1( , ). 0 f x dx by L 0 1 Consider integration t (Fz)(t) = z(r)dr 0

as a map from L1(0, 1) to L1(0, 1). Integration can also be defined as an operator from C([0, 1]) to C1([0, 1]), the space of continuously differentiable functions, and of course between many other normed spaces. Integration can be used to define a wide class of operators. For a function k ∈ C([0, 1]×[0, 1]), define the operator from C([(0, 1)] to itself 1 (Kz)(t) = k(r, t)z(r)dr. (A.13) 0

(Any operator of the form (A.13) is known as a Volterra operator.) Integration is a special case of (A.13) with 1 r ≤ t k(r, t) = 0 r > t.

Deﬁnition A.36 For linear spaces Y and Z consider an operator T : Y → Z.The operator T is linear if, for any y1, y2 ∈ Y and any scalar α,

T (y1 + αy2) = Ty1 + αTy2.

Setting y2 = 0 in the above deﬁnition shows that T (0) = 0 for any linear operator. It is straightforward to verify that any operator of the form (A.13) is a linear operator.

Deﬁnition A.37 An operator T : Y → Z where Y and Z are normed linear spaces is bounded if there is a constant c such that for all y ∈ Y

Ty≤cy.

Example A.38 Deﬁne the operator T to be integration from C([0, 1]) to itself with norm on C([0, 1]) f = max | f (t)|. t∈[0,1] Appendix: Functional Analysis and Operators 269

For any f ∈ C([0, 1]), t Tf= max | f (r)dr| ∈[ , ] t 0 1 0 t ≤ max | f (r)|dr ∈[ , ] t 0 1 0 ≤ max | f (r)| r∈[0,1] =f .

Thus, T is a bounded linear operator.

Denote the linear space of real-valued p-times continuously differentiable functions on ⊂ Rn by C p().

Example A.39 Consider now differentiation as an operator D from C1([0, 1]) to C([0, 1]) with norm f = max | f (t)| t∈[0,1] on both spaces. Like integration, differentiation is a linear operator. Consider the functions sin(nπx) for any positive integer n. For all n, sin(nπx) = 1but D sin(nπx)=nπ.

There is no constant M so that

Df≤M f for all f ∈ C1([0, 1]). On the other hand, if the norm

f 1 = max | f (t)|+|f (t)| t∈[0,1] is used to make C1([0, 1]) a normed linear space, then the differentiation operator D : C1[0, 1]→C([0, 1]) is bounded.

The idea of continuity for functions from R to R generalizes to operators on normed linear spaces.

Deﬁnition A.40 Consider an operator T : Y → Z where Y, Z are normed linear spaces. The operator T is continuous at y0 ∈ Y if for any > 0 there is δ > 0 such that if y − y0Y < δ then Ty− Ty0Z < .

The following theorem states that a linear operator is continuous if and only if it is bounded. 270 Appendix: Functional Analysis and Operators

Theorem A.41 Consider any T : Y → Z. If T is a linear operator then the following properties are equivalent: 1. T is a bounded operator. 2. T is continuous at the zero element in 0 ∈ Y. 3. T is continuous at all y ∈ Y.

This theorem and Example A.39 imply that differentiation is a discontinuous operator when the usual norms are used. Discontinuity means that small errors in the data can lead to large errors in the calculated derivative. This is reflected in the difficulty of computations that involve derivatives. Define the dual space Y to be the linear space of all bounded linear operators from a Banach space to the scalars. Every element y0 of a Hilbert space Y defines a scalar-valued bounded linear operator through the inner product:

Ty =y, y0 .

The converse statement is also true: any bounded linear operator from a Hilbert space to a scalar value can be uniquely deﬁned using an element of the Hilbert space. It is stated below for a complex inner product; the result for a Hilbert space with a different ﬁeld of scalars is identical.

Theorem A.42 (Riesz Representation Theorem) Let T ∈ Y . There exists a unique y0 ∈ Y so that for all y ∈ Y, Ty =y, y0 .

