• Communications of the Korean Mathematical Society
• /
• v.11 no.4
• /
• pp.1055-1059
• /
• 1996

We characterize the left invariant Riemannian metrics on SO(1,2) which give rise to 3- or 4-dimensional isometry groups.

• The Pure and Applied Mathematics
• /
• v.16 no.3
• /
• pp.277-281
• /
• 2009

A Hilbert space operator T is a 2-isometry if $T^{{\ast}2}T^2\;-\;2T^{\ast}T+I$ = O. We shall study some properties of 2-isometries, in particular spectra of a non-unitary 2-isometry and give an example. Also we prove with alternate argument that the Weyl's theorem holds for 2-isometries.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.653-659
• /
• 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.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.4
• /
• pp.623-633
• /
• 2009

The classical Mazur-Ulam theorem states that every surjective isometry between real normed spaces is affine. In this paper, we study 2-isometries in probabilistic 2-normed spaces.

• Journal of the Korean Mathematical Society
• /
• v.45 no.1
• /
• pp.205-219
• /
• 2008

It is shown that if $A{\geq}0$ and T are two bounded linear operators on a complex Hilbert space H satisfying the inequality $T^*\;AT{\leq}A$ and the condition $AT=A^{1/2}TA^{1/2}$, then there exists the maximum reducing subspace for A and $A^{1/2}T$ on which the equality $T^*\;AT=A$ is satisfied. We concretely express this subspace in two ways, and as applications, we derive certain decompositions for quasinormal contractions. Also, some facts concerning the quasi-isometries are obtained.

• Bulletin of the Korean Mathematical Society
• /
• v.39 no.1
• /
• pp.141-151
• /
• 2002

Let X and Y be real Banach spaces and $\varepsilon$, p $\geq$ 0. A mapping T between X and Y is called an ($\varepsilon$, p)-isometry if ｜∥T(x)-T(y)∥-∥x-y∥｜$\leq$ $\varepsilon$∥x-y∥$^{p}$ for x, y$\in$X. Let H be a real Hilbert space and T : H longrightarrow H an ($\varepsilon$, p)-isometry with T(0) = 0. If p$\neq$1 is a nonnegative number, then there exists a unique isometry I : H longrightarrow H such that ∥T(x)-I(y)∥$\leq$ C($\varepsilon$)(∥x∥$^{ 1+p)/2}$+∥x∥$^{p}$ ) for all x$\in$H, where C($\varepsilon$) longrightarrow 0 as $\varepsilon$ longrightarrow 0.

• Bulletin of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.485-495
• /
• 2017

The aim of this paper is twofold. Firstly, we introduce the concept of semi-partial isometry in a Banach algebra and carry out a comparison and a classification study for this concept. In particular, we show that in the context of $C^*$-algebras this concept coincides with the notion of partial isometry. Our results encompass several earlier ones concerning partial isometries in Hilbert spaces, Banach spaces and $C^*$-algebras. Finally, we study the notion of (m, p)-semi partial isometries.

• Journal of the Korean Mathematical Society
• /
• v.33 no.2
• /
• pp.455-468
• /
• 1996

Certain analytic behavior of geometric objects defined on a Riemannian manifold depends on some very crude properties of the manifold. Some of those crude invariants are the volume growth rate, isoperimetric constants, and the likes. However, these crude invariants sometimes exercise surprising control over the analytic behavior.

• Journal of the Chungcheong Mathematical Society
• /
• v.22 no.1
• /
• pp.65-71
• /
• 2009

Let f be an 2-isometry on a non-Archimedean 2-normed space. In this paper, we prove that the barycenter of triangle is invariant for f up to the translation by f(0), in this case, needless to say, we can imply naturally the Mazur-Ulam theorem in non-Archimedean 2-normed spaces.

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