• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.50 no.5
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• pp.1631-1637
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• 2013

This paper generalizes the Aleksandrov problem and Mazur Ulam theorem to the case of $n$-normed spaces. For real $n$-normed spaces X and Y, we will prove that $f$ is an affine isometry when the mapping satisfies the weaker assumptions that preserves unit distance, $n$-colinear and 2-colinear on same-order.

• 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 $\mathcal{H}$.

• Bulletin of the Korean Mathematical Society
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• v.43 no.2
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• pp.309-318
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• 2006

Let X and Y be real Banach spaces and ${\varepsilon}\;>\;0$, p > 1. Let f : $X\;{\to}\;Y$ be a bijective mapping with f(0) = 0 satisfying $$|\;{\parallel}f(x)-f(y){\parallel}-{\parallel}{x}-y{\parallel}\;|\;{\leq}{\varepsilon}{\parallel}{x}-y{\parallel}^p$$ for all $x\;{\in}\;X$ and, let $f^{-1}\;:\;Y\;{\to}\;X$ be uniformly continuous. Then there exist a constant ${\delta}\;>\;0$ and N(${\varepsilon},p$) such that lim N(${\varepsilon},p$)=0 and a unique surjective isometry I : X ${\to}$ Y satisfying ${\parallel}f(x)-I(x){\parallel}{\leq}N({\varepsilon,p}){\parallel}x{\parallel}^p$ for all $x\;{\in}\;X\;with\;{\parallel}x{\parallel}{\leq}{\delta}$.

• Journal of Broadcast Engineering
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• v.17 no.6
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• pp.954-963
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• 2012

As a greedy algorithm reconstructing the sparse signal from underdetermined system, orthogonal matching pursuit (OMP) algorithm has received much attention. In this paper, we multiple candidate matching pursuit (MuCaMP), which builds up candidate support set in every iteration and uses the minimum residual at last iteration. Using the restricted isometry property (RIP), we derive the sufficient condition for MuCaMP to recover the sparse signal exactly. The MuCaMP guarantees to reconstruct the K-sparse signal when the sensing matrix satisfies the RIP constant ${\delta}_{N+K}<\frac{\sqrt{N}}{\sqrt{K}+3\sqrt{N}}$. In addition, we show a recovery performance both noiseless and noisy measurements.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.39A no.12
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• pp.739-745
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• 2014

It has been shown in compressive sensing that every s-sparse $x{\in}R^N$ can be recovered from the measurement vector y=Ax or the noisy vector y=Ax+e via ${\ell}_1$-minimization as soon as the 3s-restricted isometry constant of the sensing matrix A is smaller than 1/2 or smaller than $1/\sqrt{3}$ by applying the Iterative Hard Thresholding (IHT) algorithm. However, recovery can be guaranteed by practical algorithms for some certain assumptions of acquisition schemes. One of the key assumption is that the sensing matrix must satisfy the Restricted Isometry Property (RIP), which is often violated in the setting of many practical applications. In this paper, we studied a generalization of RIP, called Restricted Biorthogonality Property (RBOP) for anisotropic cases, and the new recovery algorithms called oblique pursuits. Then, we provide an analysis on the success of sparse recovery in terms of restricted biorthogonality constant for the IHT algorithms.

• Journal of the Institute of Electronics Engineers of Korea SP
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• v.49 no.2
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• pp.122-129
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• 2012

As a greedy algorithm reconstructing the sparse signal from underdetermined system, orthogonal matching pursuit (OMP) algorithm has received much attention in recent years. In this paper, we present an extension of OMP for pursuing efficiency of the index selection. Our approach, referred to as generalized OMP (gOMP), is literally a generalization of the OMP in the sense that multiple (N) columns are identified per step. Using the restricted isometry property (RIP), we derive the condition for gOMP to recover the sparse signal exactly. The gOMP guarantees to reconstruct sparse signal when the sensing matrix satisfies the RIP constant ${\delta}_{NK}$ < $\frac{\sqrt{N}}{\sqrt{K}+2\sqrt{N}}$. In addition, we show recovery performance and the reduced number of iteration required to recover the sparse signal.

• The Pure and Applied Mathematics
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• v.10 no.3
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• pp.193-198
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• 2003

Let X and Y be n-dimensional Euclidean spaces with $n\;{\geq}\;3$. In this paper, we generalize a classical theorem of Bookman and Quarles by proving that if a mapping, from a half space of X into Y, preserves a distance $\rho$, then the restriction of f to a subset of the half space is an isometry.

• Communications of the Korean Mathematical Society
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• v.10 no.3
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• pp.609-620
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• 1995

In this paper, we introduce the generalization $A^{(2)}_{2n}$ of tridiagonal algebras $A_{2n}$ and investigate the isometries of such algebras.