加权Fock空间F_α~2上的测不准原理的推广（英文）
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摘要
对α> 0,本文主要研究了复平面上的加权Fock空间F_α~2上的自伴算子和线性算子的测不准原理.利用泛函分析中的一般性原理,在F_α~2上构造了两个线性算子Tf=(f′)/α和T*=zf.进一步,构造了满足条件的两个自伴算子A和B,使得[A,B]为恒等算子的常数倍,得到了F_α~2上更精确的算子的测不准原理形式,其中T*是T的对偶算子,[A,B]=AB-BA为A和B的换位置.本文的结果推广并完善了屈非非和朱克和在文献[1]和[2]中的结果.
In this article, for α > 0, we characterize several versions uncertainty principles of self-adjoint operators and linear operators for the α-fock space F_α~2 in the complex plane. By using the general result from functional analysis, we find two linear operators Tf =(f′)/α and T*= zf to construct two self-adjoint operators A and B such that[A,B] is a scalar multiple of the identity operator on F_α~2 and obtain some more accurate results about the uncertainty principles for theα-fock space F_α~2, where T* is the adjoint of T, [A, B] = AB-BA is the commutator of A and B,which extends and completes the results of Qu [1] and Zhu [2].
In this article, for α > 0, we characterize several versions uncertainty principles of self-adjoint operators and linear operators for the α-fock space F_α~2 in the complex plane. By using the general result from functional analysis, we find two linear operators Tf =(f′)/α and T*= zf to construct two self-adjoint operators A and B such that[A,B] is a scalar multiple of the identity operator on F_α~2 and obtain some more accurate results about the uncertainty principles for theα-fock space F_α~2, where T* is the adjoint of T, [A, B] = AB-BA is the commutator of A and B,which extends and completes the results of Qu [1] and Zhu [2].
引文
[1] Qu F F, Deng G T. Further discussion on uncertainty principles for the fock space[J]. J. Beijing Normal Univ.(Nat. Sci.), 2017, 2, 53(1):1-5.
[2] Chen Y, Zhu K H. Uncertainty principles for the Fock space[J]. Sci. Sin. Math., 2015, 45(11):1847-1854.
[3] Gabor D. Theory of communication[J]. lee Proc. London, 1946, 93(73):58.
[4] Qu F F, Deng G T, Wen Z H. Uncertainty principles of linear operators for Fock space[J]. J. Beijing Normal Univ.(Nat. Sci.), 2017, 53(2):133-136.
[5] Benedetto J J. Frame decompositions, sampling, and uncertainty principle inequalities[M].Benedetto J J, Frazier M W, eds. Math. Appl., Wavelets:CRC Press, 1994.
[6] Dym H, Mckean H P. Fourier series and integrals[M]. New York:Academic Press, 1972.
[7] Folland G B, Sitaram A. The uncertainty principle:a mathematical survey[J]. J. Fourier Anal.Appl., 1997, 3(3):207-238.
[8] Faris W G. Inequalities and uncertainty principles[J]. J. Math. Phys., 1978, 19(2):461-466.
[9] Selig K K. Uncertainty principles revisited[J]. Electr. Trans. Numer. Anal., 2002, 21(359):165-177.
[10] Wel H. The theory of groups and quantum mechanics[J]. Dover Publ., 1950, 75:586.
[11] Tawfik A N, Diab A M. Review on generalized uncertainty principle[J]. Egyptian Center The. Phys.,2015, 01, arXiv:1509.02436.
[12] Breitenberger E. Uncerainty measure and uncertainty relations for angle observables[J]. Found.Phys., 1985, 15:353-364.
[13] Havin V. The uncertainty principle in harmonic analysis[M]. Berlin:Springer-Verlag, 1994.
[14] Grochenifg K. Foundations of time-frequency analysis[M]. Boston:Birkhauser, 2001, 33(6):464-466.
[15] Cohen L. Time-frequency analysis:theory and applications[M]. Digital Signal Processing with Matlab Examples, Vol. 1, Madrid:Universidad Complutense de Madrid, 1994.
[16] Pan W Y, Yang C L, Zhao J. Uncertainty principles for theα-Fock spaceF_α~2[J]. Pure Math., 2018,3(2):149-163.
[17] Goh S S, Micchelli C A. Uncertainty principles in Hilbert spaces[J]. J. Fourier Anal., 2002, 8(4):335-374.
[2] Chen Y, Zhu K H. Uncertainty principles for the Fock space[J]. Sci. Sin. Math., 2015, 45(11):1847-1854.
[3] Gabor D. Theory of communication[J]. lee Proc. London, 1946, 93(73):58.
[4] Qu F F, Deng G T, Wen Z H. Uncertainty principles of linear operators for Fock space[J]. J. Beijing Normal Univ.(Nat. Sci.), 2017, 53(2):133-136.
[5] Benedetto J J. Frame decompositions, sampling, and uncertainty principle inequalities[M].Benedetto J J, Frazier M W, eds. Math. Appl., Wavelets:CRC Press, 1994.
[6] Dym H, Mckean H P. Fourier series and integrals[M]. New York:Academic Press, 1972.
[7] Folland G B, Sitaram A. The uncertainty principle:a mathematical survey[J]. J. Fourier Anal.Appl., 1997, 3(3):207-238.
[8] Faris W G. Inequalities and uncertainty principles[J]. J. Math. Phys., 1978, 19(2):461-466.
[9] Selig K K. Uncertainty principles revisited[J]. Electr. Trans. Numer. Anal., 2002, 21(359):165-177.
[10] Wel H. The theory of groups and quantum mechanics[J]. Dover Publ., 1950, 75:586.
[11] Tawfik A N, Diab A M. Review on generalized uncertainty principle[J]. Egyptian Center The. Phys.,2015, 01, arXiv:1509.02436.
[12] Breitenberger E. Uncerainty measure and uncertainty relations for angle observables[J]. Found.Phys., 1985, 15:353-364.
[13] Havin V. The uncertainty principle in harmonic analysis[M]. Berlin:Springer-Verlag, 1994.
[14] Grochenifg K. Foundations of time-frequency analysis[M]. Boston:Birkhauser, 2001, 33(6):464-466.
[15] Cohen L. Time-frequency analysis:theory and applications[M]. Digital Signal Processing with Matlab Examples, Vol. 1, Madrid:Universidad Complutense de Madrid, 1994.
[16] Pan W Y, Yang C L, Zhao J. Uncertainty principles for theα-Fock spaceF_α~2[J]. Pure Math., 2018,3(2):149-163.
[17] Goh S S, Micchelli C A. Uncertainty principles in Hilbert spaces[J]. J. Fourier Anal., 2002, 8(4):335-374.