Czechoslovak Mathematical Journal, first online, pp. 1-14


On $m$-expansive tuples of commuting operators on a Banach space

Muneo Cho, Injo Hur, Ji Eun Lee

Received May 20, 2024.   Published online January 24, 2025.

Abstract:  We present $m$-expansive tuples of commuting operators in a complex Banach space, expanding upon the concept of $m$-isometric tuples. We provide a characterization of the joint approximate point spectrum of these tuples. Furthermore, we investigate a multivariable extension of these single-variable $[m,C]$-expansive operators discussed in M. Cho, I. Hur, J. E. Lee (2024) and delve into several fundamental properties associated with them.
Keywords:  $[m,C]$-expensive operator; $[m,C]$-expansive tuple of operator; Banach space
Classification MSC:  47B01, 47A11

PDF available at:  Springer   Institute of Mathematics CAS

References:
[1] J. Agler, M. Stankus: $m$-isometric transformations of Hilbert space. I. Integral Equations Oper. Theory 21 (1995), 383-429. DOI 10.1007/BF01222016 | MR 1321694 | Zbl 0836.47008
[2] C.-G. Ambrozie, M. Engliš, V. Müller: Operator tuples and analytic models over general domains in $\Bbb{C}^n$. J. Oper. Theory 47 (2002), 287-302. MR 1911848 | Zbl 1019.47015
[3] A. Ben Amor: An extension of Henrici theorem for the joint approximate spectrum of commuting spectral operators. J. Aust. Math. Soc. 75 (2003), 233-245. DOI 10.1017/S1446788700003748 | MR 2000431 | Zbl 1060.47007
[4] T. Bermúdez, A. Martinón, V. Müller: $(m,q)$-isometries on metric spaces. J. Oper. Theory 72 (2014), 313-328. DOI 10.7900/jot.2013jan29.1996 | MR 3272034 | Zbl 1363.54041
[5] F. F. Bonsall, J. Duncan: Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. London Mathematical Society Lecture Note Series 2. Cambridge University Press, London (1971). DOI 10.1017/CBO9781107359895 | MR 0288583 | Zbl 0207.44802
[6] M. Cho, I. Hur, J. E. Lee: Generalization of expansive operators. Math. Inequal. Appl. 27 (2024), 159-171. DOI 10.7153/mia-2024-27-12 | MR 4702249 | Zbl 07926904
[7] M. Cho, E. Ko, J. E. Lee: On $(m,C)$-isometric operators. Complex Anal. Oper. Theory 10 (2016), 1679-1694. DOI 10.1007/s11785-016-0549-0 | MR 3558362 | Zbl 1373.47006
[8] M. Cho, J. E. Lee, H. Motoyoshi: On $[m,C]$-isometric operators. Filomat 31 (2017), 2073-2080. DOI 10.2298/FIL1707073C | MR 3635243 | Zbl 1484.47011
[9] M. Cho, K. Tanahashi: On conjugations for Banach spaces. Sci. Math. Jpn. 81 (2018), 37-45. DOI 10.32219/isms.81.1_37 | MR 3792440 | Zbl 06902693
[10] M. Cho, W. Żelazko: On geometric spectral radius of commuting $n$-tuples of operators. Hokkaido Math. J. 21 (1992), 251-258. DOI 10.14492/hokmj/1381413680 | MR 1169792 | Zbl 0784.47004
[11] M. D. Choi, C. Davis: The spectral mapping theorem for joint approximate point spectrum. Bull. Am. Math. Soc. 80 (1974), 317-321. DOI 10.1090/S0002-9904-1974-13481-6 | MR 0333780 | Zbl 0276.47001
[12] J. Gleason, S. Richter: $m$-isometric commuting tuples of operators on a Hilbert space. Integral Equations Oper. Theory 56 (2006), 181-196. DOI 10.1007/s00020-006-1424-6 | MR 2264515 | Zbl 1112.47003
[13] C. Gu: The $(m,q)$-isometric weighted shifts on $l_p$ spaces. Integral Equations Oper. Theory 82 (2015), 157-187. DOI 10.1007/s00020-015-2234-5 | MR 3345637 | Zbl 1314.47048
[14] P. H. W. Hoffmann, M. Mackey: $(m,p)$-and $(m,\infty)$-isometric operator tuples on normed spaces. Asian-Eur. J. Math. 8 (2015), Article ID 1550022, 32 pages. DOI 10.1142/S1793557115500229 | MR 3354483 | Zbl 1325.47014
[15] S. A. O. A. Mahmoud, M. Cho, J. E. Lee: On $(m,C)$-isometric commuting tuples of operators on a Hilbert space. Result. Math. 73 (2018), Article ID 51, 31 pages. DOI 10.1007/s00025-018-0810-0 | MR 3770897 | Zbl 1512.47042
[16] H. Motoyashi: Linear operators and conjugations on a Banach space. Acta Sci. Math. 85 (2019), 325-336. DOI 10.14232/actasm-018-801-y | MR 3967893 | Zbl 1449.47041
[17] K. B. Laursen, M. M. Neumann: An Introduction to Local Spectral Theory. London Mathematical Society Monographs. New Series 20. Clarendon Press, Oxford (2000). DOI 10.1093/oso/9780198523819.002.0001 | MR 1747914 | Zbl 0957.47004
[18] J. L. Taylor: A joint spectrum for several commuting operators. J. Funct. Anal. 6 (1970), 172-191. DOI 10.1016/0022-1236(70)90055-8 | MR 0268706 | Zbl 0233.47024
[19] J. L. Taylor: The analytic-functional calculus for several commuting operators. Acta Math. 125 (1970), 1-38. DOI 10.1007/BF02392329 | MR 0271741 | Zbl 0233.47025

Affiliations:   Muneo Cho, Department of Mathematics, Kanagawa University, 3-27-1 Rokkakubashi, Yokohama 221-8686, Japan, 15-3-1113, Tsutsui-machi Yahatanishi-ku, Kita-kyushu 806-0032, Japan, e-mail: muneocho0105@gmail.com; Injo Hur, Department of Mathematics Education, Chonnam National University, 77 Yongbong-ro, Buk-gu, Gwangju 61186, Republic of Korea, e-mail: injohur@jnu.ac.kr; Ji Eun Lee (corresponding author), Department of Mathematics and Statistics, Sejong University, Yeongsil-gwan 313, Seoul, 05006, Republic of Korea, e-mail: jieunlee7@sejong.ac.kr


 
PDF available at: