Spectra of the constant Jacobi matrices on Banach sequence spaces
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Spectra of the constant Jacobi matrices on Banach sequence spaces Saad R. El-Shabrawy1
· Asmaa M. Shindy1
Received: 4 December 2019 / Accepted: 23 July 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract This paper presents an investigation of the spectra of the infinite tridiagonal Jacobi matrices with constant entries as operators acting on Banach sequence spaces. It is shown that the spectra vary depending on the Banach sequence space under consideration. However, the methods presented in this paper are flexible to be adapted to study the spectral problem of the Jacobi matrices in many sequence spaces. Keywords Infinite matrices · Jacobi operators · Sequence spaces · Spectrum Mathematics Subject Classification Primary 47A10 · 47B36; Secondary 47B37 · 46B45
1 Introduction and the main results We specify the spectra of infinite-dimensional Jacobi matrices, that are, symmetric (infinite) tridiagonal matrices; ⎡ ⎤ a b 0 0 ··· ⎢b a b 0 ···⎥ ⎢ ⎥ ⎢ ⎥ J (a, b) = ⎢ 0 b a b · · · ⎥ , ⎢0 0 b a ···⎥ ⎣ ⎦ .. .. .. .. . . . . . . . where a and b are given real numbers with b = 0. For simplicity, we assume that b > 0. However, there is no limitation caused by the condition b > 0 since the case b < 0 can be treated by considering −J (a, b); see Remark 1.1 below. Let μ be a Banach sequence space. The matrix J (a, b) may be identified with the operator J (a, b) : μ −→ μ; (J (a, b) x)n = bxn−1 + axn + bxn+1 ,
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n = 0, 1, 2, . . . ,
Saad R. El-Shabrawy [email protected] Asmaa M. Shindy [email protected]
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Department of Mathematics, Faculty of Science, Damietta University, New Damietta 34517, Egypt 0123456789().: V,-vol
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S. R. El-Shabrawy, A. M. Shindy
where x = (xk ) = (xk )∞ k=0 ∈ μ and x −1 = 0. We call it the constant Jacobi operator (or simply Jacobi operator). The J (a, b) has also appeared under the name of tridiagonal symmetric matrix in [1] or second order difference operator in [15]. The free Jacobi operator J0 is the Jacobi operator J (0, 21 ). It is clear that the operator J0 generates J (a, b) with the help of identity operator; see Remark 4.1. The term Jacobi matrix unfortunately has more than one meaning in the literature. For example, we refer to [7, p. 99], [10, p. 36] and [11, p. 86] for different meanings of this term. Furthermore, in [6], a generalized Jacobi matrix is considered, where it is shown that its inverse can be raised in terms of the resolution of a boundary value problem associated with a second order linear difference equation. Amon others, several applications are presented related with Schrödinger equations and boundary value problems; like classification, regularity and resolvent kernels. A definition like ours has been used, for example, in [4, p. 503], where the case we consider here is the Jacobi matrix with constant entries. Let 0 = 0 (N0 ) be the set of all sequences x = (xk ) = (xk )∞ k=0 of complex numbers. By ∞ , c and c0 , we denote the Banach spaces of bounded, convergent and null sequences of complex numbers with the supremum norm,
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