摘 要 利用算子理论方法证明了Hilbert C*-模上的可伴算子序列是g-Riesz基且有唯一对偶g-框架当且仅当相应的合成算子是一线性同胚,这修正了已有的一个结论.进一步,作为该结果的直接应用,給出了Hilbert C*-模中的g-Riesz基具有唯一对偶g-框架的保界等价刻画.
关键词 Hilbert C*-模;g-框架;g-Riesz基;对偶g-框架
中图分类号 O177.1文献标识码 A文章编号 1000-2537(2017)06-0080-07
Abstract The present paper proves, by utilizing the method of operator theory, that a sequence of adjointable operators on a Hilbert C*-module is a g-Riesz basis with unique dual g-frame if and only if the corresponding synthesis operator is a homeomorphism, which provides a correction to one existing conclusion and further, as a direct application of this result, it gives an equivalent characterization for g-Riesz bases with unique dual g-frames in Hilbert C*-modules, which preserves the g-frame bounds.
Key words Hilbert C*-module; g-frame; g-Riesz basis; dual g-frame
参考文献:
[1] DUFFIN R J, SCHAEFFER A C. A class of nonharmonic Fourier series [J]. Trans Am Math Soc, 1952,72(2):341-366.
[2] DAUBECHIES I, GROSSMANN A, MEYER Y. Painless nonorthogonal expansions [J]. J Math Phys, 1986,27(5):1271-1283.
[3] CHRISTENSEN O. An introduction to frames and Riesz bases [M]. Boston: Birkhuser, 2002.
[4] 郭训香. Hilbert空间上的预框架算子[J].应用数学学报, 2012,35(5):795-803.
[5] 杨守志,郑贤伟.L2(Rd)上的半正交多小波框架[J].中国科学:数学, 2014,44(3):249-262.
[6] 吴国昌,曹怀信,鲁大勇.波包Parseval框架的刻画及应用[J].数学学报, 2015,58(1):91-102.
[7] SUN W C. G-frames and g-Riesz bases [J]. J Math Anal Appl, 2006,322(1):437-452.
[8] FRANK M, LARSON D. Frames in Hilbert C*-modules and C*-algebras [J]. J Operator Theory, 2002,48(2):273-314.
[9] KHOSRAVI A, KHOSRAVI B. Fusion frames and g-frames in Hilbert C*-modules [J]. Int J Wavelets Multiresolut Inf Process, 2008,6(3):433-446.
[10] ARAMBASIC L, BAKIC D. Frames and outer frames for Hilbert C*-modules [J]. Linear Multilinear A, 2017,65(2):381-431.
[11] HAN D G, JING W, LARSON D, et al. Dilation of dual frame pairs in Hilbert C*-modules [J]. Results Math, 2013,63(1-2):241-250.
[12] XIANG Z Q. A note on the stability of g-frames in Hilbert C*-modules [J]. Int J Wavelets Multiresolut Inf Process, 2016,14(4):ID1650031.
[13] XIANG Z Q, LI Y M. G-frames for operators in Hilbert C*-modules [J]. Turk J Math, 2016,40(2):453-469.
[14] ALIJANI A. Generalized frames with C*-valued bounds and their operator duals [J]. Filomat, 2015,29(7):1469-1479.
[15] 相中启,简辉华. Hilbert C*-模中对偶g-框架的稳定性[J].湖南师范大学自然科学学报, 2015,38(2):68-73.
[16] XIAO X C, ZENG X M. Some properties of g-frames in Hilbert C*-modules [J]. J Math Anal Appl, 2010,363(2):399-408.
[17] ARAMBASIC L. On frames for countably generated Hilbert C*-modules [J]. Proc Am Math Soc, 2007,135(2):469-478.
[18] XU Q X, SHENG L J. Positive semi-definite matrices of adjointable operators on Hilbert C*-modules [J].Linear Algebra Appl, 2008,428(4):992-1000.