Weyl's theorem holds for algebraically hyponormal operators

Authors:
Young Min Han and Woo Young Lee

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2291-2296

MSC (2000):
Primary 47A10, 47A53; Secondary 47B20

DOI:
https://doi.org/10.1090/S0002-9939-00-05741-5

Published electronically:
March 29, 2000

MathSciNet review:
1756089

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this note it is shown that if is an ``algebraically hyponormal" operator, i.e., is hyponormal for some nonconstant complex polynomial , then for every , Weyl's theorem holds for , where denotes the set of analytic functions on an open neighborhood of .

**1.**S.K. Berberian,*An extension of Weyl's theorem to a class of not necessarily normal operators*, Michigan Math. J.**16**(1969), 273-279. MR**40:3335****2.**S.K. Berberian,*The Weyl spectrum of an operator*, Indiana Univ. Math. J.**20**(1970), 529-544. MR**43:5344****3.**J.B. Conway and B.B. Morrel,*Roots and Logarithms of bounded operators on Hilbert space*, J. Funct. Anal.**70**(1987), 171-193. MR**87m:47044****4.**L.A. Coburn,*Weyl's theorem for nonnormal operators*, Michigan Math. J.**13**(1966), 285-288. MR**34:1846****5.**P.R. Halmos,*A Hilbert Space Problem Book*, Springer, New York, 1982. MR**84e:47001****6.**R.E. Harte,*Fredholm, Weyl and Browder theory*, Proc. Royal Irish Acad.**85A (2)**(1985), 151-176. MR**87h:47029****7.**R.E. Harte,*Invertibility and Singularity for Bounded Linear Operators*, Dekker, New York, 1988. MR**89d:47001****8.**R.E. Harte and W.Y. Lee,*Another note on Weyl's theorem*, Trans. Amer. Math. Soc.**349**(1997), 2115-2124. MR**98j:47024****9.**W.Y. Lee and S.H. Lee,*A spectral mapping theorem for the Weyl spectrum*, Glasgow Math. J.**38(1)**(1996), 61-64. MR**97c:47023****10.**K.K. Oberai,*On the Weyl spectrum (II)*, Illinois J. Math.**21**(1977), 84-90. MR**55:1102****11.**C. Schmoeger,*Ascent, descent and the Atkinson region in Banach algebras II*, Ricerche di Matematica**vol. XLII, fasc.**(1993), 249-264. MR**95g:46093****12.**H. Weyl,*Über beschränkte quadratische Formen, deren Differenz vollsteig ist*, Rend. Circ. Mat. Palermo**27**(1909), 373-392.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47A10,
47A53,
47B20

Retrieve articles in all journals with MSC (2000): 47A10, 47A53, 47B20

Additional Information

**Young Min Han**

Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea

**Woo Young Lee**

Affiliation:
Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Korea

Email:
wylee@yurim.skku.ac.kr

DOI:
https://doi.org/10.1090/S0002-9939-00-05741-5

Keywords:
Weyl's theorem,
algebraically hyponormal operators,
unilateral weighted shifts

Received by editor(s):
August 22, 1998

Published electronically:
March 29, 2000

Additional Notes:
This work was partially supported by the BSRI-97-1420 and the KOSEF through the GARC at Seoul National University.

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society