Thus, for a Hilbert space we may identify its dual space with itself; this is sometimes written Y = Y.

Deﬁnition A.43 For a bounded linear operator T : Y → Z where Y and Z are normed linear spaces, the operator norm of T is the smallest M such that for all y ∈ Y TyZ ≤ MyY and is indicated by T . Equivalently,

TyZ T = = TyZ . sup sup (A.14) y=0 y Y yY =1

The sum of 2 linear operators is linear, and multiplying a linear operator by a scalar creates another linear operator.

Deﬁnition A.44 The space of all bounded linear operators between two (not necessarily different) normed linear spaces Y and Z is itself a linear space, denoted B(Y, Z). The operator norm is a norm on this linear space.

For any bounded linear operators S ∈ B(X , Y), T ∈ B(Y, Z) and any x ∈ X , Appendix: Functional Analysis and Operators 271

TSx≤T Sx≤T Sx,

TS≤T S.

Thus, an operator norm has the special property of being sub-multiplicative This is not true of a general norm.

Deﬁnition A.45 For Y, Z normed linear spaces, consider a sequence {Tn}⊂ B(Y, Z) and T ∈ B(Y, Z).

– If for all z ∈ Z, limn→∞ Tn z − Tz=0 then {Tn} converges strongly to T . – If limn→∞ Tn − T =0 then {Tn} converges uniformly or in operator norm to T .

Example A.46 For any k ∈ C([0, 1]×[0, 1]), deﬁne the Volterra operator T : C([0, 1]) → C([0, 1]) by 1 (Tz)(t) = k(s, t)z(s)ds. 0

By approximating the kernel k by a simpler function we may obtain an operator that is simpler to work with. For example, suppose

k(s, t) = cos(st) and deﬁne kn to be the truncated Taylor series:

1 1 (−1)n k (st) = 1 − (st)2 + (st)4 ···+ (st)2n. n 2 4! (2n)!

Use kn to deﬁne an approximation to T : 1 1 1 1 2 2 (Tn z)(t) = kn(st)z(s)ds = z(s)ds − t s z(s)ds +··· . 0 0 2 0

Taylor’s Remainder Theorem can be used to bound Tn z − Tz as follows. For any z ∈ C([0, 1]), 1 Tn z − Tz= max | [kn(s, t) − k(s, t)]z(s)ds| 0≤t≤1 0 1 ≤ max |k (s, t) − k(s, t)|dsz ≤ ≤ n 0 t 1 0 1 ≤ z (2n + 1)! 272 Appendix: Functional Analysis and Operators by Taylor’s Remainder Theorem. Thus,

1 T − T ≤ n (2n + 1)! and limn→∞ T − Tn=0. The operator Tn uniformly approximates T . For any desired error , there is N so that all operators Tn with n > N, will provide the uniform error Tn z − Tz < z for all functions z ∈ C([0, 1]). This sequence of operators converges uniformly, or in operator norm.

Strong convergence and uniform convergence of a sequence of operators are different. The deﬁnition of the operator norm can be used to show that uniform convergence implies strong convergence. But strong convergence does not imply uniform convergence. This is illustrated by this example.

Example A.47 (Riemann Sums) Deﬁne simple integration T : C([0, 1]) → R, 1 Tz = z(x)dx 0 and also the Riemann sums Tn : C([0, 1]) → R,

− m1 1 i T z = z( ). m m m i=0

But,

− m1 1 T z = | sin(π)| m m m i=0 = 0.

< 2 = Thus for any π and any n, there is z (choose z zn) so that

Tz− Tn z > .

This illustrates that for some functions z, a large number of points are needed to get a satisfactory error when approximating the integral by a Riemann sum. Thus, the Riemann sums {Tn} do not converge uniformly, although they do converge strongly.

Deﬁnition A.48 For any linear operator T mapping a linear space Y to a linear space Z, deﬁne the range of T :

Range (T ) ={Ty| y ∈ Y}⊂Z.

