Operators with 𝐶*-algebra generated by a unilateral shift

Conway, John B.; McGuire, Paul

doi:10.1090/S0002-9947-1984-0742417-7

Operators with $C\sp{\ast}$ -algebra generated by a unilateral shift

Authors: John B. Conway and Paul McGuire
Journal: Trans. Amer. Math. Soc. 284 (1984), 153-161
MSC: Primary 47B20; Secondary 47C15
DOI: https://doi.org/10.1090/S0002-9947-1984-0742417-7
MathSciNet review: 742417
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: If is an operator on a Hilbert space $\mathcal{H}$ , this paper gives necessary and sufficient conditions on such that ${C^{\ast} }(T)$ , the ${C^{\ast} }$ -algebra generated by , is generated by a unilateral shift of some multiplicity. This result is then specialized to the cases in which is a hyponormal or subnormal operator. In particular, it is shown how to prove a recent conjecture of C. R. Putnam as a consequence of our result.

[1] C. A. Berger and B. I. Shaw, Selfcommutators of multicyclic hyponormal operators are always trace class, Bull. Amer. Math. Soc. 79 (1973), 1193–1199, (1974). MR 0374972, https://doi.org/10.1090/S0002-9904-1973-13375-0
[2] L. G. Brown, R. G. Douglas, and P. A. Fillmore, Unitary equivalence modulo the compact operators and extensions of 𝐶*-algebras, Proceedings of a Conference on Operator Theory (Dalhousie Univ., Halifax, N.S., 1973) Springer, Berlin, 1973, pp. 58–128. Lecture Notes in Math., Vol. 345. MR 0380478
[3] Richard W. Carey and Joel D. Pincus, An invariant for certain operator algebras, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 1952–1956. MR 0344925
[4] Kevin Clancey, Seminormal operators, Lecture Notes in Mathematics, vol. 742, Springer, Berlin, 1979. MR 548170
[5] Stuart Clary, Equality of spectra of quasi-similar hyponormal operators, Proc. Amer. Math. Soc. 53 (1975), no. 1, 88–90. MR 0390824, https://doi.org/10.1090/S0002-9939-1975-0390824-7
[6] L. A. Coburn, The 𝐶*-algebra generated by an isometry. II, Trans. Amer. Math. Soc. 137 (1969), 211–217. MR 0236720, https://doi.org/10.1090/S0002-9947-1969-0236720-9
[7] John B. Conway, On quasisimilarity for subnormal operators, Illinois J. Math. 24 (1980), no. 4, 689–702. MR 586807
[8] John B. Conway, Subnormal operators, Research Notes in Mathematics, vol. 51, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. MR 634507
[9] Ronald G. Douglas, Banach algebra techniques in operator theory, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 49. MR 0361893
[10] William F. Osgood, A Jordan curve of positive area, Trans. Amer. Math. Soc. 4 (1903), no. 1, 107–112. MR 1500628, https://doi.org/10.1090/S0002-9947-1903-1500628-5
[11] C. R. Putnam, An inequality for the area of hyponormal spectra, Math. Z. 116 (1970), 323–330. MR 0270193, https://doi.org/10.1007/BF01111839
[12] C. R. Putnam, The spectra of subnormal operators, Proc. Amer. Math. Soc. 28 (1971), 473–477. MR 0275215, https://doi.org/10.1090/S0002-9939-1971-0275215-8
[13] C. R. Putnam, Hyponormal operators with minimal spectral area, Houston J. Math. 6 (1980), no. 1, 135–139. MR 575921
[14] -, Singly generated hyponormal ${C^{\ast}}$ -algebras, preprint.
[15] M. Radjabalipour, Ranges of hyponormal operators, Illinois J. Math. 21 (1977), no. 1, 70–75. MR 0448140
[16] Marc Raphael, Quasisimilarity and essential spectra for subnormal operators, Indiana Univ. Math. J. 31 (1982), no. 2, 243–246. MR 648174, https://doi.org/10.1512/iumj.1982.31.31021