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SUPERCYCLICITY OF ℓ<sup>p</sup>-SPHERICAL AND TORAL ISOMETRIES ON BANACH SPACES
Ansari, Mohammad,Hedayatian, Karim,Khani-Robati, Bahram Korean Mathematical Society 2017 대한수학회논문집 Vol.32 No.3
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.
ROBATI, B. KHANI 호남수학회 2001 한국수학학술지 Vol.23 No.1
Let B be a Banach space of analytic functions defined on the open unit disk. Assume S is a bounded operator defined on B such that S is in the commutant of M_z^n or SM_z^n=-M_z^nS for some positive integer n. We give necessary and sufficient condition between compactness of SM_z+cM_zS where c=1, -1, i, -i, and the structure of S. Also we characterize the commutant of M_z,^n for some positive integer n.
ON THE COMMUTANT OF MULTIPLICATION OPERATORS WITH ANALYTIC POLYNOMIAL SYMBOLS
Robati, B. Khani Korean Mathematical Society 2007 대한수학회보 Vol.44 No.4
Let $\mathcal{B}$ be a certain Banach space consisting of analytic functions defined on a bounded domain G in the complex plane. Let ${\varphi}$ be an analytic polynomial or a rational function and let $M_{\varphi}$ denote the operator of multiplication by ${\varphi}$. Under certain condition on ${\varphi}$ and G, we characterize the commutant of $M_{\varphi}$ that is the set of all bounded operators T such that $TM_{\varphi}=M_{\varphi}T$. We show that $T=M_{\Psi}$, for some function ${\Psi}$ in $\mathcal{B}$.
ROBATI, B. KHANI The Honam Mathematical Society 2001 호남수학학술지 Vol.23 No.1
Let $\mathcal{B}$ be a Banach space of analytic functions defined on the open unit disk. Assume S is a bounded operator defined on $\mathcal{B}$ such that S is in the commutant of $M_zn$ or $SM_zn=-M_znS$ for some positive integer n. We give necessary and sufficient condition between compactness of $SM_z+cM_zS$ where c = 1, -1, i, -i, and the structure of S. Also we characterize the commutant of $M_zn$ for some positive integer n.
On the commutant of multiplication operators with analytic polynomial symbols
B. Khani Robati 대한수학회 2007 대한수학회보 Vol.44 No.4
LetB be a certain Banach space consisting of analytic func-tions dened on a bounded domain G in the complex plane. Let ' be ananalytic polynomial or a rational function and let M ' denote the operatorof multiplication by '. Under certain condition on ' and G, we charac-terize the commutant ofM ' that is the set of all bounded operators Tsuch that TM' = M ' T. We show that T = M Ψ for some function Ψ inB.
SOME PROPERTIES OF INVARIANT SUBSPACES IN BANACH SPACES OF ANALYTIC FUNCTIONS
Hedayatian, K.,Robati, B. Khani The Honam Mathematical Society 2007 호남수학학술지 Vol.29 No.4
Let $\cal{B}$ be a reflexive Banach space of functions analytic on the open unit disc and M be an invariant subspace of the multiplication operator by the independent variable, $M_z$. Suppose that $\varphi\;\in\;\cal{H}^{\infty}$ and $M_{\varphi}$ : M ${\rightarrow}$ M, defined by $M_{\varphi}f={\varphi}f$, is the operator of multiplication by ${\varphi}$. We would like to investigate the spectrum and the essential spectrum of $M_{\varphi}$ and we are looking for the necessary and sufficient conditions for $M_{\varphi}$ to be a Fredholm operator. Also we give a sufficient condition for a sequence $\{w_n\}$ to be an interpolating sequence for $\cal{B}$. At last the commutant of $M_{\varphi}$ under certain conditions on M and ${\varphi}$ is determined.
ON THE CLOSED RANGE COMPOSITION AND WEIGHTED COMPOSITION OPERATORS
Keshavarzi, Hamzeh,Khani-Robati, Bahram Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.1
Let ψ be an analytic function on 𝔻, the unit disc in the complex plane, and φ be an analytic self-map of 𝔻. Let 𝓑 be a Banach space of functions analytic on 𝔻. The weighted composition operator W<sub>φ,ψ</sub> on 𝓑 is defined as W<sub>φ,ψ</sub>f = ψf ◦ φ, and the composition operator C<sub>φ</sub> defined by C<sub>φ</sub>f = f ◦ φ for f ∈ 𝓑. Consider α > -1 and 1 ≤ p < ∞. In this paper, we prove that if φ ∈ H<sup>∞</sup>(𝔻), then C<sub>φ</sub> has closed range on any weighted Dirichlet space 𝒟<sub>α</sub> if and only if φ(𝔻) satisfies the reverse Carleson condition. Also, we investigate the closed rangeness of weighted composition operators on the weighted Bergman space A<sup>p</sup><sub>α</sub>.
Some Properties of Invariant Subsoaces in Banach Spaces of Analytic Functions
K.Hedayatian,B. Khani Robati 호남수학회 2007 호남수학학술지 Vol.29 No.4
LetB be a reexive Banach space of functions analyticon the open unit disc and M be an invariant subspace of the multi-plication operator by the independent variable, M z. Suppose that' 2 H1 and M ' :M ! M , dened byM ' f = 'f , is the operatorof multiplication by '. We would like to investigate the spectrumand the essential spectrum ofM ' and we are looking for the nec-essary and sucient conditions forM ' to be a Fredholm operator.Also we give a sucient condition for a sequencef! n g to be aninterpolating sequence forB. At last the commutant ofM ' undercertain conditions on M and ' is determined.