Tensor products of subnormal operators

Author:
Nathan S. Feldman

Journal:
Proc. Amer. Math. Soc. **127** (1999), 2685-2695

MSC (1991):
Primary 47B20; Secondary 47A80

Published electronically:
April 9, 1999

MathSciNet review:
1625745

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We shall use a -algebra approach to study operators of the form where is subnormal and is normal. We shall determine the spectral properties for these operators, and find the minimal normal extension and the dual operator. We also give a necessary condition for to contain a compact operator and a sufficient condition for the algebraic equivalence of and .

We also consider the existence of a homomorphism satisfying . We shall characterize the operators such that exists for every operator .

The problem of when is unitarily equivalent to is considered. Complete results are given when and are positive operators with finite multiplicity functions and has compact self-commutator. Some examples are also given.

**1.**I. David Berg,*An extension of the Weyl-von Neumann theorem to normal operators*, Trans. Amer. Math. Soc.**160**(1971), 365–371. MR**0283610**, 10.1090/S0002-9947-1971-0283610-0**2.**Arlen Brown,*On a class of operators*, Proc. Amer. Math. Soc.**4**(1953), 723–728. MR**0059483**, 10.1090/S0002-9939-1953-0059483-2**3.**Arlen Brown and Carl Pearcy,*Spectra of tensor products of operators*, Proc. Amer. Math. Soc.**17**(1966), 162–166. MR**0188786**, 10.1090/S0002-9939-1966-0188786-5**4.**John Bunce,*Characters on singly generated 𝐶*-algebras*, Proc. Amer. Math. Soc.**25**(1970), 297–303. MR**0259622**, 10.1090/S0002-9939-1970-0259622-4**5.**J. W. Bunce and J. A. Deddens,*On the normal spectrum of a subnormal operator*, Proc. Amer. Math. Soc.**63**(1977), no. 1, 107–110. MR**0435912**, 10.1090/S0002-9939-1977-0435912-3**6.**J.B. Conway,*Towards a functional calculus for subnormal tuples: the minimal normal extension and approximation in several complex variables*, Proc. Symp. Pure Math., vol. 51, part 1, Amer. Math. Soc., Providence, R.I., 1990, p. 105-112. CMP**91:03****7.**John B. Conway,*The theory of subnormal operators*, Mathematical Surveys and Monographs, vol. 36, American Mathematical Society, Providence, RI, 1991. MR**1112128****8.**Kenneth R. Davidson,*𝐶*-algebras by example*, Fields Institute Monographs, vol. 6, American Mathematical Society, Providence, RI, 1996. MR**1402012****9.**N.S. Feldman,*The self-commutator of a subnormal operator*, Ph.D. Thesis, University of Tennessee, 1997.**10.**N.S. Feldman,*Essentially subnormal operators*, to appear in Proc. Amer. Math. Soc.**11.**Richard V. Kadison and I. M. Singer,*Three test problems in operator theory*, Pacific J. Math.**7**(1957), 1101–1106. MR**0092123****12.**Jan Stochel,*Seminormality of operators from their tensor product*, Proc. Amer. Math. Soc.**124**(1996), no. 1, 135–140. MR**1286008**, 10.1090/S0002-9939-96-03017-1**13.**Dan Voiculescu,*A non-commutative Weyl-von Neumann theorem*, Rev. Roumaine Math. Pures Appl.**21**(1976), no. 1, 97–113. MR**0415338**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
47B20,
47A80

Retrieve articles in all journals with MSC (1991): 47B20, 47A80

Additional Information

**Nathan S. Feldman**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Email:
feldman@math.msu.edu

DOI:
https://doi.org/10.1090/S0002-9939-99-05054-6

Keywords:
Tensor product,
subnormal operator,
dual operator

Received by editor(s):
November 20, 1997

Published electronically:
April 9, 1999

Additional Notes:
This paper was written while the author was a graduate student at the University of Tennessee working under the direction of Professor John B. Conway. He received partial support from the NSF grant DMS–9401027.

Communicated by:
David R. Larson

Article copyright:
© Copyright 1999
American Mathematical Society