On completely hyperexpansive operators

Author:
Ameer Athavale

Journal:
Proc. Amer. Math. Soc. **124** (1996), 3745-3752

MSC (1991):
Primary 47B20; Secondary 47B39

MathSciNet review:
1353373

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce and discuss a class of operators, to be referred to as the class of completely hyperexpansive operators, which is in some sense antithetical to the class of contractive subnormals. The new class is intimately related to the theory of negative definite functions on abelian semigroups. The known interplay between positive and negative definite functions from the theory of harmonic analysis on semigroups can be exploited to reveal some interesting connections between subnormals and completely hyperexpansive operators.

**[A]**Jim Agler,*Hypercontractions and subnormality*, J. Operator Theory**13**(1985), no. 2, 203–217. MR**775993****[At1]**Ameer Athavale,*Holomorphic kernels and commuting operators*, Trans. Amer. Math. Soc.**304**(1987), no. 1, 101–110. MR**906808**, 10.1090/S0002-9947-1987-0906808-6**[At2]**Ameer Athavale,*Some operator-theoretic calculus for positive definite kernels*, Proc. Amer. Math. Soc.**112**(1991), no. 3, 701–708. MR**1068114**, 10.1090/S0002-9939-1991-1068114-8**[B-C-R1]**Christian Berg, Jens Peter Reus Christensen, and Paul Ressel,*Positive definite functions on abelian semigroups*, Math. Ann.**223**(1976), no. 3, 253–274. MR**0420150****[B-C-R2]**Christian Berg, Jens Peter Reus Christensen, and Paul Ressel,*Harmonic analysis on semigroups*, Graduate Texts in Mathematics, vol. 100, Springer-Verlag, New York, 1984. Theory of positive definite and related functions. MR**747302****[B-S]**C. A. Berger and B. I. Shaw,*Selfcommutators of multicyclic hyponormal operators are always trace class*, Bull. Amer. Math. Soc.**79**(1973), 1193–1199, (1974). MR**0374972**, 10.1090/S0002-9904-1973-13375-0**[Co]**John B. Conway,*The theory of subnormal operators*, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR**1112128****[R]**Stefan Richter,*Invariant subspaces of the Dirichlet shift*, J. Reine Angew. Math.**386**(1988), 205–220. MR**936999**, 10.1515/crll.1988.386.205**[S]**Allen L. Shields,*Weighted shift operators and analytic function theory*, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49–128. Math. Surveys, No. 13. MR**0361899****[St]**J. G. Stampfli,*Which weighted shifts are subnormal?*, Pacific J. Math.**17**(1966), 367–379. MR**0193520**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47B20,
47B39

Retrieve articles in all journals with MSC (1991): 47B20, 47B39

Additional Information

**Ameer Athavale**

Affiliation:
Department of Mathematics, University of Pune, Pune - 411007, India

Email:
athavale@math.unipune.ernet.in

DOI:
https://doi.org/10.1090/S0002-9939-96-03609-X

Keywords:
Positive definite,
negative definite,
completely monotone,
completely alternating,
subnormal,
completely hyperexpansive

Received by editor(s):
May 17, 1995

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1996
American Mathematical Society