Hyperinvariant subspaces of contractions

Author:
Pei Yuan Wu

Journal:
Proc. Amer. Math. Soc. **75** (1979), 53-58

MSC:
Primary 47A45; Secondary 47A15

DOI:
https://doi.org/10.1090/S0002-9939-1979-0529212-2

MathSciNet review:
529212

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Abstract | References | Similar Articles | Additional Information

Abstract: For an operator *T* on a Hilbert space let Hyperlat *T* denote its hyperinvariant subspace lattice. Assume that *T* is a completely nonunitary contraction with finite defect indices. In this note we characterize the elements of Hyperlat *T* among invariant subspaces for *T* in terms of their corresponding regular factorizations and show that elements in Hyperlat *T* are exactly the spectral subspaces of *T* defined by Sz.-Nagy and Foiaş. As a corollary, if are two such operators which are quasi-similar to each other, then Hyperlat is (lattice) isomorphic to Hyperlat .

**[1]**C. Apostol,*Spectral decompositions and functional calculus*, Rev. Roumaine Math. Pures Appl.**13**(1968), 1481-1528. MR**0250092 (40:3333)****[2]**R. G. Douglas,*On the operator equation**and related topics*, Acta Sci. Math. Szeged**30**(1969), 19-32. MR**0250106 (40:3347)****[3]**R. G. Douglas and C. Pearcy,*On a topology for invariant subspaces*, J. Functional Analysis**2**(1968), 323-341. MR**0233224 (38:1547)****[4]**R. V. Kadison and I. M. Singer,*Three test problems in operator theory*, Pacific J. Math.**7**(1957), 1101-1106. MR**0092123 (19:1066e)****[5]**B. Sz.-Nagy and C. Foiaş,*Harmonic analysis of operators on Hilbert space*, North-Holland, Amsterdam; Akadémiai Kiadó, Budapest, 1970. MR**0275190 (43:947)****[6]**M. Uchiyama,*Hyperinvariant subspaces of operators of class*, Acta Sci. Math. (Szeged)**39**(1977), 179-184. MR**0482299 (58:2372)****[7]**-,*Hyperinvariant subspaces of contractions of class*, Hokkaido Math. J.**6**(1977), 260-272. MR**0482300 (58:2373)****[8]**P. Y. Wu,*Hyperinvariant subspaces of the direct sum of certain contractions*, Indiana Univ. Math. J.**27**(1978), 267-274. MR**0493437 (58:12444)****[9]**-,*Jordan model for weak contractions*, Acta Sci. Math. (Szeged)**40**(1978), 189-196. MR**0487521 (58:7144)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1979-0529212-2

Keywords:
Hyperinvariant subspace,
contraction,
quasi-similarity,
Jordan model for contractions

Article copyright:
© Copyright 1979
American Mathematical Society