Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On strictly cyclic algebras, $ \mathcal{P}$-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
MathSciNet review: 0328619
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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\Vert {A{x_j}} \right\Vert \leq \left\Vert {A{x_0}} \right\Vert$, 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 [Enhancements On Off] (What's this?)

  • [1] J. A. Deddens, Reflexive operators, Indiana Univ. Math. J. 20 (1971), 887-889. MR 0412847 (54:968)
  • [2] R. Gellar, Cyclic vectors and parts of the spectrum of a weighted shift, Trans. Amer. Math. Soc. 146 (1969), 69-85. MR 41 #4277b. MR 0259642 (41:4277b)
  • [3] -, Two sublattices of weighted shift invariant subspaces (to appear).
  • [4] D. A. Herrero, Eigenvectors and cyclic vectors for bilateral weighted shifts, Rev. Un. Mat Argentina (to appear). MR 0336395 (49:1170)
  • [5] -, Álgebras de operadores que contienen una subálgebra de multiplicidad estricta finita, Rev. Un. Mat. