Growth conditions and similarity orbits

Ghatage, P.

doi:10.1090/S0002-9939-1977-0448132-3

Growth conditions and similarity orbits
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by P. Ghatage

Proc. Amer. Math. Soc. 65 (1977), 127-130

DOI: https://doi.org/10.1090/S0002-9939-1977-0448132-3

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Abstract:

It is proved that an operator T which acts on a separable Hilbert space and whose resolvent satisfies a first-order growth condition inside and outside the unit circle, can be approximated in norm by operators which are similar to a single unitary operator. In particular the ’Markus-operator’ can be approximated by similarities of a diagonal unitary operator and is a compact perturbation of the same.

References

R. G. Douglas and Carl Pearcy, Invariant subspaces of non-quasitriangular operators, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Lecture Notes in Math., Vol. 345, Springer, Berlin, 1973, pp. 13–57. MR 0358391
L. A. Fialkow, The similarity orbit of a normal operator, Trans. Amer. Math. Soc. 210 (1975), 129–137. MR 374956, DOI 10.1090/S0002-9947-1975-0374956-X
Domingo A. Herrero, Closure of similarity orbits of Hilbert space operators. I, Rev. Un. Mat. Argentina 27 (1976), no. 4, 244–260 (Spanish). MR 512768
A. S. Markus, Certain criteria for the completeness of a system of root-vectors of a linear operator in a Banach space, Mat. Sb. (N.S.) 70 (112) (1966), 526–561 (Russian). MR 0216316
A. S. Markus, L. N. Nikol′skaja, and N. K. Nikol′skiĭ, The unitary spectrum of a contraction in a Banach space, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 65–74 (Russian). MR 0290152
N. K. Nikol′skiĭ, On spectral analysis on the unitary spectrum. Point spectrum, Dokl. Akad. Nauk SSSR 199 (1971), 544–547 (Russian). MR 0288609
C. R. Putnam, The spectra of operators having resolvents of first-order growth, Trans. Amer. Math. Soc. 133 (1968), 505–510. MR 229073, DOI 10.1090/S0002-9947-1968-0229073-2
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
J. G. Stampfli, A local spectral theory for operators. III. Resolvents, spectral sets and similarity, Trans. Amer. Math. Soc. 168 (1972), 133–151. MR 295114, DOI 10.1090/S0002-9947-1972-0295114-0