• Title, Summary, Keyword: supercyclicity

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HEREDITARILY HYPERCYCLICITY AND SUPERCYCLICITY OF WEIGHTED SHIFTS

  • Liang, Yu-Xia;Zhou, Ze-Hua
    • Journal of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.363-382
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    • 2014
  • In this paper we first characterize the hereditarily hypercyclicity of the unilateral (or bilateral) weighted shifts on the spaces $L^2(\mathbb{N},\mathcal{K})$ (or $L^2(\mathbb{Z},\mathcal{K})$) with weight sequence {$A_n$} of positive invertible diagonal operators on a separable complex Hilbert space $\mathcal{K}$. Then we give the necessary and sufficient conditions for the supercyclicity of those weighted shifts, which extends some previous results of H. Salas. At last, we give some conditions for the supercyclicity of three different weighted shifts.

N-SUPERCYCLICITY OF AN A-m-ISOMETRY

  • HEDAYATIAN, KARIM
    • Honam Mathematical Journal
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    • v.37 no.3
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    • pp.281-285
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    • 2015
  • An A-m-isometric operator is a bounded linear operator T on a Hilbert space $\mathcal{H}$ satisfying $\sum\limits_{k=0}^{m}(-1)^{m-k}T^{*^k}AT^k=0$, where A is a positive operator. We give sufficient conditions under which A-m-isometries are not N-supercyclic, for every $N{\geq}1$; that is, there is not a subspace E of dimension N such that its orbit under T is dense in $\mathcal{H}$.

Supercyclicity of Convex Operators

  • Hedayatian, Karim;Karimi, Lotfollah
    • Kyungpook Mathematical Journal
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    • v.58 no.1
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    • pp.81-90
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    • 2018
  • A bounded linear operator T on a Hilbert space ${\mathcal{H}}$ is convex, if for each $x{\in}{\mathcal{H}}$, ${\parallel}T^2x{\parallel}^2-2{\parallel}Tx{\parallel}^2+{\parallel}x{\parallel}^2{\geq}0$. In this paper, it is shown that if T is convex and supercyclic then it is a contraction or an expansion. We then present some examples of convex supercyclic operators. Also, it is proved that no convex composition operator induced by an automorphism of the disc on a weighted Hardy space is supercyclic.

DISJOINT SUPERCYCLIC WEIGHTED COMPOSITION OPERATORS

  • Liang, Yu-Xia;Zhou, Ze-Hua
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.1137-1147
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    • 2018
  • In this paper, we discovered a sufficient condition ensuring the weighted composition operators $C_{{\omega}_1,{\varphi}_1},{\cdots},C_{{\omega}_N,{\varphi}_N}$ were disjoint supercyclic on $H({\Omega})$ endowed with the compact open topology. Besides, we provided a condition on inducing symbols to guarantee the disjoint supercyclicity of non-constant adjoint multipliers $M^*_{{\varphi}_1},M^*_{{\varphi}_2},{\cdots},M^*_{{\varphi}_N}$ on a Hilbert space ${\mathcal{H}}$.

SUPERCYCLICITY OF JOINT ISOMETRIES

  • ANSARI, MOHAMMAD;HEDAYATIAN, KARIM;KHANI-ROBATI, BAHRAM;MORADI, ABBAS
    • Bulletin of the Korean Mathematical Society
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    • v.52 no.5
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    • pp.1481-1487
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    • 2015
  • Let H be a separable complex Hilbert space. A commuting tuple $T=(T_1,{\cdots},T_n)$ of bounded linear operators on H is called a spherical isometry if $\sum_{i=1}^{n}T^*_iT_i=I$. The tuple T is called a toral isometry if each $T_i$ is an isometry. In this paper, we show that for each $n{\geq}1$ there is a supercyclic n-tuple of spherical isometries on $\mathbb{C}^n$ and there is no spherical or toral isometric tuple of operators on an infinite-dimensional Hilbert space.

ON SUBSPACE-SUPERCYCLIC SEMIGROUP

  • El Berrag, Mohammed;Tajmouati, Abdelaziz
    • Communications of the Korean Mathematical Society
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    • v.33 no.1
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    • pp.157-164
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    • 2018
  • A $C_0$-semigroup ${\tau}=(T_t)_{t{\geq}0}$ on a Banach space X is called subspace-supercyclic for a subspace M, if $\mathbb{C}Orb({\tau},x){\bigcap}M=\{{\lambda}T_tx\;:\;{\lambda}{\in}\mathbb{C},\;t{\geq}0\}{\bigcap}M$ is dense in M for a vector $x{\in}M$. In this paper we characterize the notion of subspace-supercyclic $C_0$-semigroup. At the same time, we also provide a subspace-supercyclicity criterion $C_0$-semigroup and offer two equivalent conditions of this criterion.

SUPERCYCLICITY OF TWO-ISOMETRIES

  • Ahmadi, M. Faghih;Hedayatian, K.
    • Honam Mathematical Journal
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    • v.30 no.1
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    • pp.115-118
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    • 2008
  • A bounded linear operator T on a complex separable Hilbert space H is called a two-isometry, if $T^{*2}T^2-2T^*T+1=0$. In this paper it is shown that every two-isometry is not supercyclic. This generalizes a result due to Ansari and Bourdon.

SUPERCYCLICITY OF ℓp-SPHERICAL AND TORAL ISOMETRIES ON BANACH SPACES

  • Ansari, Mohammad;Hedayatian, Karim;Khani-Robati, Bahram
    • Communications of the Korean Mathematical Society
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    • v.32 no.3
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    • pp.653-659
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    • 2017
  • Let $p{\geq}1$ be a real number. A tuple $T=(T_1,{\ldots},T_n)$ of commuting bounded linear operators on a Banach space X is called an ${\ell}^p$-spherical isometry if ${\sum_{i=1}^{n}}{\parallel}T_ix{\parallel}^p={\parallel}x{\parallel}^p$ for all $x{\in}X$. The tuple T is called a toral isometry if each Ti is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n{\geq}1$, there is a supercyclic ${\ell}^2$-spherical isometric n-tuple on ${\mathbb{C}}^n$ but there is no supercyclic ${\ell}^2$-spherical isometry on an infinite-dimensional Hilbert space. In this article, we investigate the supercyclicity of ${\ell}^p$-spherical isometries and toral isometries on Banach spaces. Also, we introduce the notion of semicommutative tuples and we show that the Banach spaces ${\ell}^p$ ($1{\leq}p$ < ${\infty}$) support supercyclic ${\ell}^p$-spherical isometric semi-commutative tuples. As a result, all separable infinite-dimensional complex Hilbert spaces support supercyclic spherical isometric semi-commutative tuples.