On strictly cyclic algebras, $\mathcal {P}$-algebras and reflexive operators
HTML articles powered by AMS MathViewer
- by Domingo A. Herrero and Alan Lambert
- Trans. Amer. Math. Soc. 185 (1973), 229-235
- DOI: https://doi.org/10.1090/S0002-9947-1973-0328619-5
- PDF | Request permission
Abstract:
An operator algebra $\mathfrak {A} \subset \mathcal {L}(\mathcal {X})$ (the algebra of all operators in a Banach space $\mathcal {X}$ over the complex field C) is called a “strictly cyclic algebra” (s.c.a.) if there exists a vector ${x_0} \in \mathcal {X}$ such that $\mathfrak {A}({x_0}) = \{ A{x_0}:A \in \mathfrak {A}\} = \mathcal {X};{x_0}$ is called a “strictly cyclic vector” for $\mathfrak {A}$. If, moreover, ${x_0}$ separates elements of $\mathfrak {A}$ (i.e., if $A \in \mathfrak {A}$ and $A{x_0} = 0$, then $A = 0$), then $\mathfrak {A}$ is called a “separated s.c.a." $\mathfrak {A}$ is a $\mathcal {P}$-algebra if, given ${x_1}, \ldots ,{x_n} \in \mathcal {X}$, there exists ${x_0} \in \mathcal {X}$ such that $\left \| {A{x_j}} \right \| \leq \left \| {A{x_0}} \right \|$, for all $A \in \mathfrak {A}$ and for $j = 1, \ldots ,n$. Among other results, it is shown that if the commutant $\mathfrak {A}’$ of the algebra $\mathfrak {A}$ is an s.c.a., then $\mathfrak {A}$ is a $\mathcal {P}$-algebra and the strong and the uniform operator topology coincide on $\mathfrak {A}$; these results are specialized for the case when $\mathfrak {A}$ and $\mathfrak {A}’$ are separated s.c.a.’s. (Here, and in what follows, algebra means strongly closed subalgebra on $\mathcal {L}(\mathcal {X})$ containing the identity I on $\mathcal {X}$.) 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 $\mathfrak {A}$ is called reflexive if, whenever $T \in \mathcal {L}(\mathcal {X})$ and the lattice of invariant subspaces of T contains the corresponding lattice of $\mathfrak {A}$, then $T \in \mathfrak {A}$.References
- James A. Deddens, Reflexive operators, Indiana Univ. Math. J. 20 (1971), no. 10, 887–889. MR 412847, DOI 10.1512/iumj.1971.20.20072
- 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 —, Two sublattices of weighted shift invariant subspaces (to appear).
- Domingo A. Herrero, Eigenvectors and cyclic vectors for bilateral weighted shifts, Rev. Un. Mat. Argentina 26 (1972/73), 24–41. MR 336395 —, Álgebras de operadores que contienen una subálgebra de multiplicidad estricta finita, Rev. Un. Mat. Argentina (to appear).
- 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
- Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717–726. MR 310664
- Alan Lambert, Strictly cyclic weighted shifts, Proc. Amer. Math. Soc. 29 (1971), 331–336. MR 275213, DOI 10.1090/S0002-9939-1971-0275213-4
- Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717–726. MR 310664
- 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
- D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517. MR 192365
- 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
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- 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