On asymptotic properties of several classes of operators
Authors:
Stephen L. Campbell and Ralph Gellar
Journal:
Proc. Amer. Math. Soc. 66 (1977), 7984
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.
 [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
(52 #1391), http://dx.doi.org/10.1090/S0002993919740380491X
 [2]
Stephen
L. Campbell, Linear operators for which 𝑇*𝑇 and
𝑇𝑇*\
commute. II, Pacific J. Math. 53
(1974), 355–361. MR 0361886
(50 #14328)
 [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
(53 #8963)
 [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 (54 #5889), http://dx.doi.org/10.1090/S00029939197604178413
 [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
(55 #8856), http://dx.doi.org/10.1090/S00029947197704359050
 [6]
P. R. Halmos, Finitedimensional 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
(51 #6474), http://dx.doi.org/10.1090/S00029939197503702477
 [1]
 J. Bastian and K. J. Harrison, Subnormal weighted shifts and asymptotic properties of normal operators, Proc. Amer. Math. Soc. 42 (1974), 475479. MR 0380491 (52:1391)
 [2]
 S. L. Campbell, Linear operators for which and commute. II, Pacific J. Math. 53 (1974), 355361. MR 0361886 (50:14328)
 [3]
 , Linear operators for which and commute, Pacific J. Math. 61 (1975), 5358. 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), 197202. MR 0417841 (54:5889)
 [5]
 , Linear operators for which and commute. II, Trans. Amer. Math. Soc. 226 (1977), 305319. MR 0435905 (55:8856)
 [6]
 P. R. Halmos, Finitedimensional 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), 138142. MR 0370247 (51:6474)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704611875
PII:
S 00029939(1977)04611875
Keywords:
Normal,
quasinormal,
hyponormal,
approximation,
asymptotic,
algebraic
Article copyright:
© Copyright 1977 American Mathematical Society
