Orthogonal Polynomials, Operators and Commutation Relations

Mälardalen University Press Licentiate Theses No. 260

ORTHOGONAL POLYNOMIALS, OPERATORS AND COMMUTATION RELATIONS

John Musonda

2017

This thesis is about orthogonal polynomials, operators and commutation relations, and these appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal functions in general are central to the development of Fourier series and wavelets which are essential to signal processing. In particular, as demonstrated in this thesis, orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. The Lp- convergence of Fourier series is closely related to the Lp-boundedness of singular integral operators. Many important relations in physical sciences are represented by operators satisfying various commutation relations. Such commutation relations play key roles in such areas as quantum mechanics, wavelet analysis, spectral theory, representation theory, and many others.

This thesis consists of four chapters. Chapter 1 is the thesis introduction.

Chapter 2 presents a new system of orthogonal polynomials, and establishes its relation to the previously studied systems in the class of Meixner-Pollaczek polynomials. Boundedness properties of two singular integral operators of convolution type are investigated in the Hilbert spaces related to the relevant orthogonal polynomials. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved that the operators are bounded on L2-spaces, and estimates of the norms are obtained.

Chapter 3 extends the investigation of the boundedness properties of the two operators to Lp-spaces (1 < p < ) on the real line, both in the weighted ∞ and unweighted spaces. It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by ﬁrst proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p 2. To obtain ≤ boundedness for 2 p < , duality is used in the translation invariant case, ≤ ∞ while the weighted case is partly based on the methods developed by M. Riesz in his paper of 1928 for the conjugate function operator.

Chapter 4 derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas.

Acknowledgements

I would like to thank all my supervisors for their valuable and constructive suggestions throughout this research work. Their willingness to give their time so generously has been very much appreciated. In a special way, I thank my main supervisor, Professor Sergei Silvestrov, for his patience and guidance throughout the research and writing of this thesis. His enthusiasm when working with the commutation relations and exceptional teaching abil- ities in courses such as Applied Algebraic Structues provided inspiration and motivation to me. To my co-supervisor, Professor Sten Kaijser, I say thank you very much for having introduced me to the fascinating field of orthogonal polynomials. From the time we met in Uppsala in 2012, you have guided and encouraged me to carry on through these years and you have contributed to this thesis with a major impact. And to my other co-supervisor, Professor Anatoliy Malyarenko, I say thank you for graciously proofreading the earliest version of this work. I would also like to thank all my PhD colleagues at MälardalenUniversity who have taken some time to discuss and enrich my work. I have found the environment at the School of Education, Culture and Communication at MärdalenUniversity friendly and stimulating, and this experience will leave marks beyond this thesis. I am very grateful to the International Science Programme (ISP) and the Eastern Africa Universities Mathematics Programme (EAUMP) for the financial support. In a special way, I thank Leif Abrahamson, Pravina Gajjar and all the ISP staff for being helpful during my stay in Sweden. My heartfelt thanks go to my family, especially my loving mother and my late father, who have always supported and encouraged me to complete my studies. This thesis is dedicated to them.

V¨aster˚as, May, 2017 John Musonda

List of Papers

This thesis is partly based on the following papers:

Paper A. Musonda, J., Kaijser, S. (2015), Three systems of orthogonal polynomials and L2-boundedness of two associated operators. Submitted to an international journal. http://urn.kb.se/resolve?urn=urn:nbn:se:mdh:diva-29566

Paper B. Musonda, J., Kaijser, S. (2016), Lp-boundedness of two singular integral operators of convolution type. In: Silvestrov S., Ranˇci´cM. (eds) Engineering Mathematics II. Springer Proceedings in Mathematics & Statistics, vol 179. Springer, Cham.

Paper C. Musonda, J., Silvestrov, S. D., Kaijser, S. (2017), Reordering, centralizers and centers in an algebra with three generators and Lie type relations. To be submitted to an international journal.

