Abstract
We present an extensive analysis ofpositively quadratically hyponormal weighted shiftsW α with α0 = α1 = 1. Our main result states that such weighted shifts abound! Specifically, by focusing on recursively generated weighted shifts of the form
, we establish that the planar set
is positively quadratically hyponormal} is a closed convex, set with nonempty interior. In addition, we are able to describe in detail the boundary of\(\mathcal{R}\)
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[Ath] A. Athavale, On joint hyponormality of operators, Proc. Amer. Math. Soc. 103(1988), 417–423.
[Ch] Y.B. Choi, A propagation of quadratically hyponormal weighted shifts, preprint 1998.
[Cu1] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations and Operator Theory 13(1990), 49–66.
[Cu2] R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symposia Pure Math. 51(1990), Part II, 69–91.
[Cu3] R. Curto, Polynomially hyponormal operators on Hilbert space, in Proceedings of ELAM VII, Revista Unión Mat. Arg. 37(1991), 29–56.
[CuFi1] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, I, Integral Equations and Operator Theory 17(1993), 202–246.
[CuFi2] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem II, Integral Equations and Operator Theory 17(1993), 202–246.
[CMX] R. Curto, P. Muhly and J. Xia, Hyponormal pairs of commuting operators, Operator Theory: Adv. Appl. 35(1988), 1–22.
[CuPu1] R. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc., 25(1991), 373–378.
[CuPu2] R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Funct. Anal., 115(1993), 480–497.
[Fan] P. Fan, A note on hyponormal weighted shifts, Proc. Amer. Math. Soc. 92(1984), 271–272.
[Ioh] I. Iohvidov, Hankel and Toeplitz Matrices and Forms, Birkhäuser Boston, 1982.
[Jo1] A. Joshi, Hyponormal polynomials of monotone shifts, Ph.D. dissertation, Purdue University, 1971.
[Jo2] A. Joshi, Hyponormal polynomials of monotone shifts, Indian J. Pure Appl. Math., 6(1975), 681–686.
[JuPa] I.B. Jung and S.S. Park, Quadratically hyponormal weighted shifts and their examples, Integral Equations and Operator Theory, to appear.
[McCP] S. McCullough and V. Paulsen, A note on joint hyponormality, Proc. Amer. Math. Soc. 107(1989), 187–195.
[Shi] A. Shields, Weighted Shift Operators and Analytic Function Theory, Math. Surveys 13(1994), 49–128.
[Sta] J. Stampfli, Which weighted shifts are subnormal, Pacific J. Math. 17(1996), 367–379.
[Wol] Wolfram Research, Inc. mathematica, Version 3.0, Wolfram Research Inc., Champaign, IL, 1996.
Author information
Authors and Affiliations
Additional information
Research partially supported by NSF grants DMS-9401455 and DMS-9800931
Research partially supported by KOSEF grant 971-0102-006-2 and by TGRC-KOSEF
Rights and permissions
About this article
Cite this article
Curto, R.E., Jung, I.B. Quadratically hyponormal weighted shifts with two equal weights. Integr equ oper theory 37, 208–231 (2000). https://doi.org/10.1007/BF01192423
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01192423