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The Numerical Radius of a Weighted Shift Operator∗ 1 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 944-963, December 2015 ELA THE NUMERICAL RADIUS OF A WEIGHTED SHIFT OPERATOR∗ BATZORIG UNDRAKH†, HIROSHI NAKAZATO‡, ADIYASUREN VANDANJAV§, AND MAO-TING CHIEN¶ Abstract. In this paper, the point spectrum of the real Hermitian part of a weighted shift operator with weight sequence a1, a2,...,an, 1, 1,... is investigated and the numerical radius of the weighted shift operator in terms of the weighted shift matrix with weights a1, a2,...,an is formulated explicitly. Key words. Numerical radius, Weighted shift operator, Weighted shift matrix. AMS subject classifications. 47A12, 15A60. 1. Introduction. Let A be an operator on a separable Hilbert space H. The numerical range of A is defined to be the set W (A)= Ax, x : x =1, x H . {h i k k ∈ } The numerical radius w(A) is the supremum of the modulus of W (A). It is a classical result due to Toeplitz and Hausdorff that the numerical range is a convex set. For references on the theory of numerical range, see, for instance, [1, 10, 11, 12]. We consider a weighted shift operator A with weights (a1,a2,...) on the Hilbert space ∗Received by the editors on December 7, 2014. Accepted for publication on November 28, 2015. Handling Editor: Michael Tsatsomeros. †Institute of Mathematics, National University of Mongolia, Ulaanbaatar, Mongolia. Current address: School of Mathematics and Statistics, Newcastle University, Newcastle upon and Tyne, NE1 7RU, United Kingdom (batzorig [email protected]). ‡Department of Mathematical Sciences, Faculty of Science and Technology, Hirosaki University, Hirosaki 036-8561, Japan ([email protected]). §Department of Mathematics, National University of Mongolia, Ulaanbaatar, Mongo- lia, and Department of Mathematics, New Mongol Institute of Technology, Mongolia (v [email protected]). ¶Department of Mathematics, Soochow University, Taipei 11102, Taiwan ([email protected]). Partially supported by Taiwan Ministry of Science and Technology under NSC 102-2115-M-031-001, and the J.T. Tai & Co Foundation Visiting Research Program. 944 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 944-963, December 2015 ELA The Numerical Radius of a Weighted Shift Operator 945 ℓ2(N) defined by 0 0 0 ...... .. .. a1 0 0 . .. .. A = A(a1,a2,...)= 0 a2 0 . , .. .. 0 0 a3 . . .. .. .. .. . where a is a bounded sequence. From the operator-theoretic view point, the class { n} of weighted shift operators contains a typical non-unitary isometry A(1, 1,...) (cf. [2]). The real part Hermitian operator (A) = (A + A∗)/2 of a weighted operator A ℜ is interpreted as the adjacency matrix of weighted A∞-graph (cf. [8]). In addition, weighted shift operators are closely related to numerical analysis and information theory (cf. [4]). Shields [16] proved that the numerical range W (A) of a weighted shift operator A is a circular disk centered at the origin. In this case, the radius of the disk equals its numerical radius w(A) which is the maximal spectrum of the self-adjoint operator (A). Ridge [15] computed the radius for a weighted shift operator with periodic ℜ weights. Computations of the radii of weighted shift operators with typical weights such as (r, 1, 1,...), (1, s, 1, 1,...), (r, s, 1, 1,...) and (r, r2, r3,...) were carried out in [2, 5, 6, 18, 19]. Stout [17] provided a method to obtain the numerical radius of a weighted shift operator A(a1,a2,...) with square summable weights by introducing the analytic function F (z) = det(I z (A(a ,a ,...))). A − ℜ 1 2 It is shown in [17] that the analytic function is given by ∞ 1 k F (z)=1+ c z2k, A −4 k Xk=1 where 2 2 2 ck = a a a , i1 i2 ··· ik the sum being taken over X 1 i <i < <i < , i i 2, i i 2,..., i i − 2, ≤ 1 2 ··· k ∞ 2 − 1 ≥ 3 − 2 ≥ k − k 1 ≥ and the radius w(A(a1,a2,...)) = 1/λ, where λ is the minimal positive root of FA(z) = 0. Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 944-963, December 2015 ELA 946 B. Undrakh, H. Nakazato, A. Vandanjav, and M.T. Chien In the finite-dimensional case, an n n weighted shift matrix with weights × (a1,a2,...,an−1) is defined by 0 0 0 ... 