Contents

1 Introduction 13

1.1 Motivation, notation and overview ...... 13

1.2 Orthogonal polynomials ...... 16

1.2.1 Elementary theory ...... 16

1.2.2 Meixner–Pollaczek polynomials ...... 18

1.3 Interpolation of Lp-spaces ...... 19

1.3.1 Distribution functions and weak Lp ...... 19

1.3.2 Interpolation theorems ...... 20

1.4 Commutation in noncommutative algebras ...... 22

1.4.1 General notions ...... 22

1.4.2 Commutation relations ...... 24

1.5 Summary of Chapters 2–4 ...... 26

1.5.1 Chapter 2 ...... 26

1.5.2 Chapter 3 ...... 28

1.5.3 Chapter 4 ...... 29

9 Orthogonal Polynomials, Operators and Commutation Relations

2 Three systems of orthogonal polynomials and L2-boundedness of associated operators 33 2.1 Overview ...... 33 2.2 The Kaijser–Araaya systems ...... 34 2.3 The ρ-system ...... 35 2.4 Some connections between the systems ...... 40 2.5 L2-boundedness of B and S ...... 43

3 Lp-boundedness of two singular integral operators of convolution type 51 3.1 Overview ...... 51 3.2 Main results ...... 52 3.3 Weak boundedness for p =1...... 52 3.4 The case 1 < p 2...... 55 ≤ 3.5 The case 2 p < ...... 56 ≤ ∞ 4 Reordering, centralizers and centers in an algebra with three generators and Lie type relations 61 4.1 Overview ...... 61 4.2 J, R-polynomials in ...... 62 A 4.2.1 An expression for [Q, p(J, R)] ...... 62 4.2.2 An expression for [Qn,J]...... 64 4.2.3 An expression for [Qn,R]...... 66 4.3 Normalised elements in ...... 68 A 4.4 Centralizers of J, R and Q in ...... 68 A 4.5 S,T -polynomials in ...... 71 A 4.5.1 An expression for [Q, SmT n]...... 72 4.5.2 An expression for [Qk,SmT n]...... 74

References 77

10 Chapter 1

This chapter is the thesis introduction.

Chapter 1

Introduction

1.1 Motivation, notation and overview

In his article [24], Sten Kaijser presented two systems of orthogonal polynomials belonging to the class of Meixner–Pollaczek polynomials [38, 46] together with some operators connecting them. One of the systems was the special (λ) case of the symmetric Meixner–Pollaczek polynomials, Pn (x/2; π/2), with parameter λ = 1/2, a system that can also be described as the polynomials orthogonal on the real line R with respect to the weight function 1 ω1(x) = π . (1.1) 2 cosh 2 x The other system was a limiting case of the symmetric Meixner–Pollaczek polynomials with the parameter λ tending to 0. That system could also be described as the polynomials orthogonal in the strip S = z C : Im z < 1 { ∈ | | } with respect to ω1. These polynomials were studied in a series of papers by Tsehaye Araaya [2, 3, 4, 5]. The monic polynomials in the ﬁrst system were denoted by τn(x) and those from the second system by σn(z). Both systems turned out to have simple exponential generating functions given by

∞ n x arctan s ∞ n s e s z arctan s τn(x) = and σn(z) = e . n! √ 2 n! n=0 1 + s n=0 X X From these generating functions, it could be concluded that the normalized polynomialsτ ˜n(x) = τn(x)/n! were orthonormal, while the normalized poly- nomialsσ ˜n(z) = σn(z)/n! were orthogonal with norms 1 for σ0 and √2 for

13 Orthogonal Polynomials, Operators and Commutation Relations the others. Araaya [4] also discovered a very interesting connection of these polynomials to the umbral calculus developed by Rota [50]. The systems studied by Araaya are connected by two operators, f(x + i) + f(x i) f(x + i) f(x i) Rf(x) = − and Jf(x) = − − , 2 2i mapping functions in the strip S to funcions on the real line R (see [5, 24]). Later on, it was observed that the operator R acting on polynomials from the ﬁrst system gives rise to another system of orthogonal polynomials, also belonging to the class of Meixner–Pollaczek polynomials. Furthermore, this third system is connected to the second system by the operator

Qf(x) = xf(x).

The third system, therefore, turns out to fill a gap related to the two Araaya systems and it is the motivation behind this thesis. We denote the monic polynomials in this system by ρn(x). In as much as Araaya might have known that the ρ-polynomials were orthogonal by Favard’s condition (See Remark 2 on page 15), he did not know the weight function. In 2012, Lars Holst [21] presented a new way to 2 2 calculate the Euler sum, n∞=1 1/n = π /6. His calculations inspired us to find the weight function as given by the convolution P x ω2(x) = (ω1 ω1)(x) = π , (1.2) ∗ 2 sinh 2 x where ω1 is defined by (1.1). 1 Besides the aforementioned operators J, R and Q, the operators B = R− 1 and S = JR− turn out to have interesting properties with respect to these polynomials. Both operators can be represented as convolution operators

∞ f(t)dt f(t)dt Bf(z) = π and Sf(x) = lim π , ε 0+ 2 cosh 2 (z t) x t >ε 2 sinh 2 (x t) Z−∞ − → Z| − | − leading to the Fourier transforms

Bf(t) = sech tfˆ(t) and Sf(t) = i tanh tfˆ(t).