Definition A.49 An operator T ∈ B(Y, Z) is finite-rank if the range of T , Range (T ), lies in a finite-dimensional subspace of X .

Clearly, every operator into Rn is finite-rank, but finite-rank operators can exist in infinite-dimensional spaces. A simple example is B ∈ B(R, L2(0, 1)) defined by for some b(x) ∈ L2(0, 1), Bu = b(x)u which has one-dimensional range.

Deﬁnition A.50 Let T ∈ B(X , Y) be a bounded linear operator between Banach spaces X and Y. Indicating the unit ball in X by B, if the closure of T (B) is a compact set in Y, then T is a compact operator.

Clearly, every bounded ﬁnite-rank operator is a compact operator. There is a close link between more general compact operators and ﬁnite-rank operators.

Theorem A.51 Let T ∈ B(X , X ) be a bounded linear operator on a Banach space X . If there exists a sequence of ﬁnite-rank operators Tn ∈ B(X , X ) so that limn→∞ T − Tn=0, then T is a compact operator or compact. If X is a Hilbert space, then T is a compact operator if and only if T is the uniform limit of a sequence of ﬁnite rank operators. 274 Appendix: Functional Analysis and Operators

Example A.52 Not every operator on an infinite-dimensional space is a compact operator. For example, consider the identity operator I on a separable infinite- 2 dimensional Hilbert space Z such as L (a, b) with orthonormal basis {en}. Let {Tn} be any sequence of finite-rank operators. It is not difficult to find sequences that converge strongly to the identity; for instance

n Tn z = z, en en. k=1

However, for any finite-rank operator Tn, it is possible to select em that is not in the range of Tn and so Tnem − em =em =1. Thus, no sequence of finite rank operators converges uniformly to I and the identity is not a compact operator. Example A.53 For any k ∈ C([0, 1]×[0, 1]), define the Volterra operator T : C([0, 1]) → C([0, 1]) by 1 (Tz)(t) = k(s, t)z(s)ds. 0

It was shown in Example A.46 that this class of operators can be uniformly approximated by a sequence of ﬁnite-rank operators. Thus, these operators are compact.

Theorem A.54 Let T be a compact operator and Sn a sequence of bounded operators on a Banach space X strongly convergent to a bounded operator S. Then

lim TSn − TS=0, lim Sn T − ST=0. n→∞ n→∞

The following deﬁnition is a generalization of the complex conjugate transpose of a matrix (or the simple transpose on a real space) to general Hilbert spaces. Deﬁnition A.55 Consider a linear operator A : dom(A) ⊂ Z → Y where Z, Y are Hilbert spaces. The domain dom(A∗) of the adjoint operator A∗ : Y → Z is the set of all y ∈ Y where exists z˜ ∈ Z so that for all z ∈ dom(A),

Az, y Y =z, z˜ Z .

Furthermore, A∗ is deﬁned by A∗ y =˜z so the above expression can be written

∗ Az, y Y =z, A y Z .

If A is a matrix with real entries, then A∗ is simply the transpose of A;for a matrix with complex entries, A∗ is the complex conjugate transpose. For general operators the adjoint needs to be calculated. For differential operators, the calculation of the adjoint is generally done using “integration-by-parts” or the multi-dimensional generalization. Appendix: Functional Analysis and Operators 275

Example A.56 Deﬁne a derivative operator

A : dom(A) ⊂ L2(−1, 1) → L2(−1, 1), as ∂z ∂z Az = , dom(A) ={z ∈ L2(−1, 1)| ∈ L2(−1, 1), z(1) = 0}. ∂x ∂x The domain of A can also be written

dom(A) ={z ∈ H1(−1, 1)| z(1) = 0}.

