Weighted composition operator on quaternionic Fock space
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Banach J. Math. Anal. https://doi.org/10.1007/s43037-020-00087-6 ORIGINAL PAPER
Weighted composition operator on quaternionic Fock space Pan Lian1 · Yuxia Liang1 Received: 18 March 2020 / Accepted: 6 August 2020 © Tusi Mathematical Research Group (TMRG) 2020
Abstract This paper is concerned with several important properties of weighted composition operator acting on the quaternionic Fock space F2 (ℍ) . Complete equivalent characterizations for its boundedness and compactness are established. As corollaries, the descriptions for composition operator and multiplication operator on F2 (ℍ) are presented, which can indicate some well-known existing theories in complex Fock space. Finally, as an appendix the closed expression for the kernel function of F2 (ℍ) is exhibited, which can deepen the understanding of F2 (ℍ). Keywords weighted composition operator · quaternionic Fock space · boundedness · compactness · closed expression Mathematics Subject Classification 30G35 · 47B38 · 30H20
1 Introduction It is known that many mathematicians have been in creating a theory of slice regular functions of a quaternionic variable, which would somehow resemble the classical theory of holomorphic functions of one complex variable. Indeed the beginning of this theory is due to two important work by Gentili and Struppa [9, 10] and equally it gets a lot of interests. And then several significant theories have been developed systematically and found a wide range of applications, especially promising for the study of quaternionic quantum mechanics, see e.g. [4]. For more about slice regular Communicated by Manuel Maestre. * Yuxia Liang [email protected] Pan Lian [email protected] 1
School of Mathematical Science, Tianjin Normal University, Tianjin 300387, People’s Republic of China Vol.:(0123456789)
P. Lian and Y. Liang
function theory, we refer the readers into the detailed monographs [5, 11] and their reference therein. In Sect. 2, some preliminaries for quaternions and slice regular functions will be recalled for our further use. Up to now, there appear various slice regular function spaces along with the welldevelopment of the theory of regular functions. We refer the readers to [1, 8] for Fock space, [18] for Bloch, Besov and Dirichlet spaces and [3] for Bergman space, in the slice regular settings. These quaternionic spaces have attracted interest in the past decade for their various applications especially in operator theory. Among them, Fock spaces play important role in quantum mechanics and quaternionic formulation, and a full account of properties for the Fock spaces in the holomorphic setting have been formulated in [19]. Fortunately, the recent study of slice regular Fock spaces over quaternions is carried out in [1] and it also presents the Fock space of slice monogenic functions with values in a Clifford algebra, which generalize the corresponding case in the excellent book [19]. The flavor of these results gives a great impulse to the theory of quaternionic function spaces. For a long time, one of the
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