In this paper, the notion of an $ \mathcal{A} $-group-like unitary system on a Hilbert $ C^{\ast} $-module is introduced and some basic properties are studied, where $ \mathcal{A} $ is a unital $ C^{\ast} $-algebra. Let $ \mathcal{U} $ be such a unitary system. We prove that a complete Parseval frame vector for $ \mathcal{U} $ can be dilated to a complete wandering vector. Also, it is shown that the set of all the complete Bessel vectors for $ \mathcal{U} $ can be parameterized by the set of the adjointable operators in the double commutant of $ \mathcal{U} $, and that the frame multiplicity of $ \mathcal{U} $ is always finite.
Citation: Xiujiao Chi, Pengtong Li. Characterization of frame vectors for $ \mathcal{A} $-group-like unitary systems on Hilbert $ C^{\ast} $-modules[J]. AIMS Mathematics, 2023, 8(9): 22112-22126. doi: 10.3934/math.20231127
In this paper, the notion of an $ \mathcal{A} $-group-like unitary system on a Hilbert $ C^{\ast} $-module is introduced and some basic properties are studied, where $ \mathcal{A} $ is a unital $ C^{\ast} $-algebra. Let $ \mathcal{U} $ be such a unitary system. We prove that a complete Parseval frame vector for $ \mathcal{U} $ can be dilated to a complete wandering vector. Also, it is shown that the set of all the complete Bessel vectors for $ \mathcal{U} $ can be parameterized by the set of the adjointable operators in the double commutant of $ \mathcal{U} $, and that the frame multiplicity of $ \mathcal{U} $ is always finite.
| [1] |
S. T. Ali, J. P. Antoine, J. P. Gazeau, Continuous frames in Hilbert space, Ann. Phys., 222 (1993), 1–37. https://doi.org/10.1006/aphy.1993.1016 doi: 10.1006/aphy.1993.1016
|
| [2] |
L. Aramba${\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{{\rm{s}}}}$ić, On frames for countably generated Hilbert space $C^{\ast}$-modules, Proc. Amer. Math. Soc., 135 (2007), 469–478. https://doi.org/10.1090/S0002-9939-06-08498-X doi: 10.1090/S0002-9939-06-08498-X
|
| [3] |
J. J. Benedetto, S. Li, The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmon. Anal., 5 (1998), 389–427. http://dx.doi.org/10.1006/acha.1997.0237 doi: 10.1006/acha.1997.0237
|
| [4] | O. Christensen, An Introduction to Frames and Riesz Bases, Boston: Birkhäuser, 2003. http://dx.doi.org/10.1007/978-3-319-25613-9 |
| [5] |
S. Dahlke, M. Fornasier, T. Raasch, Adaptive frame methods for elliptic operator equations, Adv. Comput. Math., 27 (2007), 27–63. http://dx.doi.org/10.1007/s10444-005-7501-6 doi: 10.1007/s10444-005-7501-6
|
| [6] |
X. Dai, D. Larson, Wandering vectors for unitary systems and orthogonal wavelets, Memoirs Amer. Math. Soc., 134 (1998), 27–63. http://dx.doi.org/10.1090/memo/0640 doi: 10.1090/memo/0640
|
| [7] |
R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366. http://dx.doi.org/10.2307/1990760 doi: 10.2307/1990760
|
| [8] |
M. Frank, D. Larson, Modular frames for Hilbert $C^{\ast}$-modules and symmetric approximation of frames, Proc. SPIE, 4119 (2000), 325–336. http://dx.doi.org/10.1117/12.408617 doi: 10.1117/12.408617
|
| [9] | M. Frank, D. Larson, Frames in Hilbert $C^{\ast}$-modules and $C^{\ast}$-algebras, J. Operator Theory, 48 (2002), 273–314. |
| [10] | J. P. Gabardo, D. Han, Frame representations for group-like unitary operator systems, J. Operator Theory, 49 (2003), 1–22. |
| [11] | D. Han, D. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc., 147 (2000), 1–94. |
| [12] |
W. Jing, D. Han, R. Mohapatra, Structured Parseval frames in Hilbert $C^{\ast}$-modules, Contemp. Math., 414 (2006), 275–287. https://doi.org/10.48550/arXiv.math/0603091 doi: 10.48550/arXiv.math/0603091
|
| [13] |
A. Khorsavi, B. Khorsavi, Fusion frames and $g$-frames in Hilbert $C^{\ast}$-modules, Int. J. Wavelets Multi. Inf. Proc., 6 (2008), 433–446. https://doi.org/10.1142/S0219691308002458 doi: 10.1142/S0219691308002458
|
| [14] | E. C. Lance, Hilbert $C^{\ast}$-Modules: A Toolkit for Operator Algebraists, Cambridge: Cambridge University Press, 1995. http://dx.doi.org/10.1017/CBO9780511526206 |
| [15] |
M. Mahmoudieh, H. Hosseinnezhad, G. A. Tabadkan, Multi-Frame Vectors for Unitary Systems in Hilbert $C^{\ast}$-modules, Sahand Commun. Math. Anal., 15 (2019), 1–18. https://doi.org/10.22130/scma.2018.77908.356 doi: 10.22130/scma.2018.77908.356
|
| [16] |
J. Packer, Applications of the work of Stone and von Neumann to wavelets, Contemp. Math., 365 (2004), 253–279. http://dx.doi.org/10.1090/conm/365/06706 doi: 10.1090/conm/365/06706
|
| [17] |
W. Paschke, Inner product modules over $B^{\ast}$-algebras, Trans. Am. Math. Soc., 182 (1973), 443–468. http://dx.doi.org/10.1090/S0002-9947-1973-0355613-0. doi: 10.1090/S0002-9947-1973-0355613-0
|
| [18] |
M. Rossafi, F. D. Nhari, Controlled $K$-$g$-fusion frames in Hilbert $C^{\ast}$-modules, Int. J. Anal. Appl., 20 (2022). https://doi.org/10.28924/2291-8639-20-2022-1 doi: 10.28924/2291-8639-20-2022-1
|
| [19] | N. Wegge-Olsen, $K$-Theory and $C^{\ast}$-Algebras, A Friendly Approach, Oxford: Oxford University Press, 1993. http://doi.org/10.1112/blms/27.2.196 |
| [20] |
P. Wood, Wavelets and Hilbert modules, J. Fourier Anal. Appl., 10 (2004), 573–598. http://dx.doi.org/10.1007/s00041-004-0828-4 doi: 10.1007/s00041-004-0828-4
|
| [21] |
Z. Q. Xiang, On $K$-frame generators for unitary systems in Hilbert $C^{\ast}$-modules, J. Pseudo-Differ. Oper. Appl., 14 (2021). https://doi.org/10.1007/s11868-021-00377-z doi: 10.1007/s11868-021-00377-z
|
Xiujiao Chi, Pengtong Li. Characterization of frame vectors for $ \mathcal{A} $-group-like unitary systems on Hilbert $ C^{\ast} $-modules[J]. AIMS Mathematics, 2023, 8(9): 22112-22126. doi: 10.3934/math.20231127
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