\(g\)-Bessel Sequences and Operators
DOI:
https://doi.org/10.26713/cma.v7i2.398Keywords:
g-Bessel sequence, g-Orthonormal basis, g-Riesz basis, Bounded linear operatorAbstract
Let $H$ be a separable Hilbert space, and let \(g\mathfrak{B}\) be the set of all \(g\)-Bessel sequences for \(H\). We show that \(g\mathfrak{B}\) is a \(C^*\)-algebra isometrically isomorphic to \(L(H)\) (the algebra of all bounded linear operators of \(H\)). Also, we classify \(g\)-Bessel sequences in \(H\) in terms of different kinds of operators in \(L(H)\). Using operator theory tools, we investigate geometry of \(g\)-Bessel sequences.Downloads
References
A.A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of g-frames, Turk. J. Math. 37(2013), 71–79.
M.L. Arias, G. Corach and M. Pacheco, Characterization of Bessel sequences, Extracta Mathematicae 22 (2007), 55–66.
D. Bakic and T. Beri, Finite extensions of Bessel sequences, Banach J. Math. Anal. 9 (2015), 1–13.
O. Christensen, An Introduction to Frames and Riesz Bases, Boston, Birkhaüser (2003).
J.B. Conway, Subnormal Operators, Pitman, London (1981).
G. Corach, M. Pacheco and D. Stojanoff, Geometry of epimorphisms and frames, Proc. Amer. Math. Soc. 132 (2004), 2039–2050.
M.A. Dehghan and M. Mesbah, Operators, frames and convergence of sequences of Bessel sequences, U.P.B. Sci. Bull., Series A, 77 (2015), 75–86.
M.L. Ding and Y.C. Zhu, g-Besselian frames in Hilbert spaces, Acta Math. Sin. (Engl. ser.) 26 (2010), 2117–2130.
R.G. Douglas, Banach Algebra Techniques in Operator Theory, 2nd edition, Springer-Verlag, New York (1998).
P.R. Halmos, A Hilbert Space Problem Book, 2nd edition, Springer-Verlag, New York (1982).
P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–993.
D. Han and D.R. Larson, Frames, basis and group representations, Mem. Amer. Math. Soc. 697 (2000).
R.Q. Jia, Bessel sequences in Sobolev spaces, Appl. Comput. Harmon. Anal. 20 (2006), 298–311.
N.J. Kalton, N.T. Peck and J.W. Roberts, An F-space sampler, London Math. Soc. Lect. Notes, 89, Cambridge University Press, Cambridge (1984).
A. Khosravi and K. Musazadeh, Fusion frames and g-frames, J. Math. Anal. Appl. 342 (2008), 1068–1083.
J. Li and Y. Zhu, Some equalities and inequalities for g-Bessel sequences in Hilbert spaces, Appl. Math. Lett. 25 (2012), 1601-1607.
J. Spurní½, A note on compact operators on normed linear spaces, Expositiones Mathematicae 25 (2007), 261–263.
W. Sun, g-frames and g-Riesz bases, J. Math. Anal. Appl. 322 (2006), 437–452.
Y.J. Wang and Y.C. Zhu, g-frames and g-frame sequences in Hilbert spaces, Acta Math. Sin. (Engl. ser.) 25 (2009), 2093–2106.
Y.C. Zhu, Characterization of g-frames and g-Riesz bases in Hilbert spaces, Acta Math. Sin. (Engl. ser.) 24 (2008), 1727–1736.
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