0 .. a1 0 0 . A(a ,a ,...,a − )= . 1 2 n 1 0 a2 0 . . .. .. .. . 0 0 an−1 0 It is easy to see that a weighted shift matrix or operator A is unitarily equivalent to its entry-wise modulus operator A . Hence we may assume that the weights are | | nonnegative for the discussion of the numerical range of a weight shift matrix or operator. There have been a number of interesting papers on the properties of the numerical ranges of weighted shift matrices ([3, 6, 7, 9, 16, 17, 20]). The numerical range of a weighted matrix A is a closed disk centered at the origin. Various radii of the disk were studied, e.g.,[6, 13, 19]. In particular, w(A(1, 1,..., 1)) = cos(π/(n+1)) for the n n shift matrix A(1, 1,..., 1) (cf. [13]). The numerical radii of the modi- × fied shift matrices A(1,..., 1, r, 1,..., 1) and A(1,..., 1, r, r, 1,..., 1) were computed respectively in [6] and [19]. In this paper, we consider weighted shift operators A(a1,a2,...,an, 1, 1,...) which perturb the canonical shift operator A(1, 1,...), and investigate the point spectrum of the Hermitian operator (A(a ,a ,...,a , 1, 1,...)) which gives the numerical ℜ 1 2 n radius of the operator A(a1,a2,...,an, 1, 1,...). Furthermore, we explicitly formu- late the radius w(A(a1,a2,...,an, 1, 1,...)) in terms of the weighted shift matrix A(a1,a2,...,an). 2. Weighted shift matrices. Let A(a ,a ,...,a − ) be an n n weighted 1 2 n 1 × shift matrix with nonnegative weights. The spectral analysis of a real symmetric tridiagonal matrix (A) is related to numerical analysis (cf. [4]). The Hermitian ℜ matrix 2I 2 (A(1,..., 1)) is also discussed as the discrete Laplacian in applied n − ℜ mechanics ([14]). The characteristic polynomial p (t) = det tI (A(a ,a ,...,a − )) n n −ℜ 1 2 n 1 has the recurrence 1 2 p (t)= tp − (t) a p − (t). n n 1 − 4 n−1 n 2 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 944-963, December 2015 ELA The Numerical Radius of a Weighted Shift Operator 947 By the formula [17, Lemma 1], k n 1 n−2k (2.1) p (t)= t + S (a ,...,a − )t , n −4 k 1 n 1 1≤Xk≤n/2 where the circularly symmetric functions 2 2 2 Sk(a1,...,an−1)= a a a , j1 j2 ··· jk ≤ ··· k ≤ − 1 j1<j2<X<j n 1 over all 1 j < j < < j n 1 satisfying j j 2, j j 2,..., ≤ 1 2 ··· k ≤ − 2 − 1 ≥ 3 − 2 ≥ jk jk−1 2. The circularly symmetric function Sk(a1,...,an−1) is abbreviated to (n−−1) ≥ Sk if there is no confusion. We will use Chebyshev polynomials of the second kind to find α = w(A(a1,a2,...,an−1)). If the weights a1,...,an−1 are positive and relatively small, e.g., less than 1, we have 0 <α< 1. Then 0 < θ0 = arccos(α) < π is the minimal zero of the trigonometric polynomial n (2.2) 2 sin θ det cos θI (A(a ,a ,...,a − )) . n −ℜ 1 2 n 1 On the other hand, if the weights a1,...,an−1 are relatively large, say, greater than sec(π/(n + 1)), then α > 1 and θ0 = arccosh(α) is the minimal zero of the trigono- metric polynomial n 2 sinh θ det cosh θI (A(a ,a ,...,a − )) . n −ℜ 1 2 n 1 Let C , 0 k ℓ, denote the binomial coefficients with boundary values C = ℓ k ≤ ≤ ℓ 0 C = 1. For 1 k ℓ 1, C = ℓ!/(k!(ℓ k)!). The following lemma is essential ℓ ℓ ≤ ≤ − ℓ k − to expand the trigonometric polynomial (2.2). Lemma 2.1. n−1 n n 2 sin θ cos θ = − C sin((n +1 2k)θ) n 1 k − (2.3) Xk=0 = ( − C − C − ) sin((n +1 2k)θ), n 1 k − n 1 k 2 − 0≤Xk≤n/2 and n−1 n n 2 sinh θ cosh θ = − C sinh((n +1 2k)θ) n 1 k − (2.4) Xk=0 = ( − C − C − ) sinh((n +1 2k)θ). n 1 k − n 1 k 2 − 0≤Xk≤n/2 Electronic Journal of Linear Algebra ISSN 1081-3810 A publication of the International Linear Algebra Society Volume 30, pp. 944-963, December 2015 ELA 948 B. Undrakh, H. Nakazato, A. Vandanjav, and M.T. Chien Proof. We compute n−1 n−1 i(n+1−2k)θ − C sin((n +1 2k)θ)= − C e n 1 k − ℑ n 1 k Xk=0 kX=0 n−1 i(n+1)θ −2ikθ = e − C e ℑ n 1 k Xk=0 n−1 = ei(n+1)θ 1+ e−2iθ ℑ n−1 = e2iθ eiθ + e−iθ ℑ =2n−1 cos n−1 θ (e2iθ) ℑ =2n sin θ cosn θ, where (z) denotes the imaginary part of a complex number.
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