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Shift Operator in L2 Space

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Shift Operator in L2 Space Shift Operator in `2 Space Johan Balkare (890801-1672) [email protected] SA104X Degree Project in Engineering Physics, First Level Department of Mathematics Royal Institute of Technology (KTH) Supervisor: Serguei Shimorin May 20, 2014 Abstract A Hilbert space H is the abstraction of a finite-dimensional Eu- clidean space. The spectrum of a bounded linear operator A : H ! H , denoted σ(A), is given by all numbers λ 2 C such that (A − λI) is not invertible. The shift operators are one type of bounded linear op- erators. In this report we prove five claims regarding the spectrum of the shifts. We work in the Hilbert space `2 which consists of all square summable sequences, both single sided (x0; x1; x2; :::) and double sided (:::x−1; x0; x1; :::). One of the most general results proved applies to 2 the weighted unilateral shift Sα defined for (x0; x1; x2; :::) 2 ` by Sα(x0; x1; x2; :::) = (0; α0x0; α1x1; α2x2; :::) where fαng is a bounded arbitrary weight sequence with αn > 0 for all n ≥ 0. Theorem. Let r(Sα) be the radius of the smallest disc which contain σ(Sα). Then σ(Sα) = fλ : jλj ≤ r(Sα)g: Contents 1 Introduction 1 2 Preliminaries 2 2.1 Hilbert spaces . 2 2.2 Unilateral and bilateral shift operators . 3 2.3 Bounded linear operators in Hilbert spaces . 3 2.4 Adjoint of an operator . 5 2.5 The Spectrum . 6 2.6 Other concepts . 7 3 Spectrum of shift operators 8 3.1 Unilateral forward and backward shifts . 8 3.2 Weighted unilateral shift . 11 3.3 Weighted bilateral shift . 16 3.4 Weighted unilateral shift with arbitrary weight sequence . 19 References 22 1 Introduction The theory of the so called shift operators, or simply the shifts, is one of the major areas of research in mathematics. The shift is used as a mathe- matical tool in several areas. In quantum mechanics we recognize it as the lowering/raising operator. It acts by decreasing/increasing the eigenvalues of other operators (Harmonic oscillator and Angular momentum operators). We can also see the shifts in control theory where it works as a translator of time in Tustin's formula and in the Euler backward/forward methods. There are many different shift operators but one thing they have in com- mon is translations of some scale. In this report we will study shifts of general form and characterise their spectrum, a generalisation of the concept of eigenvalues. The purpose of introducing the spectrum is to construct an analogy between diagonalization of linear operators in finite-dimensional spaces to the infinite-dimensional case. We will not study diagonalization in this report. We will also to restrict our work to shifts in Hilbert spaces, especially to the so called `2 spaces. A Hilbert space is the abstraction of the finite-dimensional Euclidean spaces. The properties are very regular but the presence of an infinity of dimensions makes it spectacular. Historically, it was the properties of Hilbert spaces that guided mathematicians when they began to generalize the methods of vector algebra and calculus to any finite or infite-dimensional space. The goal is to provide the reader with understanding of spectral theory, a branch of functional analysis, with focus on this operator. In section 2 we deal with some theory to give the reader necessary foun- dation about operators in Hilbert spaces and the spectrum of an operator. Some of the proofs are taken from the literature and some are written by own hand. If a proof is admitted we always give an reference. It is in section 3 most of the work have been done. Here we do the reasoning and calculations to obtain the spectrum of certain shifts. The report may also act as an encouragement for further studies and research in the subject. We mainly target third year students in engineering or mathematics. 1 2 Preliminaries 2.1 Hilbert spaces Definition 1. A Hilbert space is a vector space H over F (R or C) together with an inner product h·; ·i such that relative to the metric d(x; y) = kx−yk induced by the norm k · k2 = h·; ·i, H is a complete metric space. 