2017
8
1
0
106
$G$Frames for operators in Hilbert spaces
2
2
$K$frames as a generalization of frames were introduced by L. Gu{a}vruc{t}a to study atomic systems on Hilbert spaces which allows, in a stable way, to reconstruct elements from the range of the bounded linear operator $K$ in a Hilbert space. Recently some generalizations of this concept are introduced and some of its difference with ordinary frames are studied. In this paper, we give a new generalization of $K$frames. After proving some characterizations of generalized $K$frames, new results are investigated and some new perturbation results are established. Finally, we give several characterizations of $K$duals.
1

1
21


Bahram
Dastourian
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, P.O. Box 115991775, Iran.
Department of Pure Mathematics, Ferdowsi
Iran
bdastorian@gmail.com


Mohammad
Janfada
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, P.O. Box 115991775, Iran.
Department of Pure Mathematics, Ferdowsi
Iran
janfada@um.ac.ir
$g$atomic system
$g$$K$frame
$g$$K$dual
Perturbation
[[1] A. Abdollahi and E. Rahimi, Some results on gframes in Hilbert spaces, Turk. J. Math., 35 (2011) 695704.##[2] M.R. Abdollahpour, M.H. Faroughi, and A. Rahimi, Pgframes in Banach spaces, Methods Funct. Anal. Topology, 13 (2007) no. 3, 201210.##[3] A.A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of gframes, Turk. J. Math., 37 (2013) 7179.##[4] M.S. Asgari and H. Rahimi, Generalized frames for operators in Hilbert spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17 (2014) no. 2, 1450013 (20 pages).##[5] H. Bolcskei, F. Hlawatsch and H.G. Feichtinger, Frametheoretic analyssis of oversampled filter banks, IEEE Trans. Signal Process., 46 (1998) 32563268.##[6] O. Christensen, An Introduction to Frame and Riesz Bases, Birkhäuser, 2002.##[7] B. Dastourian and M. Janfada, *frames for operators on Hilbert modules, Wavelets and Linear algebras., 3 (2016) 2743.##[8] B. Dastourian and M. Janfada, Frames for operators in Banach spaces via semiinner products, Int. J. Wavelets Multiresult. Inf. Process., 14 (2016) no. 3, 1650011 (17 pages).##[9] I. Daubechies, A. Grossmann, and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys., 27 (1986) 12711283.##[10] R.G. Douglas, On majorization, factorization and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc., 17 (1966) no. 2, 413415.##[11] N.E. Dudey Ward and J.R. Partington, A construction of rational wavelets and frames in HardySobolev space with applications to system modelling, SIAM J. Control Optim., 36 (1998) 654679.##[12] J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952) 341366.##[13] Y.C. Eldar, Sampling with arbitrary sampling and reconstruction spaces and oblique dual frame vectors, J. Fourier. Anal. Appl., 9 (2003) no. 1, 7796.##[14] Y.C. Eldar and T. Werther, General framework for consistent sampling in Hilbert spaces, Int. J. Wavelets Multi. Inf. Process., 3 (2005) no. 3, 347359.##[15] H.G. Feichtinger and T. Werther, Atomic systems for subspaces, in: L. Zayed (Ed.), Proceedings SampTA 2001, Orlando, FL, (2001) 163165.##[16] P.J.S.G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In: Byrnes, J.S. (ed.) Signal processing for multimedia, IOS Press, Amsterdam (1999) 3554.##[17] L. Gávruta, Frames for operators, Appl. Comput. Harmon. Anal., 32 (2012) 139144.##[18] A. Khosravi and K. Musazadeh, Fusion frames and gframes, J. Math. Anal. Appl., 342 (2008) 10681083.##[19] A. Najati, M.H. Faroughi, and A. Rahimi, Gframes and stability of gframes in Hilbert space, Methods Funct. Anal. Topology, 14 (2008) 271286.##[20] F.A. Neyshaburi and A.A. Arefijamaal, Some constructions of Kframes and their duals, To appear in Rocky Mountain J. Math.##[21] S. Obeidat, S. Samarah, P.G. Casazza, and J. C. Tremain, Sums of Hilbert space frames, J. Math. Anal. Appl., 351 (2009) 579585.##[22] T. Strohmer and R. Jr. Heath, Grassmanian frames with applications to coding and communications, Appl. Comput. Harmon. Anal., 14 (2003) 257275.##[23] W. Sun, gframes and gRiesz bases, J. Math. Anal. Appl., 322 (2006) 437–452.##[24] Y.J. Wang and Y.C. Zhu, GFrames and gFrame sequences in Hilbert spaces, Acta Mathematica Sinica, 25 (2009) no. 12, 20932106.##[25] X.C. Xiao and X.M. Zeng, Some properties of gframes in Hilbert C*modules, J. Math. Anal. Appl., 363 (2010) 399408.##[26] X. Xiao, Y. Zhu, and L. Gávruta, Some properties of $K$frames in Hilbert spaces, Results. Math., 63 (2013) no. 34, 12431255.##[27] X. Xiao, Y. Zhu, Z. Shu, and M. Ding, Gframes with bounded linear operators, Rocky Mountain J. Math., 45 (2015) no. 2, 675693.##]
Generalized Ritt type and generalized Ritt weak type connected growth properties of entire functions represented by vector valued Dirichlet series
2
2
In this paper, we introduce the idea of generalized Ritt type and generalised Ritt weak type of entire functions represented by a vector valued Dirichlet series. Hence, we study some growth properties of two entire functions represented by a vector valued Dirichlet series on the basis of generalized Ritt type and generalised Ritt weak type.
1

