Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A formula for $k$-hyponormality of backstep extensions of subnormal weighted shifts


Authors: Il Bong Jung and Chunji Li
Journal: Proc. Amer. Math. Soc. 129 (2001), 2343-2351
MSC (2000): Primary 47B37
DOI: https://doi.org/10.1090/S0002-9939-00-05844-5
Published electronically: December 28, 2000
MathSciNet review: 1823917
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

Let ${\alpha }: {\alpha }_{0}, {\alpha }_{1}, \cdots $ be a weight sequence of positive real numbers and let $W_{{\alpha }}$ be a subnormal weighted shift with a weight sequence $\alpha $. Consider an extended weight sequence ${\alpha }(x) : x, {\alpha }_{0}, {\alpha }_{1}, \cdots $ with $0<x \le \alpha _{0}$and let $HE({\alpha ,k}):= \{ x >0: W_{\alpha (x)} \text{is} k \text{-hyponormal}\}$for $k \in {\mathbb{N}}\cup \{\infty \}$, where $\mathbb{N}$ is the set of natural numbers. We obtain a formula to find the interval $HE({\alpha ,k})\setminus HE({\alpha ,k+1})$, which provides several examples to distinguish the classes of $k$-hyponormal operators from one another.


References [Enhancements On Off] (What's this?)

  • [Br] J. Bram, Subnormal operators, Duke Math. J. 22 (1955), 75-94. MR 16:835a
  • [Cu1] R. Curto, Quadratically hyponormal weighted shifts, Integral Equations and Operator Theory 13 (1990), 49-66. MR 90k:47061
  • [Cu2] R. Curto, Joint hyponormality: A bridge between hyponormality and subnormality, Proc. Symposia Pure Math. 51 (1990), 69-91. MR 91k:47049
  • [CuF1] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, Integral Equations and Operator Theory 17 (1993), 202-246. MR 94h:47050
  • [CuF2] R. Curto and L. Fialkow, Recursively generated weighted shifts and the subnormal completion problem, II, Integral Equations and Operator Theory 18 (1994), 369-426. MR 94m:47044
  • [CuF3] R. Curto and L. Fialkow, Recursiveness, positivity, and truncated moment problems, Houston J. Mathematics 17 (1991), 603-635. MR 93a:47016
  • [CuJ1] R. Curto and I. Jung, Quadratically hyponormal weighted shifts with first two equal weights, Integral Equations and Operator Theory 37 (2000), 208-231.
  • [CuL] R. Curto and W. Lee, Joint hyponormality of Toeplitz pairs, Memoirs Amer. Math. Soc., to appear.
  • [CuP1] R. Curto and M. Putinar, Existence of non-subnormal polynomially hyponormal operators, Bull. Amer. Math. Soc. 25 (1991), 373-378. MR 93e:47028
  • [CuP2] R. Curto and M. Putinar, Nearly subnormal operators and moment problems, J. Functional Analysis 115 (1993), 480-497. MR 95d:47024
  • [Gan] F. R. Gantmacher, The theory of matrices, Vol. 1, Chelsea Publ. Co., 1960. MR 21:6372c
  • [Hal] P. Halmos, Normal dilations and extensions of operators, Summer Bras. Math. 2 (1950), 124-134. MR 13:359b
  • [Li] X. Li, Moment sequences and their applications, Dissertation, Virginia Polytechnic Institute and State University, 1994.
  • [ShT] J. A. Shohat and J. D. Tarmakin, The Problem of Moments, Math. Surveys I, American Math. Soc., Providence, 1943. MR 55:5c
  • [Sta] J. Stampfli, Which weighted shifts are subnormal, Pacific J. Math. 17 (1966), 367-379. MR 33:1740
  • [Wol] Stephen Wolfram, The Mathematica Book, 3rd ed., Wolfram Media/Cambridge University Press, 1996. MR 97d:68001

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 47B37

Retrieve articles in all journals with MSC (2000): 47B37


Additional Information

Il Bong Jung
Affiliation: Department of Mathematics, Kyungpook National University, Taegu 702–701, Korea
Email: ibjung@kyungpook.ac.kr

Chunji Li
Affiliation: Department of Mathematics, Yanbian University, Yanji 133-002, People’s Republic of China
Address at time of publication: TGRC, Kyungpook National University, Taegu 702-701, Korea
Email: chunjili@hanmail.com

DOI: https://doi.org/10.1090/S0002-9939-00-05844-5
Keywords: Subnormal weighted shifts, $k$-hyponormal weighted shifts
Received by editor(s): January 22, 1999
Received by editor(s) in revised form: December 7, 1999
Published electronically: December 28, 2000
Additional Notes: The first author was partially supported by KOSEF grant 971-0102-006-2 and the Korea Research Foundation made in the program year of 1998, 1998-015-D00019. The second author was partially supported by TGRC-KOSEF
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society