q -Difference Operator and Its q -Cohyponormality
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Complex Analysis and Operator Theory
q-Difference Operator and Its q-Cohyponormality Meltem Sertba¸s1 · Co¸skun Saral2 Received: 30 January 2020 / Accepted: 1 October 2020 © Springer Nature Switzerland AG 2020
Abstract In this study the minimal and maximal operator generated by q-difference expression and their adjoint operators are introduced in L q2 (0, 1). Any closed extension of the minimal operator is studied. Also, the structure of the spectrum of the minimal operator and the maximal operator are investigated. Keywords Minimal and maximal operators · q-difference operator · q-hyponormal operator · q-cohyponormal operator · Spectrum Mathematics Subject Classification 39A13 · 47A05 · 47A10
1 Introduction and Preliminaries The q-calculus was started in the eighteenth century by Euler [1], but the definition of q-integral was given by Jackson in 1910 [2]. The q-calculus is often called calculus without limits. It allows the substitution of the classical derivative with the qderivative operator to deal with sets of nondifferentiable functions. The q-calculus has an unexpected role in several mathematical areas such as fractal geometry, quantum theory, hypergeometric functions, orthogonal polynomials, the calculus of variation and theory of relativity. Also, researches in the q-calculus, especially focused on approximation by positive linear operators, have been continually coming out such as
Communicated by Ilwoo Cho.
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Meltem Sertba¸s [email protected] Co¸skun Saral [email protected]
1
Department of Mathematics, Faculty of Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey
2
Institute of Natural Sciences, Karadeniz Technical University, 61080 Trabzon, Turkey 0123456789().: V,-vol
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M. Sertba, C. Saral
[3,4]. The main books [5,6] and [7] can be cited for some results related to the history of quantum calculus, its basic concepts and q-differential equations. In [7], Annaby and Mansour investigated a q-analogue of Sturm–Liouville problems in L q2 (0, a), 0 < a < +∞. However, they need to extend the domains of functions in L q2 (0, a) to [0, q −1 a], because they can write the formal adjoint operator of qdifference operator. This is not necessary, since it is well known that a dense define operator has always the adjoint operator. We deal with this problem in this study. Let λ ∈ R be fixed. A subset M of C is called λ-geometric if λt ∈ M for every t ∈ M. If a subset M of C is λ-geometric then it contains all geometric sequences {λn t}∞ n=0 , t ∈ M. If u is a function, real or complex valued and defined on a q-geometric set M, |q| = 1. The q-difference operator or q-derivative, which was reintroduced by Jackson[8], is defined by
Dq u (t) =
u (t) − u (qt) , t ∈ M\{0}. (1 − q)t
The product formula of q-derivative is Dq (uv)(t) = u(t)Dq v(t) + u(qt)Dq v(t). The q-difference operator sometimes is called Jackson q-difference operator, Euler– Jackson q-difference operator or Euler–Heine–Jackson q-difference operator. If 0 ∈ M the q-derivative at zero is defined for |q|
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