These two operatorsc can be studied inc the context of either real or complex analysis, and in this thesis we consider the operator B as an operator from

14 Motivation, notation and overview functions on the real line R to functions in the strip S, while the operator S is studied as an operator on functions on R. Throughout this thesis, function spaces on R are denoted by L and those on S by H. For 1 p < , and for an arbitrary nonnegative and locally ≤ ∞ integrable function ω on R, Lp(ω) denotes measurable functions on R with

p ∞ p f p = f(x) ω(x) dx < , (1.3) || ||L (ω) | | ∞ Z−∞ and Hp(ω) analytic functions on S with

p ∞ p f Hp(ω) = sup f(x + ia) ω(x) dx < . (1.4) || || 1recurrence relation so that its exponential generating function is easily computed. Using this, the orthogonality of the system is proved. Investigated also in this chapter 1 1 are boundedness properties of the operators B = R− and S = JR− in the Hilbert spaces related to the three systems. Orthogonal polynomials are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that these two operators are bounded on the L2-spaces, and estimates of the norms are obtained. In Chapter 3, we extend the investigation of the boundedness properties of B and S to Lp spaces (1 < p < ) on the real line, both as convolution p ∞ p operators on L (R) and on the spaces L (ω1), where ω1 is deﬁned by (1.1). It is proved that both operators are bounded on these spaces and estimates of the norms are obtained. This is achieved by ﬁrst proving boundedness for p = 2 and weak boundedness for p = 1, and then using interpolation to obtain boundedness for 1 < p 2. To obtain boundedness also for 2 p < , we use duality in the translation≤ invariant case, while the weighted≤ case is∞ based on the methods developed by Riesz [49] for the conjugate function operator.

15 Orthogonal Polynomials, Operators and Commutation Relations

Finally, in Chapter 4 we derive simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Cen- tralizers and centers are computed as an example of an application of the formulas. This is motivated by the fact that J, R and Q satisfy the relations

QJ JQ = R, (1.5) − − QR RQ = J, (1.6) − JR RJ = 0. (1.7) − In the sequel, we consider the eﬀect of adding the constraint J 2 + R2 = I, where I is the identity operator. For a more general setting, we also consider the eﬀect of replacing RJ with qRJ in relation (1.7), where q is a scalar.

1.2 Orthogonal polynomials

Orthogonal polynomials appear in many areas of mathematics, physics and engineering where they play a vital role. For instance, orthogonal polynomials are central to the development of Fourier series and wavelets which are essential to signal processing. In our particular case, in Chapter 2, we show that orthogonal polynomials can be used to establish the L2-boundedness of singular integral operators which is a fundamental problem in harmonic analysis and a subject of extensive investigations. But ﬁrst we give some elementary theory regarding orthogonal polynomials, and then introduce a special class of orthogonal polynomials, the Meixner–Pollaczek polynomials, to which all the three systems studied in Chapter 2 belong.

1.2.1 Elementary theory We review diﬀerent aspects of the theory of orthogonal polynomials of one real variable that are particularly relevant to the computations in this thesis. For the more general theory, we refer the interested reader to [11, 13, 17, 22, 26, 57, 61, 62, 66, 71] and the references therein. A function ω on R is called a polynomially bounded weight function if it is nonnegative, integrable and all its moments are ﬁnite. These requirements may be written, respectively, as

ω 0, 0 < ω(x)dx < and 0 < x nω(x)dx < , ≥ ∞ | | ∞ ZR ZR

16 Orthogonal polynomials where n is a nonnegative integer. The finite moment property implies that all polynomials are included in L2(ω), the space of measurable functions on with f(x) 2ω(x) dx < . For functions f, g L2(ω), we can define an R R inner product| | ∞ ∈ R (f, g)ω = f(x)g(x)ω(x) dx ZR and the corresponding norm f = (f, f) . The Cauchy-Schwarz in- || ||ω ω equality enables us to bound inner products by norms: (f, g)ω f ω g ω. p 2 ≤ || || || || A system pn n∞=0 of polynomials in L (ω), where every polynomial pn has degree n, is{ called} orthogonal if (p , p ) = 0 for n = m. An orthogonal n m ω 6 system is called orthonormal if (pn, pn)ω = 1 for all n. If p is a polynomial of degree m and m m 1 2 p(x) = cmx + cm 1x − + + c2x + c1x + c0, − ··· then cm is called the leading coefficient of p. If cm = 1, we say that p is a monic polynomial. A useful property of real orthogonal polynomials is that they obey a three-term recurrence relation as described in the following. Proposition 1.2.1. For any given weight function ω, there exists a unique system pn n∞=0 of monic orthogonal polynomials. More precisely, we can construct{ the} monic orthogonal polynomials as follows:

