On a Singular Sylvester Equation with Unbounded Self-Adjoint A and B
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Complex Analysis and Operator Theory
On a Singular Sylvester Equation with Unbounded Self-Adjoint A and B Bogdan D. Djordjevi´c1 Received: 25 January 2020 / Accepted: 9 April 2020 © Springer Nature Switzerland AG 2020
Abstract In this paper we solve the Sylvester equation AX − X B = C, where A and B are closed densely defined self-adjoint operators, and C is a linear operator. We obtain sufficient conditions for the existence of infinitely many solutions and we manage to classify them. These results generalize the previously known results regarding singular Sylvester equations. Finally, we illustrate our results on an operator equation that appears in quantum mechanics, called the position-momentum operator equation. Keywords Sylvester equations · Closed operators · Spectral theory of self-adjoint operators Mathematics Subject Classification Primary 47A62 · 47A60; Secondary 47A70 · 47A75
1 Introduction Equations of the form AX − X B = C
(1.1)
Communicated by Vladimir Bolotnikov. This article is part of the topical collection “Linear Operators and Linear System” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz. The author is supported by the Ministry of Science, Republic of Serbia, Grant No. 174007.
B 1
Bogdan D. Djordjevi´c [email protected]; [email protected] Mathematical Institute of Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Beograd, p.p. 367, Serbia 0123456789().: V,-vol
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B. D. Djordjevi´ c
are called Sylvesters’ equations, where, in general, A ∈ L(K ) B ∈ L(H ) and C ∈ L(H , K ), for given Banach spaces H and K . They were introduced by J. J. Sylvester in 1884, when he proved the fundamental result concerning such equations in matrices. Theorem 1.1 [21] Let A, B and C be matrices of appropriate dimensions. The equation (1.1) has a unique solution if and only if σ (A) ∩ σ (B) = ∅. It wasn’t until the mid 1900s when Rosenblum extended the results to bounded linear operators. Theorem 1.2 [19] Let A, B and C be bounded linear operators on the corresponding Banach spaces. If σ (A) ∩ σ (B) = ∅, then the equation (1.1) has a unique bounded solution. Remark In the bounded operator setting, the converse statement does not hold. A trivial counterexample was provided in [10]. For this reason, if σ (A) ∩ σ (B) = ∅, then the equation (1.1) is said to be regular. Regular equations have been studied extensively so far, with various applications in theoretical and applied mathematics, physics and engineering. An interested reader is referred to the survey [2] and rich references therein, articles [3,6,11,15,22] and their numerous references. In addition, there are several results regarding a unique bounded solution to (1.1), while the operators at hand are unbounded, consult [13,16] and [18]. These results have a huge impact on mathematical physics, quantum mechanics and abstract differential equations. If the equation (1.1) is not uniquely solvable for the afore-given A, B and C, then it is called singular. Contrar
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