Consider any z ∈ dom(A) and y ∈ L2(−1, 1). Formally integrating by parts so z is without an operation in the integral, 1 Az, y =−z(−1)y(−1) − z(x)y (x)dx. −1

In order for the previous step to be justiﬁed, y should be differentiable in some sense. Also, in order for there to exist z˜ ∈ L2(−1, 1) so that for all z ∈ dom(A), 1 z, z˜ = z(x)z˜(x)dx − 1 1 =−z(−1)y(−1) + z(x)(−y (x))dx, −1 also y(−1) = 0. Thus,

∂y dom(A∗) ={y ∈ L2(−1, 1)| ∈ L2(−1, 1), y(−1) = 0}, ∂x or dom(A∗) ={y ∈ H1(−1, 1)| y(−1) = 0}, and for y ∈ domA∗, z˜ = A∗ y =−y .

The following deﬁnition generalizes the deﬁnition of a symmetric matrix on Rn, or Hermitian matrix on Cn to a general linear space.

Deﬁnition A.57 If for a linear operator A : dom(A) ⊂ Z → Z, its adjoint operator has dom(A∗) = dom(A) and A∗z = Az for all z ∈ dom(A), the operator A is self- adjoint. 276 Appendix: Functional Analysis and Operators

Example A.58 Consider the operator A : L2(−1, 1) → L2(−1, 1) deﬁned by

d2z Az = dx2 with domain dom(A) ={z ∈ H2(−1, 1) ; z(−1) = z(1) = 0}.

Consider any z ∈ dom(A) and y ∈ L2(−1, 1) smooth enough so that formal integration by parts twice is justiﬁed. Integrate by parts twice to obtain 1 Az, y =z (1)y(1) − z (−1)y(−1) + z(x)y (x)dx. −1

In order that there be z˜ ∈ L2(−1, 1) so

Az, y =z, z˜ , for all z ∈ dom(A) , y(1) = y(−1) = 0 and y ∈ H2(−1, 1). In this case, set

z˜ = y and then

z˜ = A∗ y = y , dom(A∗) ={y ∈ H2(−1, 1) ; y(−1) = y(1) = 0}.

Both the deﬁnition of the operator and its domain equal that of A and so the operator A is self-adjoint.

Note that if A is not defined on the entire space, the definition of dom(A) and similarly dom(A∗) are an important part of the definition of the operator. The following definitions are straightforward extensions of the corresponding concepts in finite dimensions.

Definition A.59 Let A : dom(A) ⊂ Z → Z be a self-adjoint linear operator on a Hilbert space Z. If for all non-zero z ∈ dom(A), – Az, z > 0, Aispositive definite, – Az, z ≥0, Aispositive semi-definite. – Az, z < 0, Aisnegative definite, – Az, z ≤0, Aisnegative semi-definite. Appendix: Functional Analysis and Operators 277

A.6 Inverses

Deﬁnition A.60 A mapping F : X → Y where X , Y are linear spaces is invertible if there exists a mapping G : Y → X such that GFand FGare the identity mappings on X and Y respectively. The mapping G is said to be an inverse of F and is usually written F −1.

Let Range (F) denote the range of an operator, F : X → Y :

Range (F) ={y ∈ Y| there exists x ∈ X ;| Fx = y}.

Theorem A.61 A mapping F : X → Y where X , Y are linear spaces is invertible if and only if it is one-to-one and Range (F) = Y.

What if F : X → Y is one-to-one but not onto Y? In this case F : X → Range (F) is one-one and onto. It has an inverse defined on Y = Range (F). Consider now only linear maps. A linear map L : X → Y where X , Y are linear spaces will have an inverse defined on its range if and only if it is one-to-one. That is, Lx1 = Lx2 implies that x1 = x2. Since L is linear, this is equivalent to Lx = 0 implies that x = 0. Define the nullspace of L

N (L) ={x ∈ X|Lx = 0}.

Theorem A.62 A linear operator has an inverse deﬁned on its range if and only if N (L) = 0.

Theorem A.63 If L : dom(A) ⊂ X → X is positive definite or negative definite then it has an inverse defined on its range.

The question of whether an inverse, when it exists, is bounded is important.