p0(x) = 1, p (x) = x a , 1 − 0 pn+1(x) = xpn(x) anpn(x) bnpn 1(x), − − − 2 2 where an = (xpn, pn)ω/ pn and bn = (xpn, pn 1)ω/ pn 1 . k kω − k − kω Remark 1. This is simply the Gram–Schmidt procedure applied to the se- n 2 quence x ∞ with respect to the L (ω) inner product. { }n=0 Remark 2. The converse of this theorem is known as Favard’s theorem [15]. If ω is an even function, then its integrals with odd polynomials are all zero so that an = 0 for all n. In our paticular case, the weight function π ω1(x) = 1/(2 cosh 2 x) is even so that the three-term recurrence relation reduces to

p0(x) = 1,

p1(x) = x,

pn+1(x) = xpn(x) bnpn 1(x). − −

17 Orthogonal Polynomials, Operators and Commutation Relations

The function ω1 has three other interesting properties that make it useful as a weight function. The ﬁrst is that it a probability density function, and the second is that it is up to a dilation its own Fourier transform, that is, it is the Fourier transform of the function 1/ cosh t. The third is that it is essentially the Poisson kernel for the strip S = z C : Im z < 1 (see [24, p. 5]). These properties make it computationally{ ∈ convenient,| | and} in particular the second property makes it possible to interpret its moments as values at zero of successive derivatives.

1.2.2 Meixner–Pollaczek polynomials (λ) The Meixner–Pollaczek polynomials, denoted by Pn (x; φ) where λ > 0 and 0 < φ < π, are a special class of orthogonal polynomials that were ﬁrst discovered by Meixner [38] and later studied by Pollaczek [46]. These polynomials are deﬁned by the recurrence relation

(λ) (λ) P 1 (x; φ) = 0,P0 (x; φ) = 1, − (λ) (λ) (n + 1)Pn+1(x; φ) = 2(x sin φ + (n + λ) cos φ)Pn (x; φ) (λ) (n + 2λ 1)Pn 1(x; φ), n 1, − − − ≥ (λ) n and their generating function, Gλ(x, s) = n∞=0 Pn (x; φ)s , is given by

iφ λ+ix iφ λ ix G (x, s) = 1 se − P1 se− − − . λ − − The Meixner–Pollaczek polynomials were briefly mentioned by Erdélyiet al. [14] and Szegö[66] . Their basic properties are well studied by Askey and Wilson [6], Chihara [11] and Rahman [47] among others. Asymptotic analysis, limit relations and applications of these polynomials are also well investigated at various levels. For instance, Bender et al. [8] and Koorn- winder [29] have shown that there is a connection between the symmetric (λ) Meixner–Pollaczek polynomials, Pn (x/2; π/2), and some identities for symmetric elements in the Heisenberg algebra. Araaya [4] also discovered a very interesting connection of these polynomials to the umbral calculus developed by Rota [50]. The combinatorial interpretation of the linearization coefficients of these polynomials is discussed by Zeng [72]. The interpretation of the Meixner–Pollaczek polynomials as overlap coefficients in the positive dis- crete series representation of the Lie algebra su(1, 1) are discussed by Koelink and Van der Jeugt [26].

18 Interpolation of Lp-spaces

1.3 Interpolation of Lp-spaces

In Chapter 3, we have used interpolation to establish the Lp-boundedness of the operators B and S for 1 < p 2. Basically, there are two main results of interpolation of operators, the Marcinkiewicz≤ [37, 73] and the Riesz–Thorin [48, 69] interpolation theorems, and in this thesis we make use of the former to prove boundedness for 1 < p 2 and the latter to improve the estimates of the operator norms. ≤ The Riesz–Thorin interpolation theorem allows us to show that a linear operator that is bounded on two Lp spaces is bounded on every Lp space in between the two. The Marcinkiewicz interpolation theorem allows us to show that a sublinear operator that satisﬁes two weak-type estimates is bounded on any Lp space in between the two weak Lp spaces. Therefore, we may simplify the proof of the boundedness of an operator by proving the statement in two simpler cases (say L1 and L2) and then interpolating to prove the statement for every Lp space in between. We assume the reader is familiar with the basic notions of measure theory and integration. We recommend the following reference books as background reading: [16, 18, 51, 60, 64, 68] and the references therein.

