Supercyclicity of ℓp-spherical and toral isometries on Banach spaces
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
Let $p\ge 1$ 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 $\sum_{i=1} ^n \|T_ix\|^p =\|x\|^p$ for all $x\in X$. The tuple $T$ is called a toral isometry if each $T_i$ is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n\ge1$, 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 semi-commutative tuples and we show that the Banach spaces $\ell^p$ ($1\le p<\infty$) 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.
Abstract
Let $p\ge 1$ 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 $\sum_{i=1} ^n \|T_ix\|^p =\|x\|^p$ for all $x\in X$. The tuple $T$ is called a toral isometry if each $T_i$ is an isometry. By a result of Ansari, Hedayatian, Khani-Robati and Moradi, for every $n\ge1$, 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 semi-commutative tuples and we show that the Banach spaces $\ell^p$ ($1\le p<\infty$) 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.
참고문헌 (9)
- S. I. Ansari, "Some properties of cyclic operators", Acta Sci. Math. (Szeged), 63(1-2), pp.195~207 (1997)
- Mohammad Ansari, "SUPERCYCLICITY OF JOINT ISOMETRIES", 대한수학회보, 52(5), pp.1481~1487 (2015), 대한수학회
- A. Athavale, "On the intertwining of joint isometries", J. Operator Theory, 23(2), pp.339~350 (1990)
- N. S. Feldman, "Hypercyclic tuples of operators and somewhere dense orbits", J. Math. Anal. Appl, 346(1), pp.82~98 (2008)
- H. M. Hilden, "Some cyclic and non-cyclic vectors of certain operators", Indiana Univ. Math. J, 23, pp.557~565 (1974)
- P. Hoffmann, "(m, p)-isometric and (m,∞)-isometric operator tuples on normed spaces", Asian-Eur. J. Math, 8(2), pp.32~ (2015)
- L. Kerchy, "Operators with regular norm-sequences", Acta Sci. Math. (Szeged), 63(3-4), pp.571~605 (1997)
- L. Kerchy, "Hyperinvariant subspaces of operators with non-vanishing orbits", Proc. Amer. Math. Soc, 127(5), pp.1363~1370 (1999)
- R. Soltani, "On supercyclicity of tuples of operators", Bull. Malays. Math. Sci. Soc, 38(4), pp.1507~1516 (2015)