• Title, Summary, Keyword: isometry

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DISCUSSIONS ON PARTIAL ISOMETRIES IN BANACH SPACES AND BANACH ALGEBRAS

  • Alahmari, Abdulla;Mabrouk, Mohamed;Taoudi, Mohamed Aziz
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.2
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    • pp.485-495
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    • 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.

STABILITY Of ISOMETRIES ON HILBERT SPACES

  • Jun, Kil-Woung;Park, Dal-Won
    • Bulletin of the Korean Mathematical Society
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    • v.39 no.1
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    • pp.141-151
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    • 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.

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.

ISOMETRIES IN PROBABILISTIC 2-NORMED SPACES

  • Rahbarnia, F.;Cho, Yeol Je;Saadati, R.;Sadeghi, Gh.
    • Journal of the Chungcheong Mathematical Society
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    • v.22 no.4
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    • pp.623-633
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    • 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.

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ISOMETRY GOUP SO(1,2)

  • Kim, Sung-Sook;Shin, Joon-Kook
    • Communications of the Korean Mathematical Society
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    • v.11 no.4
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    • pp.1055-1059
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    • 1996
  • We characterize the left invariant Riemannian metrics on SO(1,2) which give rise to 3- or 4-dimensional isometry groups.

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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.

GEOMETRIC CLASSIFICATION OF ISOMETRIES ACTING ON HYPERBOLIC 4-SPACE

  • Kim, Youngju
    • Journal of the Korean Mathematical Society
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    • v.54 no.1
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    • pp.303-317
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    • 2017
  • An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic 4-space. This is a new phenomenon which occurs in hyperbolic 4-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic 4-space using the isometric spheres.

ON SPECTRA OF 2-ISOMETRIC OPERATORS

  • Yang, Young-Oh;Kim, Cheoul-Jun
    • The Pure and Applied Mathematics
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    • v.16 no.3
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    • pp.277-281
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    • 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.

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MAXIMUM SUBSPACES RELATED TO A-CONTRACTIONS AND QUASINORMAL OPERATORS

  • Suciu, Laurian
    • Journal of the Korean Mathematical Society
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    • v.45 no.1
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    • pp.205-219
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    • 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.