Example A.64 Consider L : C([0, 1]) → C([0, 1]) deﬁned by t (Lx)(t) = x(s)ds. 0

This is clearly a linear operator. With the usual norm on C([0, 1]),

f = max | f (x)|. 0≤x≤1

|(Lx)(t)|≤ max |x(s)| 0≤s≤1

Lx≤x t ( ) = and so the operator is bounded. It is also one-to-one: 0 x s ds 0 for all t implies 278 Appendix: Functional Analysis and Operators

x(t) = 0.

Thus this operator has an inverse deﬁned on its range. For y ∈ Range (L),

L−1 y(t) = y (t).

This operator is deﬁned on Range (L) = C1[0, 1], but it is not bounded.

Deﬁnition A.65 For an operator L ∈ B(Y, Z) where Y, Z are normed linear spaces, if there exists m > 0 so that

Ly≥my for all y ∈ Y, (A.15) then L is coercive.

Example A.66 L : C[0, 1]→C[0, 1] t (Lx)(t) = x(s)ds. 0

(Example A.64 cont.) Consider xn(t) = sin(nπt). Each xn has xn=1,

−1 Lx = cos(nπt) n nπ and with the usual norm on C([0, 1])

1 Lx = . n nπ

There is no constant m so that (A.15) is satisﬁed. This operator is one-to-one but not coercive. This is reﬂected in the fact, shown in Example A.64, that the inverse operator is not bounded.

Example A.67 Deﬁne L : 2 → 2 by ⎡ ⎤ a1 ... ⎢ ⎥ ⎢ 0 a2 ... ⎥ L = ⎢ ... ⎥ ⎣ 00a3 ⎦ . ..

| | < ∞. ∈ where supk ak Noting that for any z 2

[Lz]k = ak zk

= | | and deﬁning M supk ak Appendix: Functional Analysis and Operators 279

∞ 2 2 Lz = |ak zk | k=1 ∞ 2 2 ≤ M |zk | k=1 ≤ M2z2.

Thus, L is bounded with L≤M. If all ak > 0 then for any non-zero z ∈ 2,

Lz, z > 0 and L is positive deﬁnite. The nullspace of L contains only the zero element and so L is invertible. Formally, L−1 is deﬁned on Range (L) by

−1 1 [L y]k = yk ak or ⎡ ⎤ 1 ... a1 ⎢ 0 1 ... ⎥ −1 ⎢ a2 ⎥ L = ⎢ 1 ... ⎥ . ⎣ 00a ⎦ 3 . ..

For y ∈ Range (L), by deﬁnition of the range of an operator, there is x ∈ 2 such that yk = ak xk and so ∞ ∞ ∞ 1 (L−1 y)2 = | y |2 = |x |2 < ∞. k a k k k=1 k=1 k k=1

The inverse operator is a bounded operator if and only if sup | 1 | < ∞, and in k ak this case the inverse can be deﬁned for all y ∈ 2. In other words, if there is m > 0 so that 1 sup | |=inf |ak |=m > 0 k ak k then L is a coercive operator. Its inverse is a bounded operator deﬁned on all of 2 and 1 L−1= . infk |ak | 280 Appendix: Functional Analysis and Operators

Note that on an inﬁnite-dimensional space, not every positive deﬁnite operator is coercive. Every coercive operator has a bounded inverse.

Theorem A.68 The operator L : Y → Z where Y, Z are normed linear spaces has a bounded inverse deﬁned on Range (L) if and only if L is a coercive operator. In this case, letting m indicate the constant in Deﬁnition A.65, for all y ∈ Range (L),

1 L−1 y≤ y. (A.16) m A bounded linear operator deﬁned on a subset of a normed linear space can be extended to the entire space when the domain of deﬁnition is dense in the space.

Theorem A.69 (Extension Theorem) Consider a linear operator A : dom(A) ⊂ X → Y where dom(A) is a dense linear subspace of a Banach space X and Y is also a Banach space. If A is a bounded operator on dom(A) then there is a unique, bounded extension of A, Ae,toallofX and Ae=A.