23
32


Sanjib Kumar
Datta
Department of Mathematics, University of Kalyani, P.O.Kalyani, DistNadia, PIN 741235, West Bengal, India.
Department of Mathematics, University of
Iran
sanjib_kr_datta@yahoo.co.in


Tanmay
Biswas
Rajbari, Rabindrapalli, R. N. Tagore Road,
P.O.Krishnagar, DistNadia, PIN741101, West Bengal, India.
Rajbari, Rabindrapalli, R. N. Tagore Road,
P.O.Kr
Iran
tanmaybiswas_math@rediffmail.com


Jinarul Haque
Shaikh
Department of Mathematics, University of Kalyani, P.O.Kalyani, DistNadia, PIN741235, West Bengal, India.
Department of Mathematics, University of
Iran
jnrlhqshkh188@gmail.com
Vector valued Dirichlet series (VVDS)
Generalized Ritt order
Generalized Ritt lower order
Generalized Ritttype
Generalized Ritt weak type
growth
[[1] Q.I. Rahaman, The Ritt order of the derivative of an entire function, Annales Polonici Mathematici, Vol 17 (1965), pp. 137140.##[2] C.T. Rajagopal and A.R. Reddy, A note on entire functions represented by Dirichlet series, Annales Polonici Mathematici, Vol 17 (1965), pp. 199208.##[3] J.F. Ritt, On certain points in the theory of Dirichlet series, Amer. Jour. Math., Vol. 50 (1928), pp. 7386.##[4] D. Sato, On the rate of growth of entire functions of fast growth, Bull. Amer. Math. Soc., Vol. 69 (1963), pp. 411414.##[5] R.P. Srivastav and R.K. Ghosh, On entire functions represented by Dirichlet series, Annales Polonici Mathematici, Vol 13 (1963), pp. 93100.##[6] B.L. Srivastava, A study of spaces of certain classes of vector valued Dirichlet series, Thesis, I. I. T., Kanpur, (1983).##[7] G.S. Srivastava and A. Sharma, On generalized order and generalized type of vector valued Dirichlet series of slow growth, Int. J. Math. Archive, Vol. 2, No. 12 (2011), pp. 26522659.##]
Second dual space of little $alpha$Lipschitz vectorvalued operator algebras
2
2
Let $(X,d)$ be an infinite compact metric space, let $(B,parallel . parallel)$ be a unital Banach space, and take $alpha in (0,1).$ In this work, at first we define the big and little $alpha$Lipschitz vectorvalued (Bvalued) operator algebras, and consider the little $alpha$lipschitz $B$valued operator algebra, $lip_{alpha}(X,B)$. Then we characterize its second dual space.
1

