On asymptotic properties of several classes of operators

Authors:
Stephen L. Campbell and Ralph Gellar

Journal:
Proc. Amer. Math. Soc. **66** (1977), 79-84

MSC:
Primary 47B15; Secondary 47A50

DOI:
https://doi.org/10.1090/S0002-9939-1977-0461187-5

MathSciNet review:
0461187

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a polynomial in *T* and where *T* is a bounded linear operator on a separable Hilbert space. Let . Then is said to be asymptotic for *p* if for every , there exists an and function , such that if , and , then there exists such that . It is observed that the hermitian, unitary, and isometric operators are asymptotic for the obvious polynomials. It is known that the normals are not asymptotic for . An example gives several negative results including one that says the quasinormals are not asymptotic for . It is shown that if *p* is any polynomial in just one of *T* or , then is asymptotic for *p*.

**[1]**J. Bastian and K. J. Harrison,*Subnormal weighted shifts and asymptotic properties of normal operators*, Proc. Amer. Math. Soc.**42**(1974), 475-479. MR**0380491 (52:1391)****[2]**S. L. Campbell,*Linear operators for which**and**commute*. II, Pacific J. Math.**53**(1974), 355-361. MR**0361886 (50:14328)****[3]**-,*Linear operators for which**and**commute*, Pacific J. Math.**61**(1975), 53-58. MR**0405168 (53:8963)****[4]**S. L. Campbell and R. Gellar,*Spectral properties of linear operators for which**and**commute*, Proc. Amer. Math. Soc.**60**(1976), 197-202. MR**0417841 (54:5889)****[5]**-,*Linear operators for which**and**commute*. II, Trans. Amer. Math. Soc.**226**(1977), 305-319. MR**0435905 (55:8856)****[6]**P. R. Halmos,*Finite-dimensional noncommutative approximation theory*, Talk at the 1973 Conference of Theoretical Matrix Theory, University of California, Santa Barbara, Calif.**[7]**R. Moore,*An asymptotic Fuglede Theorem*, Proc. Amer. Math. Soc.**50**(1975), 138-142. MR**0370247 (51:6474)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
47B15,
47A50

Retrieve articles in all journals with MSC: 47B15, 47A50

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0461187-5

Keywords:
Normal,
quasinormal,
hyponormal,
approximation,
asymptotic,
algebraic

Article copyright:
© Copyright 1977
American Mathematical Society