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

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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. 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**, https://doi.org/10.1090/S0002-9939-1974-0380491-X**[2]**Stephen L. Campbell,*Linear operators for which 𝑇*𝑇 and 𝑇𝑇* commute. II*, Pacific J. Math.**53**(1974), 355–361. MR**0361886****[3]**Stephen L. Campbell,*Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute*, Pacific J. Math.**61**(1975), no. 1, 53–57. MR**0405168****[4]**Stephen L. Campbell and Ralph Gellar,*Spectral properties of linear operators for which 𝑇*𝑇 and 𝑇 + 𝑇* commute*, Proc. Amer. Math. Soc.**60**(1976), 197–202 (1977). MR**0417841**, https://doi.org/10.1090/S0002-9939-1976-0417841-3**[5]**Stephen L. Campbell and Ralph Gellar,*Linear operators for which 𝑇*𝑇 and 𝑇+𝑇* commute. II*, Trans. Amer. Math. Soc.**226**(1977), 305–319. MR**0435905**, https://doi.org/10.1090/S0002-9947-1977-0435905-0**[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]**Robert Moore,*An asymptotic Fuglede theorem*, Proc. Amer. Math. Soc.**50**(1975), 138–142. MR**0370247**, https://doi.org/10.1090/S0002-9939-1975-0370247-7

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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