33
41


Abbasali
Shokri
Department of Mathematics, Ahar Branch, Islamic Azad University, Ahar, Iran.
Department of Mathematics, Ahar Branch, Islamic
Iran
ashokri@iauahar.ac.ir
Second dual space
$alpha$Lipschitz operator
Vectorvalued operator
[[1] A. Abdollahi, The maximal ideal space of analutic Lipschitz algebras, Rend. Circ. Mat. Palermo(2), 47 (1998), 347352.##[2] D. Alimohammadi and A. Ebadian, Hedberg's theorem in real Lipschitz algebras, Indian J. Pure Appl. Math, 32 (2001), 14791493.##[3] H.X. Cao and Z.B. Xu, Some properties of Lipschitzα operators, Acta Mathematica Sinica, Chinese Series, 45(2) (2002), 279286.##[4] H.X. Cao, J.H. Zhang, and Z.B. Xu, Characterizations and Extensions of Lipschitzα operators, Acta Mathematica Sinica, English Series, 22(3) (2006), 671678.##[5] H.G. Dales, Banach algebras and Automatic Continuty, Clarendon Press. Oxford, 2000.##[6] A. Ebadian, Prime ideals in Lipschitz algebras of finite differentiable functions, Honam Math. J., 22 (2000), 2130.##[7] T.G. Honary and H. Mahyar, Approximation in Lipschitz algebras, Quaest. Math., 23 (2000), 1319.##[8] J.A. Johnson, Lipschitz spaces, Pacific J. Math., 51 (1975), 177186.##[9] B. Pavlovic, Automatic continuity of Lipschitz algebras, J. Funct. anal. 131 (1995), 115144.##[10] D.R. Sherbert, Banach algebras of Lipschitz functions, Pacific J. Math. 13 (1963), 13871399.##[11] N. Waver, Subalgebras of little Lipschitz algebras, Pacific J. Math., 173 (1996), 283293.##]
Generated topology on infinite sets by ultrafilters
2
2
Let $X$ be an infinite set, equipped with a topology $tau$. In this paper we studied the relationship between $tau$, and ultrafilters on $X$. We can discovered, among other thing, some relations of the Robinson's compactness theorem, continuity and the separation axioms. It is important also, aspects of communication between mathematical concepts.
1

43
53


Alireza
Bagheri Salec
Department of Mathematics, Faculty of Science, University of Qom, P.O.Box 3716146611, Qom, Iran.
Department of Mathematics, Faculty of Science,
Iran
alireza_bagheri_salec@yahoo.com
Stone$check{C}$ech compactification
Axiom of separation
Filter
[[1] R. Engelking, General Topology, Berlin, Sigma series in pure mathematics, Vol. 6, 1989.##[2] N. Hindman and I. Leader, The semigroup of ultrafilters near 0, Semigroup Forum, 59 (1999), 3355.##[3] N. Hindman and D. Strauss, Algebra in the StoneCech Compactification, Theory and Application, Springer Series in Computational Mathematics, Walter de Gruyter, Berlin, 1998.##[4] M.A. Tootkaboni and T. Vahed, The semigroup of ultrafilters near an idempotent of a semitopological semigroup, Topology and its Applications, Vol 159, Issue 16, (2012), 34943503.##[5] Y. Zelenyuk, Ultrafilters and Topologies on Groups, Walter de Gruyter, Berlin, 2011.##]
Contra $beta^{*}$continuous and almost contra $beta^{*}$continuous functions
2
2
The notion of contra continuous functions was introduced and investigated by Dontchev. In this paper, we apply the notion of $beta^{*}$closed sets in topological space to present and study a new class of functions called contra $beta^{*}$continuous and almost contra $beta^{*}$continuous functions as a new generalization of contra continuity.
1

55
71


Appachi
Vadivel
Department of Mathematics, Annamalai University, Annamalai Nagar608 002, Tamil Nadu, India.
Department of Mathematics, Annamalai University,
Iran
avmaths@gmail.com


Radhakrishnan
Ramesh
Department of Mathematics, Pope John Paul II College of Education, Reddiar Palayam, Puducherry605010, India.
Department of Mathematics, Pope John Paul
Iran
rameshroshitha@gmail.com


