On strictly cyclic algebras, -algebras and reflexive operators

Authors:
Domingo A. Herrero and Alan Lambert

Journal:
Trans. Amer. Math. Soc. **185** (1973), 229-235

MSC:
Primary 46L20; Secondary 47A15

DOI:
https://doi.org/10.1090/S0002-9947-1973-0328619-5

MathSciNet review:
0328619

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Abstract | References | Similar Articles | Additional Information

Abstract: An operator algebra (the algebra of all operators in a Banach space over the complex field **C**) is called a ``strictly cyclic algebra'' (s.c.a.) if there exists a vector such that is called a ``strictly cyclic vector'' for . If, moreover, separates elements of (i.e., if and , then ), then is called a ``separated s.c.a."

is a -algebra if, given , there exists such that , for all and for . Among other results, it is shown that if the commutant of the algebra is an s.c.a., then is a -algebra and the strong and the uniform operator topology coincide on ; these results are specialized for the case when and are separated s.c.a.'s. (Here, and in what follows, *algebra* means *strongly closed subalgebra on* *containing the identity I on* .)

In the second part of the paper, it is shown that a large class of bilateral weighted shifts (which includes all the invertible ones) in a Hilbert space are reflexive. The result is used to show that ``reflexivity'' is neither a ``restriction property'' nor a ``quotient property."

Recall that an algebra is called *reflexive* if, whenever and the lattice of invariant subspaces of *T* contains the corresponding lattice of , then .

**[1]**James A. Deddens,*Reflexive operators*, Proceedings of an International Symposium on Operator Theory (Indiana Univ., Bloomington, Ind., 1970), 1971, pp. 887–889. MR**0412847**, https://doi.org/10.1512/iumj.1971.20.20072**[2]**Ralph Gellar,*Cyclic vectors and parts of the spectrum of a weighted shift*, Trans. Amer. Math. Soc.**146**(1969), 69–85. MR**0259642**, https://doi.org/10.1090/S0002-9947-1969-0259642-6**[3]**-,*Two sublattices of weighted shift invariant subspaces*(to appear).**[4]**Domingo A. Herrero,*Eigenvectors and cyclic vectors for bilateral weighted shifts*, Rev. Un. Mat. Argentina**26**(1972/73), 24–41. MR**0336395****[5]**-,*Álgebras de operadores que contienen una subálgebra de multiplicidad estricta finita*, Rev. Un. Mat. Argentina (to appear).**[6]**Domingo Antonio Herrero and Norberto Salinas,*Analytically invariant and bi-invariant subspaces*, Trans. Amer. Math. Soc.**173**(1972), 117–136. MR**0312294**, https://doi.org/10.1090/S0002-9947-1972-0312294-9**[7]**Alan Lambert,*Strictly cyclic operator algebras*, Pacific J. Math.**39**(1971), 717–726. MR**0310664****[8]**Alan Lambert,*Strictly cyclic weighted shifts*, Proc. Amer. Math. Soc.**29**(1971), 331–336. MR**0275213**, https://doi.org/10.1090/S0002-9939-1971-0275213-4**[9]**Alan Lambert,*Strictly cyclic operator algebras*, Pacific J. Math.**39**(1971), 717–726. MR**0310664****[10]**Charles E. Rickart,*General theory of Banach algebras*, The University Series in Higher Mathematics, D. van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0115101****[11]**D. Sarason,*Invariant subspaces and unstarred operator algebras*, Pacific J. Math.**17**(1966), 511–517. MR**0192365****[12]**A. L. Shields and L. J. Wallen,*The commutants of certain Hilbert space operators*, Indiana Univ. Math. J.**20**(1970/1971), 777–788. MR**0287352**, https://doi.org/10.1512/iumj.1971.20.20062

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0328619-5

Keywords:
Banach algebras,
algebras of operators,
-algebras,
strictly cyclic algebras,
reflexive algebra,
weighted shifts,
invariant subspaces

Article copyright:
© Copyright 1973
American Mathematical Society