1.3.1 Distribution functions and weak Lp Let (X, , µ) be a measure space and 0 < p < . The set of all complex M ∞ measurable functions f : X C such that → 1/p f = f pdµ k kp | | ZX p p is finite is called the L -space, and f p is called the L -norm of f. Given a function f Lp, we define its distributionk k function m : (0, ) [0, ] by ∈ f ∞ → ∞ m (λ) = µ ( x X : f(x) > λ ) . f { ∈ | | } Using this definition, it is easy to see that

(a) mf is decreasing and right continuous, (b) if f g , then m m , | | ≤ | | f ≤ g (c) if f increases to f , then m increases to m , | n| | | fn f (d) if f = g + h, then m (λ) m (λ/2) + m (λ/2). f ≤ g h 19 Orthogonal Polynomials, Operators and Commutation Relations

The distribution function is closely connected to the Lp-norms. For instance, by observing that

p p p p f p = f dµ λ dµ = λ mf (λ) k k X | | ≥ f >λ Z Z| | for all λ > 0, one can relate the distribution function to the Lp-norms by the so called Chebyshev Inequality, 1 m (λ) f p. f ≤ λp k kp One can also use the fact that

p ∞ p dλ f = p 1 f >λλ | | | | λ Z0 and the Fubini-Tonelli theorem to obtain the formula

∞ dλ f p = p m (λ)λp , k kp f λ Z0 which shows that the Lp-norms are essentially the moments of the distribution function. A variant of the Lp-space of fundamental importance is the weak Lp which is deﬁned as the set of all measurable functions f : X C such that → 1/p p [f]p = sup λ mf (λ) λ>0 is ﬁnite. The Chebyshev Inequality implies that

[f] f and Lp Weak Lp, p ≤ k kp ⊂ that is, every function in Lp is in weak Lp. However, the converse is not 1/p always true. For example, f(x) = x− on (0, ), with Lebesgue measure, is in weak Lp but not in Lp. ∞

1.3.2 Interpolation theorems For the Riesz–Thorin interpolation theorem, we assume that (X, , µ) and (Y, , ν) are measure spaces and p , p , q , q [1, ]. If q = qM= , we N 0 1 0 1 ∈ ∞ 0 1 ∞ 20 Interpolation of Lp-spaces

1 further assume that ν is semiﬁnite . Then for 0 < θ < 1, we deﬁne pθ and qθ respectively by

1 1 θ θ 1 1 θ θ = − + and = − + . pθ p0 p1 qθ q0 q1

The assertion is that if T is a linear map from Lp0 (µ) + Lp1 (µ) into Lq0 (ν) + q1 p0 L (ν) such that T f C0 f for all f L (µ) and some C0 > 0, and k kq0 ≤ k kp0 ∈ T f C f for all f Lp1 (µ) and some C > 0, then T is bounded q1 1 p1 1 andk k ≤ k k ∈

1 θ θ T f C − C f k kqθ ≤ 0 1 k kpθ for all f Lpθ (µ). Our problem in Chapter 3 is a special case with p = q = ∈ 0 0 1 and p1 = q1 = 2. We now turn to the Marcinkiewicz interpolation theorem, which allows us to work with sublinear operators. Furthermore, we only require that the operator satisfy weak rather than strong-type estimates. Let us deﬁne what we mean by these terms before we talk about the actual theorem. Let T be a map from a vector space of measurable functions on (X, , µ) to the space of all measurable functionsV on (Y, , ν). M N (a) T is sublinear if T (f + g) T f + T g and T (af) a T f for all f, g and a > |0. | ≤ | | | | | | ≤ | | ∈ V (b) Suppose that 1 p, q . A sublinear map T is strong type (p, q) p ≤ ≤p ∞ q if L (µ) , T maps L (µ) into L (ν), and T f q C f p for all f Lp(µ⊂) and V some C > 0. We also say that Tkis boundedk ≤ k fromk Lp(µ) to∈Lq(ν) with norm at most C.

(c) Suppose that 1 p and 1 q < . A sublinear map T is weak type (p, q)≤ if Lp≤(µ) ∞ , T ≤maps L∞p(µ) into weak Lq(ν), and ⊂p V T f q C f p for all f L (µ) and some C > 0. We also say that T is |weakly| ≤ boundedk k from Lp∈(µ) to Lq(ν).