Duraisamy
Sivakumar
Department of Mathematics (DDE), Annamalai University, Annamalai Nagar608 002, Tamil Nadu, India.
Department of Mathematics (DDE), Annamalai
Iran
sivakumardmaths@yahoo.com
$beta^{*}$closed sets
Contra $beta^{*}$continuous
Almost contra $beta^{*}$continuous functions
[[1] K. AlZoubi and B. AlNashef, The topology of $omega$open subsets, AlManarah Journal, 9 (2) (2003), 169179.##[2] M. Caldas and S. Jafari, Some properties of contra$beta$continuous functions, Memoirs of the Faculty of Science Kochi University. Series A. Mathematics, 22 (2001), 1928.##[3] K. Dlaska, N. Ergun, and M. Ganster, Countably $S$closed spaces, Mathematica Slovaca, 44 (3) (1994), 337348.##[4] K. Dontchev, Contracontinuous functions and strongly $S$closed spaces, International Journal of Mathematics and Mathematical Sciences, 19 (2) (1996), 303310.##[5] J. Dontchev and T. Noiri, Contrasemicontinuous functions, Mathematica Pannonica, 10 (2) (1999), 159168.##[6] E. Ekici, Almost contraprecontinuous functions, Bulletin of the Malaysian Mathematical Sciences Society, 27 (1) (2004), 5365.##[7] H.Z. Hdeib, $omega$closed mappings, Revista Colombiana de Matematicas, 16 (12) (1982), 6578.##[8] S. Jafari and T. Noiri, On contraprecontinuous functions, Bulletin of the Malaysian Mathematical Sciences Society, 25 (2) (2002), 115128.##[9] S. Jafari and T. Noiri, Contra$alpha$continuous functions between topological spaces, Iranian International Journal of Science, 2 2 (2001), 153167.##[10] J.E. Joseph and M.H. Kwack, On $S$closed spaces, Proceedings of the American Mathematical Society, 80 (2) (1980), 341348.##[11] N. Levine, Generalized closed sets in topology, Rend. Circ. Math. Palermo, 19 (2) (1920), 8996.##[12] M. Mrsevic, On pairwise $R$ and pairwise $R_1$ bitopological spaces, Bull Math Soc Sci Math $RS$ Roumanie, 30 (1986), 141148.##[13] A.A. Nasef, Some properties of contra$gamma$continuous functions, Chaos Solitons and Fractals, 24 (2) (2005), 471477.##[14] T. Noiri and V. Popa, Some properties of almost contraprecontinuous functions, Bulletin of the Malaysian Mathematical Sciences Society, { 28} (2) (2005), 107116.##[15] P.G. Palanimani and R. Parimelazhagan, $beta^{*}$closed sets in topological spaces, ROSR Journal of Mathematics, 5 (1) (2013), 4750.##[16] R. Ramesh, A. Vadivel, and D. Sivakumar, On $beta^{*}$connectedness and $beta^{*}$disconnectedness and thier applications, Journal of advanced research in scientific computing, 7 (1) (2015), 1018.##[17] R. Ramesh, A. Vadivel, and D. Sivakumar, $beta^{*}$regular and $beta^{*}$normal spaces, Int. J. of Pure and Engg. Mathematics, 2 (III) (2014), 7880.##[18] R. Ramesh, A. Vadivel, and D. Sivakumar, Properties of $beta^{*}$homeomorphisms in topological spaces, Gen. Math. Notes, 26 (1) (2015), 17.##[19] M.K. Singal and A. Mathur, On nearlycompact spaces, Bollettino della Unione Matematica Italiana, 2 (1969), 702710.##[20] T. Soundararajan, Weakly Hausdorff spaces and the cardinality of topological spaces in General Topology and Its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968), Academia, Prague, 1971, 301306.##]
Stability of additive functional equation on discrete quantum semigroups
2
2
We construct a noncommutative analog of additive functional equations on discrete quantum semigroups and show that this noncommutative functional equation has HyersUlam stability on amenable discrete quantum semigroups. The discrete quantum semigroups that we consider in this paper are in the sense of van Daele, and the amenability is in the sense of BèdosMurphyTuset. Our main result generalizes a famous and old result due to Forti on the HyersUlam stability of additive functional equations on amenable classical discrete semigroups.
1