2 The Hilbert space we mainly will consider in further sections is ` (N) which denote the set of all square summable sequences. We will represent such sequences by vectors with infinitely many elements x = (x0; x1; x2; :::). 2 For x, y 2 ` (N) the space is endowed with the inner product 1 X hx; yi = xkyk (1) k=0 where the bar · indicates the complex conjugate. So a sequence x lies in 1 2 2 P 2 ` (N) if kxk = hx; xi = jxnj < 1. n=0 2 We will also consider the related space ` (Z). The difference is that in this space the sequences are double sided x = (:::; x−1; x0; x1; x2; :::) and the sum in the inner product (1) goes from −∞ to +1. We are going to use standard orthonormal bases for these spaces; fe0; e1; e2; :::g 2 2 for ` (N) and f:::; e−1; e0; e1; :::g for ` (Z) where en is the sequence with a 2 single "1" on place number n and the rest are zeroes. For example, in ` (N) we have e0 = (1; 0; 0; :::); e1 = (0; 1; 0; :::); e2 = (0; 0; 1; 0; :::); ::: . Our first result below is general for all Hilbert spaces. Proposition 1. Let fxng be a sequence of vectors in a Hilbert space H such that 1 X kxnk < 1: n=0 1 P Then the series xn converges in H . n=0 Proof. Let " > 0, then there is an N" 2 N such that v u v X X X kxnk − kxnk = kxnk < " n=0 n=0 n=u+1 for all v ≥ u ≥ N". The well known triangle inequality then gives v v v u X X X X " > kxnk ≥ xn = xn − xn : n=u+1 n=u+1 n=0 n=0 2 v P Hence, xn is a Cauchy sequence. By completeness of H we have n=0 1 P that xn is convergent. n=0 2.2 Unilateral and bilateral shift operators 2 The unilateral forward shift S 2 ` (N) is one of the standard shifts. 2 Definition 2. For x 2 ` (N) the unilateral forward shift is the operator 2 2 S : ` (N) ! ` (N) defined by Sx = S(x0; x1; x2; :::) = (0; x0; x1; x2; :::) (2) 2 and in terms of the standard basis in ` (N) that is Sen = en+1. 2 The corresponding operator in ` (Z) is the bilateral forward shift. 2 Definition 3. For x 2 ` (Z) the bilateral forward shift is the operator 2 2 T : ` (Z) ! ` (Z) defined by T (:::; x−1; x0; x1; :::) = (:::; x−2; x−1; x0; :::) (3) 2 and in terms of the standard basis in ` (Z) that is T en = en+1. If there is no doubt if we consider the unilateral or bilateral shift, we may just say the forward shift. 2.3 Bounded linear operators in Hilbert spaces For Hilbert spaces H and K , linearity of an operator A : H ! K are defined as usual, that is if for h1 h2 2 H , α, β 2 F, A fulfills the distrubutive law: A(αh1 + βh2) = αAh1 + βAh2: Definition 4. A bounded linear operator A on H is a linear operator for which there is a constant c > 0 such that jAhj < ckhk for all h 2 H . For a bounded linear operator A : H ! K , define kAk = supfjAhj : khk ≤ 1; h 2 H g: (4) Note that by definition, kAk < 1. kAk is called the norm of the operator A. We let B(H ; K ) denote the set of bounded linear operators from H to K . If H = K , then for short we write B(H ). Proposition 2. Let A 2 B(H ; K ) and B 2 B(K ; L ), then 3 (a) kAhk ≤ kAkkhk for every h 2 H . (b) BA 2 B(H ; L ) and kABk ≤ kAkkBk Proof. For (a), let " > 0. By definition of kAk we get that Ah ≤ kAk khk + " Hence kAhk ≤ kAk (khk + ") and letting " ! 0 we have kAhk ≤ kAkkhk For (b), if k 2 K , then (a) gives kBkk ≤ kBkkkk. Hence, if h 2 H then there is a k 2 K such that k = Ah 2 K so kBAhk ≤ kBkkAhk ≤ kBkkAkkhk. Taking the sup over all h such that khk ≤ 1 of both sides proves the assertion. For the unilateral and bilateral forward shifts S and T , defined by (2) 2 2 and (3), it is easy to see that S 2 B(` (N)) and T 2 B(` (Z)). With the use of (4) we have kSk = 1 and kT k = 1. Proposition 3. Let K 2 B(H ) with kKk < 1. Then (I − K) is invertible and the inverse is given by 1 X (I − K)−1 = I + K + K2 + K3 + ::: = Kn n=0 where the series on the right, called the Neuman series, converges uniformly in B(H ). Proof. The operator (I −K) is invertible if and only if there exists (I −K)−1 such that (I − K)−1(I − K) = (I − K)(I − K)−1 = I: We see that 1 N X X (I − K) Kn = (I − K) lim Kn N!1 n=0 n=0 N X = lim (I − K) Kn N!1 n=0 N N X X = lim Kn − Kn+1 N!1 n=0 n=0 = lim I − KN+1: N!1 4 1 Now, forming the convergent sum P kKkn we get by Proposition 2 (b) that n=0 1 1 X X kKkn ≥ kKnk n=0 n=0 1 By Proposition (1) we have that P Kn converges uniformly which says that n=0 KN+1 ! 0 as N ! 1.
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