Thus, if a bounded linear operator is defined only on a dense subspace of a Banach space, this restriction is artificial. Similarly, unbounded operators are typically only defined on a subspace. For example, differentiation on L2(0, 1) is only defined for differentiable functions. Consider T ∈ B(X , X ) where X is a Banach space. In some cases there is a formula for the inverse of I − T. An example of such an operator is b z(t) − k(s, t)z(s)ds = y(t) a Tz which can be regarded as z − Tz = y with X = C[a, b]. For real numbers x ∞ (1 − x)−1 = xn n=0 if |x| < 1. This formula extends to bounded operators on Banach spaces.

Theorem A.70 (Neumann Series) Consider T ∈ B(X , X ) where X is a Banach space and T < 1. Then (I − T )−1 exists and

∞ (I − T )−1 = T n. n=0 Appendix: Functional Analysis and Operators 281

Also, 1 (I − T )−1≤ 1 −T and deﬁning N n SN = T , n=0

T N+1 (I − T )−1 − S ≤ . N 1 −T

Theorem A.71 Let X be a Banach space and T, S and S−1 be bounded linear operators from X to X .If 1 T − S < , (A.17) S−1 then T −1 exists, is a bounded operator, and

S−12T − S T −1 − S−1≤ . 1 −S−1T − S

The above result implies that for an operator S,ifS−1 exists and is bounded, then for all operators T close in operator norm to S, T −1 also exists and is a bounded operator.

A.7 Example: Sampling Theorem

The following application illustrates many of the concepts in this appendix. Most music today is recorded and played digitally. This requires sampling a continuous time function f (t) at a discrete set of time points and saving the samples { fn}. This set of samples is then used to reconstruct the original signal (music) when it is played back. Why is it possible to listen to sound reconstructed from samples without discernible distortion? What are the limitations? In order to provide a precise answer on how fast a signal needs to be sampled, and also to show that it can be reconstructed from samples, it’s useful to use the Fourier transform. Deﬁne the operator

F : L2(−∞, ∞) → L2(−∞, ∞) known as the Fourier transform,

∞ z(ω) =[Fz](ω) = z(t)e−j2πωt dt. (A.18) −∞ 282 Appendix: Functional Analysis and Operators

This operator has an inverse, deﬁned by

∞ z(t) =[F −1z](t) = z(ω)ej2πωt dω. −∞

The variable t can be regarded as time (in seconds) while ω as frequency (in Hz). In order to distinguish one space from the other, indicate the space for the time version of the signal by Ht and the transform, or frequency space, by Hω and write F : Ht → Hω. A simple example is e−t t ≥ 0 z(t) = 0 t < 0

1 . which has Fourier transform, as deﬁned by (A.18), 1+j2πω Also consider the rect- angular function ⎧ ⎨ , |ω| < 1 , 1 2 r(ω) = 0, |ω| > 1 , ⎩ 2 1 , |ω|= 1 . 2 2

The inverse Fourier transform of r is (π ) sin t , t = 0 sinc(πt) = πt (A.19) 1 t = 0.

Since the human ear does not respond to frequencies above about 20,000 Hz and electronic devices also only respond to frequencies within a certain range, consider now functions with Fourier transform that is only non-zero on a ﬁnite-interval [−σ, σ]. Such functions are said to be band-limited. This is a useful idealization, but it is an idealization. Although the Fourier transform of continuous integrable functions is small outside of some range, it is not exactly 0. Deﬁne the linear subspace of Hω consisting of functions that are only non-zero on the interval [−σ, σ], that is band-limited, by Mσ ={z ∈ Hω, z(ω) = 0, |ω| > σ} and the set of inverse Fourier transforms of functions in Mσ

Mσ ={z ∈ Ht ; z(ω) = 0, |ω| > σ}.