73
81


Maysam
Maysami Sadr
Department of Mathematics, Institute for Advanced Studies in Basic Sciences, P.O.Box 451951159, Zanjan 4513766731, Iran.
Department of Mathematics, Institute for
Iran
sadr@iasbs.ac.ir
Discrete quantum semigroup
Additive functional equation
HyersUlam stability
Noncommutative geometry
[[1] E. Bèdos, G.J. Murphy, and L. Tuset, Amenability and coamenability of algebraic quantum groups, Int. J. Math. Math. Sci., 31 (2002) 577601.##[2] ME. Craioveanu, M. Puta, and Th.M. Rassias, Old and New Aspects in Spectral Geometry, Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.##[3] E.G. Effros and Z.J. Ruan, Discrete quantum groups $I$, the Haar measure, Int. J. Math., 5 (1994) 681723.##[4] M. Enock and J.M. Schwartz, Kac algebras and duality of locally compact groups, SpringerVerlag, BerlinHeidelbergNew York, 1992.##[5] G.L. Forti, The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg, 57 (1987) 215226.##[6] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A., 27 (1941) 222224.##[7] S.M. Jung, HyersUlamRassias Stability of Functional Equations in Nonlinear Analysis, Springer Optimization and Its Applications, New York, 2011.##[8] J. Kustermans and S. Vaes, Locally compact quantum groups, Ann. Sci. École Norm. Sup., (4) 33 (2000) 837934.##[9] C. Mortici, M.Th. Rassias, and S.M. Jung, On the stability of a functional equation associated with the Fibonacci numbers, Abstract and Applied Analysis, Hindawi Publishing Corporation, Volume 2014 (2014),Article ID 546046, 6 pages.##[10] A. Pràstaro and Th.M. Rassias, On Ulam stability in the geometry of PDEs, In: Functional Equations, Inequalities and Applications, Kluwer Academic Publishers, Dordrecht, Boston, London, 2003, pp. 139147.##[11] S.M. Ulam, A Collection of the Mathematical Problems, Interscience Publ., New York, 1960.##[12] A. Van Daele, Discrete quantum groups, J. Algebra, 180 (1996) 431444.##]
Compare and contrast between duals of fusion and discrete frames
2
2
Fusion frames are valuable generalizations of discrete frames. Most concepts of fusion frames are shared by discrete frames. However, the dual setting is so complicated. In particular, unlike discrete frames, two fusion frames are not dual of each other in general. In this paper, we investigate the structure of the duals of fusion frames and discuss the relation between the duals of fusion frames with their associated discrete frames.
1

83
96


Elnaz
Osgooei
Department of Sciences, Urmia University of Technology, P.O.Box 41957155, Urmia, Iran.
Department of Sciences, Urmia University
Iran
e.osgooei@uut.ac.ir


Ali akbar
Arefijammal
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O.Box 397, Sabzevar, Iran.
Department of Mathematics and Computer Sciences,
Iran
arefijamaal@gmail.com
Frames
fusion frames
dual fusion frames
[[1] Z. Amiri, M.A. Dehghan, and E. Rahimi, Subfusion frames, Abstr. Appl. Anal., 2012 (2012) 112.##[2] A. Arefijamaal and S. Ghasemi, On characterization and stability of alternate dual of gframes, Turk. J. Math., 37 (2013) 7179.##[3] A. Arefijamaal, E. Zekaee, Image processing by alternate dual Gabor frames, To appear in Bull. Iranian Math. Soc.##[4] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35 (2013) 535540.##[5] R. Calderbank, P.G. Casazza, A. Heinecke, G. Kutyniok, and A. Pezeshki, Sparse fusion frames: existence and construction, Adv. Comput. Math., 35 (1) (2011) 131. ##[6] P.G. Casazza and M. Fickus, Minimizing fusion frame potential, Acta, Appl. Math., 107 (2009) 724.##[7] P.G. Casazza and G. Kutyniok, Frames of subspaces, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, (2004) 87113.##[8] P.G. Casazza, G. Kutyniok, and S. Li, Fusion frames and distributed processing, Appl. Comput. Harmon. Anal., 25 (1) (2008) 114132.##[9] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston, 2008.##[10] O. Christensen and E. Osgooei, On frame properties for Fourierlike systems, J. Approx. Theory, 172 (2013) 4757. ##[11] M.A. Dehghan and M.A. Hasankhani Fard, Gdual frames in Hilbert spaces, U. P. B. Sci. Bull. Series A, 75 (1) (2013) 129140.##[12] P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl., 333 ( 2) (2007) 871879.##[13] W.H. Greub, Linear Algebra, SpringerVerlag, New York, 1981.##[14] S.K. Kaushik, A generalization of frames in Banach spaces, J. Contemp. Math. Anal., 44 (4) (2009) 212218.##[15] J. Leng, Q. Guo, and T. Huang, The duals of fusion frames for experimental data transmission coding of high energy physics, Adv. High Energy Phys., 2013 (2013) 19.##[16] P.G. Massey, M.A. Ruiz, and D. Stojanoff, The structure of minimizers of the frame potential on fusion frames, J. Fourier Anal. Appl., 16 (2010) 514543.##[17] A. Najati, A. Rahimi, and M.H. Faroughi, Continuous and discrete frames of subspaces in Hilbert spaces, Southeast Asian Bull. Math., 32 (2008) 305324.##[18] E. Osgooei and M.H. Faroughi, HilbertSchmidt sequences and dual of gframes, Acta Univ. Apulensis, 36 (2013) 115.##[19] A. Rahimi, Invariance of Fréchet frames under perturbation, Sahand Communications in Mathematical Analysis, 1 (1) (2014), 4151.##]
Subspacediskcyclic sequences of linear operators
2
2
A sequence ${T_n}_{n=1}^{infty}$ of bounded linear operators on a separable infinite dimensional Hilbert space $mathcal{H}$ is called subspacediskcyclic with respect to the closed subspace $Msubseteq mathcal{H},$ if there exists a vector $xin mathcal{H}$ such that the diskscaled orbit ${alpha T_n x: nin mathbb{N}, alpha inmathbb{C},  alpha  leq 1}cap M$ is dense in $M$. The goal of this paper is the studying of subspace diskcyclic sequence of operators like as the well known results in a single operator case. In the first section of this paper, we study some conditions that imply the diskcyclicity of ${T_n}_{n=1}^{infty}$. In the second section, we survey some conditions and subspacediskcyclicity criterion (analogue the results obtained by some authors in cite{MR1111569, MR2261697, MR2720700}) which are sufficient for the sequence ${T_n}_{n=1}^{infty}$ to be subspacediskcyclic(subspacehypercyclic).
1