(d) A map T is weak type (p, ) if and only if T is strong type (p, ). ∞ ∞ 1Given a measure space (X, , µ), a measure µ is called semiﬁnite if for each A with µ(A) = there exists B MA such that B and µ(B) < . ∈ M ∞ ⊂ ∈ M ∞ 21 Orthogonal Polynomials, Operators and Commutation Relations

Now for the Marcinkiewicz theorem, we assume that (X, , µ) and (Y, , ν) M N are measure spaces, and that p0, p1, q0, q1 are elements of [1, ] such that p q , p q and q = q . Furthermore, for 0 < θ < 1, we∞ deﬁne p and 0 ≤ 0 1 ≤ 1 0 6 1 θ qθ respectively by 1 1 θ θ 1 1 θ θ = − + and = − + . p p0 p1 q q0 q1

The assertion is that if T is a sublinear map from Lp0 (µ) + Lp1 (µ) to the space of measurable functions on Y that is weak type (p0, q0) and weak type (p1, q1), then T is strong type (p, q). More precisely, if [T f]q Cj f for j ≤ k kpj some Cj > 0 and j = 0, 1, then

T f B f (1.8) k kq ≤ pk kp where Bp depends only on p, pj, qj and Cj.

1.4 Commutation in noncommutative algebras

1.4.1 General notions

A vector space over a ﬁeld F is called an algebra if it is equipped with a A bilinear multiplication. That is for all A, B, C and p, q F, ∈ A ∈ (pA + qB)C = pAC + qBC and A(pB + qC) = pAB + qAC.

When the multiplication in an algebra has some other special properties, we get some special classes of algebras. For instance, an algebra is called associative if A (AB)C = A(BC) for all A, B, C , and it is called commutative if ∈ A AB = BA for all A, B . ∈ A A linear subspace of an algebra is called a subalgebra if AB for all A, B , and it isB called an idealA if ZB and BZ for all∈B B and Z ∈ B. Clearly every ideal is a subalgebra.∈ B ∈ B ∈ B ∈ A 22 Commutation in noncommutative algebras

Definition 1.4.1. The commutator of two elements A and B of an algebra is defined by A [A, B] = AB BA. (1.9) − Using this definition, it is easy to see that for all A, B, C and p, q F, ∈ A ∈ 1.[ A, q] = 0,

2.[ A, A] = 0,

3.[ A, B] = [B,A], − 4.[ A, pB + qC] = p[A, B] + q[A, C],

5.[ A, BC] = [A, B] C + B[A, C],

6.[ A, [B,C] ] + [B, [C,A] ] + [C, [A, B] ] = 0.

Definition 1.4.2. A vector space l over a field F is called a Lie algebra if it is equipped with a binary operation [ , ]: l l l called the Lie bracket · · × → that satisfies the following axioms for all A, B, C l and p, q F: ∈ ∈ 1.[ pA + qB, C] = p[A, C] + q[B,C], [A, pB + qC] = p[A, B] + q[A, C],

2.[ A, A] = 0,

3.[ A, [B,C]] + [B, [C,A]] + [C, [A, B]] = 0.

The ﬁrst two axioms imply anticommutativity, that is,

[A, B] = [B,A]. − The last axiom is called the Jacobi identity, and it shows that the Lie bracket is in general nonassociative. However, it is easy to see that every associative algebra is a Lie algebra with the Lie bracket given by the commutator, (1.9). A Lie algebra l is said to be commutative if

[A, B] = 0 for all A, B l. ∈ 23 Orthogonal Polynomials, Operators and Commutation Relations

Definition 1.4.3. Let be an algebra. The centralizer of A , denoted by Cen(A), is the set ofA all elements of that commute with A∈. A That is, A Cen(A) = B : AB = BA . { ∈ A } Definition 1.4.4. The center of an algebra , denoted by Z( ), is the set of all elements of that commute with everyA element of . ThatA is, A A Z( ) = A : AB = BA for all B . A { ∈ A ∈ A} It follows that the center of an algebra is the intersection of the centralizers of every element in the algebra. Note that it suffices to take the intersection of the centralizers of the generators. It is easy to see that the center of a Lie algebra l is a commutative ideal in l.

1.4.2 Commutation relations In Chapter 4, motivated by the fact that the operators J, R and Q satisfy certain commutation relations, we consider commutation relations of the form

AB = BF (A), (1.10) where A and B are operators or elements of some associative algebra, and F is some function for which the expression F (A) makes sense. In the purely algebraical context, the function F ( ) is a polynomial in one or several vari- · ables. In general, this function may belong to broader spaces of functions, whenever there exists an appropriate functional calculus appliciable for A. The importance of commutation relations (1.10) can be best seen from some well-known examples. If F (x) = x, then A and B commute, that is,

AB = BA.