The functions

1 ω + σ φn(ω) = √ exp(j2πn ), n = 0, ±1, ±2,... 2σ 2σ Appendix: Functional Analysis and Operators 283 are an orthonormal basis for L2(−σ, σ). (See Theorem A.2 with with (0, 1) rescaled to (−σ, σ).) Noting that exp(jπn) is a constant with magnitude 1, and extending the functions φn to all of the real line by setting φn(ω) = 0for|ω| > σ, leads to the basis 1 ω ω en(ω) = √ exp(jπn )r( ), n = 0, ±1, ±2,... (A.20) 2σ σ 2σ for Mσ. Anyz ∈ Mσ can be written

∞ z = z,en en (A.21) n=−∞ where 1 σ −jπnω z,en =√ z(ω) exp( )dω. (A.22) 2σ −σ σ Sincez ∈ Mσ is band-limited the inverse transform ofz is

σ z(t) = z(ω) exp(j2πωt)dω. (A.23) −σ

Comparing (A.22) and (A.23) implies

1 −n z,en =√ z( ). 2σ 2σ

Thus, from (A.20) and (A.21), and deﬁning

1 ω ω g (ω) = exp(jπn )r( ), n 2σ σ 2σ

∞ −n 1 ω ω z(ω) = z( ) exp(jπn )r( ) 2σ 2σ σ 2σ n=−∞ ∞ −n = z( )g (ω). 2σ n n=−∞

Using (A.19), −1 F gn (t) = sinc (π(n + 2σt)) ,

= 1 and deﬁning T 2σ , 284 Appendix: Functional Analysis and Operators

∞ t z(t) = z(−nT) sinc(π(n + )). (A.24) T n=−∞

Equation (A.24) implies that a band limited signal is completely deﬁned by its sam- = 1 ples as long as the samples are taken frequently enough, that is at the rate T 2σ — twice as fast as the fastest frequency present. This is known as the Nyquist rate.For example, frequencies up to 20,000Hz (the limit of most people’s hearing) in a sampled signal can be reproduced in theory, if the samples are taken at least 40,000Hz. Errors in processing mean that a slightly larger rate yields better results.

A.8 Notes and References

Some of the deﬁnitions and theorems in this appendix are more restrictive than necessary in order to simplify the exposition and focus on the main points needed. There are many good introductions to functional analysis and operator theory. The books [1, 2] are two that focus on applications. For a treatment that focuses in particular on control systems, see [3]. Theorem A.54 is from [4, Theorem 9.19] which focuses on spectral theory.

References

1. Kreyszig E (1978) Introductory functional analysis with applications. Wiley, New Jersey 2. Naylor AW, Sell GR (1982) Linear operator theory in science and engineering. Springer, New York 3. Banks HT (2012) A functional analysis framework for modeling, estimation and control in science and engineering. CRC Press, Boca Raton 4. Hislop PD, Sigal IM (1996) Introduction to spectral theory with applications to Schrödinger operators, vol 113. Applied mathematical sciences. Springer, New York Index

Symbols analytic semigroup, 86 ∗ A , 274 approximately controllable, 57 C0-semigroup, 18 approximately controllable on [0, T ], 56 G(M, ω), 105 approximately observable, 57 L2-gain, 96 approximately observable on [0, T ], 53 M(H∞), 96 assumed modes, 115 Pn, 104 asymptotically stable, 71, 80 H1( , ) 0 0 1 , 37 Range , 273 Y , 270 α , 254 B → , 40 Banach space, 264 H (B(U, Y)) ∞ , 96 band-limited, 282 H (X ), 95 2 basis, 259, 260 j, 254 bounded, 268 C+, 95 S, 25, 265 L1(), 259 L2(), 265 C L2(a, b),, 254 cantilevered, 44 L2-gain, 95 Cauchy–Schwarz Inequality, 258 L2-stable, 95 0, 253 Cholesky-ADI, 127 H∞-ARE, 171 classical solution, 19, 46 H∞-Riccati, 170 closed, 22 B(Y, Z), 270 closed operator, 22 Hm (), 266 coercive, 103, 198, 278 V-coercive, 41 compact, 81, 267, 273 compact operator, 273 compactly embedded, 81 A complete, 264 A1, 105 continuous spectrum, 72 A1∗, 106 contraction, 31 adjoint operator, 274 controllability Gramian, 71, 158 Alternating Direction Implicit (ADI), 126 controllability map, 47

