Invertible weighted shift operators which are 𝑚-isometries

Chō, Muneo; Ôta, Schôichi; Tanahashi, Kôtarô

doi:10.1090/S0002-9939-2013-11701-6

Invertible weighted shift operators which are $m$-isometries
HTML articles powered by AMS MathViewer

by Muneo Chō, Schôichi Ôta and Kôtarô Tanahashi PDF

Proc. Amer. Math. Soc. 141 (2013), 4241-4247 Request permission

Abstract:

For a bounded linear operator $T$ on a complex Hilbert space ${\mathcal H}$, let $\Delta _{T,m} = \sum _{k=0}^m (-1)^k \ \begin {pmatrix} m \\ k \end {pmatrix} T^{*m-k}T^{m-k} \ \ \mbox {for} \ m \in {\mathbb N}$. $T$ is called an $m$-isometry if $\Delta _{T,m}=0$. In this paper, for every even number $m$, we give an example of invertible $(m+1)$-isometry which is not an $m$-isometry. Next we show that if $T$ is an $m$-isometry, then the operator $\Delta _{T, m-1}$ is not invertible.

References

Jim Agler and Mark Stankus, $m$-isometric transformations of Hilbert space. I, Integral Equations Operator Theory 21 (1995), no. 4, 383–429. MR 1321694, DOI 10.1007/BF01222016
Jim Agler and Mark Stankus, $m$-isometric transformations of Hilbert space. II, Integral Equations Operator Theory 23 (1995), no. 1, 1–48. MR 1346617, DOI 10.1007/BF01261201
Jim Agler and Mark Stankus, $m$-isometric transformations of Hilbert space. III, Integral Equations Operator Theory 24 (1996), no. 4, 379–421. MR 1382018, DOI 10.1007/BF01191619
Ameer Athavale, Some operator-theoretic calculus for positive definite kernels, Proc. Amer. Math. Soc. 112 (1991), no. 3, 701–708. MR 1068114, DOI 10.1090/S0002-9939-1991-1068114-8
John V. Baxley, Some general conditions implying Weyl’s theorem, Rev. Roumaine Math. Pures Appl. 16 (1971), 1163–1166. MR 305107
Sterling K. Berberian, Approximate proper vectors, Proc. Amer. Math. Soc. 13 (1962), 111–114. MR 133690, DOI 10.1090/S0002-9939-1962-0133690-8
Teresa Bermúdez, Antonio Martinón, and Emilio Negrín, Weighted shift operators which are $m$-isometries, Integral Equations Operator Theory 68 (2010), no. 3, 301–312. MR 2735438, DOI 10.1007/s00020-010-1801-z
M. Chō, S. Ôta, K. Tanahashi, and A. Uchiyama, Spectral properties of $m$-isometric operators, Funct. Anal. Approx. Comput., 4:2 (2012), 33-39.
John B. Conway, The theory of subnormal operators, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR 1112128, DOI 10.1090/surv/036
S. M. Patel, 2-isometric operators, Glas. Mat. Ser. III 37(57) (2002), no. 1, 141–145. MR 1918101
Atsushi Uchiyama and Kotaro Tanahashi, Bishop’s property $(\beta )$ for paranormal operators, Oper. Matrices 3 (2009), no. 4, 517–524. MR 2597677, DOI 10.7153/oam-03-29
Daoxing Xia, Spectral theory of hyponormal operators, Operator Theory: Advances and Applications, vol. 10, Birkhäuser Verlag, Basel, 1983. MR 806959, DOI 10.1007/978-3-0348-5435-1