Abstract
In this article, the properties of completeness, being Riesz, being basis and minimality of a pair of g-sequences \(\Lambda =\{\Lambda _i :\mathcal {H}\longrightarrow \mathcal {H}_i\}_{i \in {\mathbb {I}}}\) and \(\Gamma =\{\Gamma _i :\mathcal {H}\longrightarrow \mathcal {H}_i\}_{i \in {\mathbb {I}}}\) are simultaneously studied as well as investigating the above-mentioned properties about the sequences of subspaces induced by them. We show that the above properties are an extension of the definitions known about a single g-sequence \(\Lambda\). The effect of invertibility of the multiplier operator of sequences \(\Lambda ,\Gamma\) on the above-mentioned properties will be investigated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balazs P, Stoeva DT (2007) Basic definition and properties of Bessel multipliers. J Math Anal Appl 325(1):571–585
Balazs P, Diana TS (2015) Representation of the inverse of a frame multiplier. J Math Anal Appl 422(2):981–994
Bilalov BT, Ismailov MI, Mamedova ZV (2018) Uncountable frames in non-separable Hilbert spaces and their characterization. Azerb J Math 8.1:151–178
Casazza PG (2000) The art of frame theory. Taiwan J Math 4:129–201
Casazza PG, Kutyniok G (2004) Frames of subspaces. Contemp Math 345:87–113
Christensen Ole (2014) A short introduction to frames, Gabor systems, and wavelet systems. Azerb J Math 4(1):25–39
Christensen O (2008) Frames and bases: an introductory course. Birkhuser, Boston MA
Christensen O, Laugesen RS (2010) Approximately dual frames in Hilbert spaces and applications to Gabor frames. Sampl Theory Signal Image Process 9:77–89
Daraby B, Delzendeh F, Rostami A, Rahimi A (2019) Fuzzy normed linear spaces and fuzzy frames. Azerb J Math 9(2):96–121
Daubechies I, Grossmann A, Meyer Y (1986) Painless nonorthogonal expansions. J Math Phys 27:1271–1283
Duffin RJ, Schaeffer AC (1952) A class of nonharmonic Fourier series. Trans Am Math Soc 72:341–366
Gabor D (1946) Theory of communication. J Inst Electr Eng 93:429–457
Gavruta P (2007) On the duality of fusion frames. J Math Anal Appl 333(2):871–879
Guo X (2015) On redundancy, dilations and canonical duals of \(g\)-frames in Hilbert spaces. Banach J Math Anal 9(4):81–99
Heil C (2010) A basis theory primer, expanded. Springer Science & Business Media, Berlin
Ismailov MI, Nasibov YI (2016) On one generalization of Banach frame. Azerb J Math 6(2):143–159
Khosravi A, Musazadeh K (2008) Fusion frames and \(g\)-frames. J Math Anal Appl 342:1068–1083
Li JZ, Zhu YC (2011) Exact g-frames in Hilbert spaces. J Math Anal Appl 374(1):201–209
Najati A, Faroughi MH, Rahimi A (2008) \(G\)-frames and stability of \(g\)-frames in Hilbert spaces. Methods Funct Anal Topol 14:271–286
Rashidi-Kouchi M, Asghar Rahimi A, Shah FA (2018) Duals and multipliers of controlled frames in Hilbert spaces. Int J Wavelets Multiresolut Inf Process 16(06):1850057
Stoeva DT, Balazs P (2012) Invertibility of multipliers. Appl Comput Harmon Anal 33:292–299
Stoeva DT, Balazs P (2013) Canonical forms of unconditionally convergent multipliers. J Math Anal Appl 399(1):252–259
Sun W (2006) \(G\)-frames and \(g\)-Riesz bases. J. Math. Anal. Appl. 322(1):437–452
Sun W (2007) Stability of g-frames. J Math Anal Appl 326(2):858–868
Young R (1980) An introduction to nonharmonic Fourier series. Academic Press, New York
Zhu YC (2008) Characterizations of \(g\)-frames and \(g\)-Riesz bases in Hilbert spaces. Acta Math Sin Springer 24(10):1727–1736
Funding
The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.
Author information
Authors and Affiliations
Contributions
The first draft of the manuscript was written by A. Fereydooni and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that there is no conflict of interest.
Rights and permissions
About this article
Cite this article
Fereydooni, A., Rahimi, A. Riesz Properties of Multiplication of a Pair of g-Sequences. Iran J Sci Technol Trans Sci 46, 927–935 (2022). https://doi.org/10.1007/s40995-022-01308-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40995-022-01308-3