© Springer Nature Switzerland AG 2020 285 K. A. Morris, Controller Design for Distributed Parameter Systems, Communications and Control Engineering, https://doi.org/10.1007/978-3-030-34949-3 286 Index controllable, 56 internally stable, 96 covariance, 193 invariant zeros of (A, B, C, D), 64 invertible, 277

D dense, 266 J detectable, 97, 177 jointly stabilizable and detectable, 97 differential Riccati equation, 104 dimension, 259 Dirichlet trace operator, 93 K dissipative, 220 Kalman filter, 192, 194, 195 Distributed Parameter Systems, 1, 13 domain, 266 dual space, 270 L Laplace transform, 60, 95 E Laplace transformable, 60 eigenfunction, 4, 26 limit, 262 eigenvalue, 4, 26, 72, 177 linear, 268 evolution operator, 104 linear space, 253 exactly controllable on [0, T ], 56 linearly dependent, 259 exactly observable on [0, T ], 53 linearly independent, 259 expectation, 192 Lyapunov equation, 71 exponentially stable, 71, 81 externally, 223 externally stable, 95, 100, 220 M matrix pair, 177 mild solution, 19, 46 F minimal, 65 Final Value Theorem, 224 mixed finite element method, 121 finite-dimensional , 259 modal truncation, 114 finite-rank, 273 fixed attenuation H∞ control problem, 170 N negative definite, 276 G negative semi-definite, 276 gap, 227, 228, 230 Neumann Series, 280 graph, 22, 226 norm, 258 growth bound, 84 normed linear space, 259 nuclear norm, 137 H nullspace, 277 Hilbert, 264 Hilbert-Schmidt operator, 157 Hille–Yosida Theorem, 25 O Hurwitz, 84, 177 observability Gramian, 158 observability map, 50 observable, 53 I operator norm, 270, 271 infinitesimal generator, 19 optimal H∞-disturbance attenuation, 174 inner product, 257 optimizable, 107 inner product space, 257 orthogonal, 259 input passive, 220 orthonormal, 260 Internal Model Principle, 225 output passive, 220 Index 287

P stable transfer function, 96 Paley–Wiener Theorem, 95 state space, 18 Parseval’s Equality, 262 storage function, 220 passive, 220 strong solution, 19, 46 Passivity Theorem, 221 strongly, 271 Poincaré Inequality, 41 strongly continuous semigroup of operators, point spectrum, 72 18 positive deﬁnite, 276 strongly stable, 72 positive real, 221 supply rate, 220 positive semi-deﬁnite, 276

T R trace, 137 range, 277 trace class, 137, 158 reachable subspace, 57 trace norm, 137 Rellich’s Theorem, 267 tracking error, 224 residual spectrum, 72 transfer function, 60 resolvent set, 26, 72 transmission zero of (A, B, C, D), 64 Riesz basis, 26 Riesz-spectral, 27, 59 U uniformly, 271 S uniformly detectable, 112 Schur Method, 124 uniformly stabilizable, 112 second-order (Vo, Ho)-system, 42 unobservable subspace, 57 self-adjoint, 275 separable, 260 sesquilinear form, 39 V single-input-single-output, 221 vector spaces, 253 singular values, 96 Volterra, 268 Small Gain Theorem, 221 Volterra operator, 271, 274 Sobolev space, 15, 266 spectral radius, 242 spectrum, 26, 72 W spectrum decomposition assumption, 97 well-posed, 19 Spectrum Determined Growth Assumption Wiener, 233 (SDGA), 84 stabilizable, 97, 177 stabilizable with attenuation γ, 170 Z stabilizable/detectable, 97 zero element, 253