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Remarks Concerning Uniformly Bounded Operators on Hilbert Space

Published online by Cambridge University Press:  20 November 2018

I. Istrǎțescu
I. Istrǎțescu*
Affiliation:
Politechnic Institute Timişoara, Timişoara, Roumania
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In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that

1

is similar to a unitary operator.

The following problem is an extension of this result: If T and S are two operators such that

  1. 1. sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)

  2. 2. TS = ST

then there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?

Also, in [7] B. Sz.-Nagy has proved that every compact operator T such that

sup ‖Tn‖<∞ (n = 1, 2, 3,…)

is similar to a contraction.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 215 - 217
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Dunford, N. and Schwartz, J., Linear operators 19 Interscience, New York, 1958.Google Scholar
2. Hill, J., Almost-convergent double sequences, Töhoku Math. J., 17 (1965), 105-116.Google Scholar
3. Istrǎțescu, I., On operators with uniformly bounded iterates, Mat. Vestnik, 21 (1969), 373-375.Google Scholar
4. Istrǎțescu, I. and Constantin, Gh., On Riesz operators with uniformly bounded iterates, Mat. Vestnik, 21 (1969), 376-378.Google Scholar
5. Ruston, A. F., Operators with Fredholm theory, J. Londo. Math. Soc. 29 (1954), 318-326.Google Scholar
6. Sz.-Nagy, B., On uniformly bounded linear transformations in Hilbert space, Acta Sci. Math. (Szeged), 11 (1946/48), 152-157.Google Scholar
7. Sz.-Nagy, B., Completely continuous operators with uniformly bounded iterates, Magyar Tud. Akad. Mat. Kutatö Int. Közl. 4 (1959), 89-93.Google Scholar