97
106


Mohammad Reza
Azimi
Department of Mathematics, Faculty of Sciences, University of Maragheh, Maragheh, Iran.
Department of Mathematics, Faculty of Sciences,
Iran
mhr.azimi@maragheh.ac.ir
Sequences of operators
Diskcyclic vectors
Subspacediskcyclicity
Subspacehypercyclicity
[[1] N. Bamerni, V. Kadets, and A. Kιlιçman, On subspaces diskcyclicity, arXiv:1402.4682 [math.FA], 111.##[2] N. Bamerni, V. Kadets, A. Kιlιçman, and M.S.M. Noorani, A review of some works in the theory of diskcyclic operators, Bull. Malays. Math. Sci. Soc., Vol. 39 (2016) 723739.##[3] F. Bayart and ´E. Matheron, Dynamics of linear operators, Cambridge Tracts in Mathematics, Vol. 179, Cambridge University Press, Cambridge, 2009.##[4] L. BernalGonz´alez and K.G. GrosseErdmann, The hypercyclicity criterion for sequences of operators, Studia Math., Vol. 157 No. 1 (2003) 1732.##[5] P.S. Bourdon, Invariant manifolds of hypercyclic vectors, Proc. Amer. Math. Soc., Vol. 118 No. 3 (1993) 845847. ##[6] G. Godefroy and J.H. Shapiro, Operators with dense, invariant, cyclic vector manifolds, J. Funct. Anal., Vol. 98 No. 2 (1991) 229269.##[7] KG. GrosseErdmann, Universal families and hypercyclic operators, Bull. Amer. Math. Soc., Vol. 36 No. 3 (1999) 345381.##[8] R.R. JiménezMunguía, R.A. MartínezAvendaño, and A. Peris, Some questions about subspacehypercyclic operators, J. Math. Anal. Appl., Vol. 408 No. 1 (2013) 209212.##[9] C. Kitai, Invariant closed sets for linear operators, ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.)University of Toronto, Canada 1982.##[10] F. LeónSaavedra and V. Müller, Hypercyclic sequences of operators, Studia Math., Vol. 175 No.1 (2006) 118.##[11] B.F. Madore and R.A. MartínezAvendaño, Subspace hypercyclicity, J. Math. Anal. Appl., Vol. 373 No.2 (2011) 502511.##[12] H. Petersson, A hypercyclicity criterion with applications, J. Math. Anal. Appl., Vol. 327 No. 2 (2007) 14311443.##[13] H. Rezaei, Notes on subspacehypercyclic operators, J. Math. Anal. Appl., Vol. 397 No. 1 (2013) 428433. ##[14] Z.J. Zeana, Cyclic Phenomena of operators on Hilbert space, Thesis, University of Baghdad, 2002.##]