Multipliers of continuous $G$-frames in Hilbert spaces

Document Type : Research Paper


Department of Mathematics‎, ‎Faculty of Sciences‎, ‎University of Mohaghegh Ardabili‎, ‎Ardabil 56199-11367‎, ‎Iran.


In this paper we introduce continuous $g$-Bessel multipliers in Hilbert spaces and investigate some of their properties. We provide some conditions under which a continuous $g$-Bessel multiplier is a compact operator. Also, we show the continuous dependency of continuous $g$-Bessel multipliers on their parameters.


Main Subjects

M.R‎. ‎Abdollahpour and F‎. ‎Bagarello‎, ‎ On some properties of g-frames and g-coherent states‎, Nuovo Cimento Soc‎. ‎Ital‎. ‎Fis‎. ‎B125 (2010)‎, ‎no‎. ‎11‎, ‎ 327--1342‎.
‎M.R‎. ‎Abdollahpour and M.H‎. ‎Faroughi‎, ‎ Continuous g-Frames in Hilbert spaces‎, ‎ Southeast Asian Bull‎. ‎Math.‎ 32 (2008) 1--19‎.
‎M.R‎. ‎Abdollahpour and A‎. ‎Najati‎, ‎ G-frames and Hilbert-Schmidt operators‎, ‎ Bull‎. ‎Iranian Math‎. ‎Soc.‎ 37 (2011)‎, ‎no‎. ‎4‎, ‎141--155‎.
‎‎M.R‎. ‎Abdollahpour and A‎. ‎Najati‎, ‎ Besselian g-frames and near g-Riesz‎ ‎bases‎, ‎ Appl‎. ‎Anal‎. ‎Discrete Math.‎ 5 (2011) 259--270‎.
‎‎S.T‎. ‎Ali‎, ‎J.P‎. ‎Antoine and J.P‎. ‎Gazeau‎, ‎ Continuous frames in Hilbert spaces‎, ‎ Ann‎.. ‎Physics‎ 222 (1993) 1--37‎.
‎‎A.A‎. ‎Arefijamaal‎, ‎R.A‎. ‎Kamyabi Gol‎, ‎R‎. ‎Raisi‎, ‎and N‎. ‎Tavallaei‎, ‎ A new approach to continuous Riesz bases‎, ‎ J‎. ‎Sci‎. ‎Islam‎. ‎Repub‎. ‎Iran24 (2013)‎, ‎no‎. ‎1‎, ‎63--69‎.
‎‎‎P‎. ‎Balazs‎, ‎ Basic definition and properties of Bessel multipliers‎, ‎ J‎. ‎Math‎. ‎Anal‎. ‎Appl.‎ 325 (2007)‎, ‎no‎. ‎1‎, ‎571--585‎.
‎‎‎P‎. ‎Balazs‎, ‎D‎. ‎Bayer and A‎. ‎Rahimi‎, ‎ Multipliers for continuous frames in Hilbert spaces‎, ‎ J‎. ‎phys‎. ‎A‎ 45 (2012)‎, ‎Article ID 244023‎, ‎20 pages‎.
‎‎‎P‎. ‎Balazs‎, ‎B‎. ‎Laback‎, ‎G‎. ‎Eckel and W.A‎. ‎Deutsch‎, ‎ Time frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking‎, ‎ IEEE Trans‎. ‎Audio Speech Lang‎. ‎Process.‎ 18 (2010) 34--49‎.
‎P‎. ‎Berg and J‎. ‎Löfström‎, ‎ Interpolation Spaces‎, ‎An Introduction‎, ‎Grundlehren der Mathmatischen Wissenschaften 223‎, ‎ Springer‎, ‎Berlin‎, ‎1976‎.
‎‎‎J.B‎. ‎Conway‎, ‎ A Course in Functional Analysis‎, ‎ Springer-Verlag‎, ‎New York‎, ‎1990‎.
‎‎Ph‎. ‎Depalle‎, ‎R‎. ‎Kronland-Martinet and B‎. ‎Torresani‎, ‎ Time-frequency multipliers for sound synthesis‎, ‎ in‎: ‎Proceedings of Wavelet XII Confrence‎, ‎SPIE Annual Symposium‎, ‎ San Diego‎, ‎August 2007‎.
‎R.J‎. ‎Duffin and A.C‎. ‎Schaeffer‎, ‎ A class of nonharmonic Fourier series‎, ‎ Trans‎. ‎Amer‎. ‎Math‎. ‎Soc.‎ 72 (1952) 341--366‎.
‎‎‎P‎. ‎Majdak‎, ‎P‎. ‎Balazs and B‎. ‎Laback‎, ‎ Multiple exponential sweep method for fast measurement of head related transfer functions‎, ‎ J‎. ‎Audio Eng‎. ‎Soc.55 (2007) 623--637‎.
‎A‎. ‎Najati and A‎. ‎Rahimi‎, ‎ Generalized frames in Hilbert spaces‎, ‎ Bull‎. ‎Iranian Math‎. ‎Soc.‎ 35 (2009)‎, ‎no‎. ‎1‎, ‎97--109‎.
‎‎‎A‎. ‎Rahimi‎, ‎ Multipliers of generalized frames in Hilbert spaces‎, ‎ Bull‎. ‎Iranian Math‎. ‎Soc.‎ 37 (2011)‎, ‎no‎. ‎1‎, ‎63--80‎.
‎A‎. ‎Rahimi and P‎. ‎Balazs‎, ‎ Multipliers for $p$-Bessel sequences in Banach spaces‎,‎ Integral Equations Operator Theory68 (2010)‎, ‎no‎. ‎2‎, ‎193--205‎.
‎W‎. ‎Sun‎, ‎ G-Frames and g-Riesz bases‎, ‎ J‎. ‎Math‎. ‎Anal‎. ‎Appl.‎ 322 (2006)‎, ‎no‎. ‎1‎, ‎437--452‎.
‎‎‎D‎. ‎Wang and G.J‎. ‎Brown‎, ‎ Computational Auditory Scene Analysis‎: ‎Principles‎, ‎Algorithms‎, ‎and Applications‎, ‎ Wiley IEEE‎, ‎New York‎, ‎2006‎.
‎‎‎K‎. ‎Zhu‎, ‎ Operator Theory in Functions Spaces‎, ‎ Marcel Dekker Inc.‎, ‎New York‎, ‎1990‎.