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

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*.

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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