Formal Taylor series and complementary invariant subspaces

Herrero, Domingo A.

doi:10.1090/S0002-9939-1974-0372662-3

Formal Taylor series and complementary invariant subspaces
HTML articles powered by AMS MathViewer

by Domingo A. Herrero PDF

Proc. Amer. Math. Soc. 45 (1974), 83-87 Request permission

Abstract:

A class of operators (which includes the unilateral weighted shifts and the noninvertible bilateral weighted shifts on Hilbert spaces), with the property that every element of the commutant of the operator is canonically associated to a formal Taylor series, is characterized. Let $T$ be one of such operators, then the following result is true: $T$ has no nontrivial complementary invariant subspaces, no roots and no logarithm. This result can be partially extended to the case when “commutant” is replaced by “double commutant", then: $T$ has no nontrivial complementary hyperinvariant subspaces.

References

R. G. Douglas and Carl Pearcy, On a topology for invariant subspaces, J. Functional Analysis 2 (1968), 323–341. MR 0233224, DOI 10.1016/0022-1236(68)90010-4
Nelson Dunford and Jacob T. Schwartz, Linear Operators. I. General Theory, Pure and Applied Mathematics, Vol. 7, Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1958. With the assistance of W. G. Bade and R. G. Bartle. MR 0117523
Ralph Gellar, Operators commuting with a weighted shift, Proc. Amer. Math. Soc. 23 (1969), 538–545. MR 259641, DOI 10.1090/S0002-9939-1969-0259641-X
Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
Domingo Antonio Herrero and Norberto Salinas, Analytically invariant and bi-invariant subspaces, Trans. Amer. Math. Soc. 173 (1972), 117–136. MR 312294, DOI 10.1090/S0002-9947-1972-0312294-9
A. L. Shields and L. J. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1970/71), 777–788. MR 287352, DOI 10.1512/iumj.1971.20.20062
Noboru Suzuki, On the irreducibility of weighted shifts, Proc. Amer. Math. Soc. 22 (1969), 579–581. MR 257793, DOI 10.1090/S0002-9939-1969-0257793-9