If F (x) = x, then A and B anti-commute, that is, − AB = BA. − If F (x) = qx + c for some scalars q and c, then A and B satisfy the commutation relation

AB qBA = cI. −

24 Commutation in noncommutative algebras

This is a deformed Heisenberg–Lie commutation relation of quantum mechanics. The famous classical Heisenberg–Lie relation AB BA = I − is obtained when q = 1 and c = 1. If c = 0, then A and B satisfy the relation AB = qBA often called the quantum plane relation in the context of noncommutative geometry and quantum groups. If F (x) = qxd, then A and B satisfy the commutation relation AB = qBAd. This reduces to the quantum plane relation for d = 1 and to the relation AB = BAd for q = 1, having important applications, for instance in wavelet analysis and in investigation of transfer operators [9, 20, 23] which are fundamental for statistical physics, dynamical systems and ergodic theory. Commutation relations of the form (1.10) play a central role in the study of crossed products and their representations, in the theory of dynamical systems and in the investigation of covariant systems and systems of imprim- itivity and thus in quantum mechanics, statistical physics and quantum ﬁeld theory [9, 10, 12, 20, 23, 32, 34, 35, 36, 42, 53, 65, 70]. Commutation relations of the form (1.10) arise in the investigations of nonlinear Poisson brackets, quantization and noncommutative analysis [25, 41]. Bounded and unbounded operators satisfying the relation (1.10) have also been considered in the context of representations of -algebras and spectral theory [52, 54, 55, 58, 59]. In the study of commutation∗ relations (1.10) and their representations, an important role is played by the dynamical system generated by the map F . A central problem in investigation of structure, representation theory and applications of noncommutative algebras is the description of commuting elements in an algebra and description of commuting operators in the represent- ing operator algebra, or in other words, the problem of explicit description of commutative subalgebras. Commutative subalgebras or commuting families of operators are a key ingredient in representation theory of many important algebras [43, 44, 45, 56, 65]. The commuting operators are also of key importance in the study of integrable systems and non-linear equations [19].

25 Orthogonal Polynomials, Operators and Commutation Relations

1.5 Summary of Chapters 2–4

1.5.1 Chapter 2 This chapter ﬁrst introduces the ρ-system of orthogonal polynomials and establishes its connection to the two systems studied by Araaya, and then 2 1 1 discusses the L -boundedness of the operators B = R− and S = JR− . One new result is the following theorem which describes the ρ-system and 2 an orthogonal basis for the Hilbert space L (ω2).

Theorem 1.5.1. Let the system ρ ∞ be given by the recurrence relation { n}n=0 ρ 1 = 0, ρ0 = 1 and ρn+1(x) = xρn(x) n(n + 1)ρn 1(x). − − − (a) The function ρn is a monic polynomial of degree n for n 0. ≥ sn ∞ (b) The exponential generating function, Gρ(x, s) = n=0 ρn(x) n! , is given by the function ex arctan s P G (x, s) = . ρ 1 + s2 ρn (c) The sequence of polynomials n! n∞=0 is an orthogonal basis for the Hilbert 2 { } space L (ω2) with norms √n + 1. Another new result is the following theorem which connects the ρ-system to the Araaya systems in terms of the operators J, R and Q. Theorem 1.5.2. The following connections between the three systems of orthogonal polynomials σ , τ and ρ hold: { n} { n} { n} Rσn = τn,

Jσn = nτn 1, − Rτn = ρn,

Jτn = nρn 1, − Qρn = σn+1. Remark 3. The ﬁrst two connections were known by Araaya (see [2, Thm 3]). The new results are the last three connections. Remark 4. These connections are used in the later part of this chapter to illustrate that orthogonal polynomials can be used to establish the L2- boundedness of singular integral operators. More precisely, the L2-boundedness of the operators B and S is established.

26 Summary of Chapters 2–4

As already mentioned, the later part of this chapter investigates boundedness properties of the operators B and S, both as convolution operators

∞ f(t)dt f(t)dt Bf(z) = π and Sf(x) = lim π , ε 0+ 2 cosh 2 (z t) x t >ε 2 sinh 2 (x t) Z−∞ − → Z| − | − in the Hilbert spaces related to the three systems. The connections between the systems in Theorem 1.5.2 are used to prove boundedness in the weighted spaces and Fourier analysis is used to prove boundedness in the translation invariant case. It is proved in both cases that the two operators are bounded on L2-spaces, and estimates of the norms are obtained.

Theorem 1.5.3. The operator S is linear and bounded with norm 1 on

(a) L2(R), 2 (b) L (ω1), 2 (c) L (ω2). Theorem 1.5.4. The operator B is linear and bounded with norm less than or equal to 2 from

(a) L2(R) to H2(S), 2 2 (b) L (ω1) to H (ω1), 2 2 (c) L (ω2) to H (ω2). Remark 5. See (1.1), (1.2), (1.3) and (1.4) for the notation.

Even though the operator R maps functions in S to functions on R so that 1 B = R− is most natural as an operator mapping functions on R to functions on S, it is of some interest to consider it also as an operator on spaces of functions on R. To do this, we use the Hadamard three lines theorem, and in order to use that, we map the weighted H2-spaces into the unweighted ones. For a similar argument, see [7, Theorem 2.1].

Lemma 1.5.1. The following hold:

(a) If f H2(ω ) and ϕ(z) = 1/(cosh π z), then the operator m deﬁned by ∈ 1 4 ϕ

mϕf(z) = ϕ(z)f(z)

is an isometric isomorphism onto H2(S).

27 Orthogonal Polynomials, Operators and Commutation Relations

(b) If f H2(ω ) and ψ(z) = √ω , then the operator m deﬁned by ∈ 2 2 ψ

mψf(z) = ψ(z)f(z)

is a bounded isomorphism onto H2(S) with norm π/2. We then have the following result. p

Theorem 1.5.5. If the operator T as an operator on functions on the real line is deﬁned by T f(x) = Bf(x), x R, ∈ then T is linear and bounded on the following spaces:

(a) L2(R) with norm 1. 2 (b) L (ω1) with norm less than or equal to 2√2. 2 (c) L (ω2) with norm less than or equal to √2π.

1.5.2 Chapter 3 This chapter extends the investigation of the boundedness properties of the operators B and S to Lp-spaces (1 < p < ) on the real line, both as p ∞ p convolution operators on L (R) and on the spaces L (ω1). It is proved that both operators are bounded on these spaces and estimates of the norms are obtained for diﬀerent values of p. Our main results are the following.

Theorem 1.5.6. For 1 < p < , the operator B is linear and bounded from ∞ (a) Lp(R) to Hp(R), p p (b) L (ω1) to H (ω1).

Theorem 1.5.7. For 1 < p < , the operator S is linear and bounded on p p ∞ the spaces L (R) and L (ω1). For p = 2, both of these results are proved in Chapter 2. In this chapter we ﬁrst prove weak boundedness for p = 1, and then use interpolation to obtain boundedness for 1 < p 2. To obtain boundedness for 2 p < , we use duality in the translation≤ invariant case, while the weighted≤ case∞ is partly based on the methods developed by M. Riesz [49] for the conjugate function operator.

28 Summary of Chapters 2–4

1.5.3 Chapter 4 This chapter derives simple and explicit formulas for reordering elements in an algebra with three generators and Lie type relations. Centralizers and centers are computed as an example of an application of the formulas. This is motivated by the fact that the operators J, R and Q satisfy the relations QJ JQ = R, (1.11) − − QR RQ = J, (1.12) − JR RJ = 0. (1.13) − In the sequel, we consider the effect of adding the constraint J 2 + R2 = I, where I is the identity operator. For a more general setting, we also consider the effect of replacing RJ with qRJ in relation (1.13), where q is a scalar. One remarkable result is that the commutation of Q with J, R-polynomials is a differential operator, that is, ∂p(J, R) ∂p(J, R) [Q, p(J, R)] = R + J . − ∂J ∂R Now writing S = J + iR and T = J iR, we have J = (S + T )/2, R = (S T )/2i and ST = J 2 + R2. Therefore,− denoting q(S,T ) = p((S + T )/2, (S− T )/2i), we see that polynomials in J and R can also be written as polynomials− in S and T . Another remarkable result is that for all nonnegative integers k, m and n, QkSmT n = SmT n (Q + (m n)i)k . − For k = 1, we have [Q, SmT n] = (m n)iSmT n. (1.14) − m n This result implies that given a polynomial p(S,T ) = cmnS T , we have m n [Q, p(S,T )] = (m n)icmnS T . It follows that [Q, p(J, R)] = 0 if and − P only if for all m, n, either m n = 0 or cmn = 0. Therefore, if p belongs to P − k the algebra center then p(S,T ) = ck(ST ) , that is, n P 2 2 k p(J, R) = ck(J + R ) . Xk=0 Finally, (1.14) implies that SmT n is an eigenvector of [Q, ] with eigenvalue · (m n)i. −

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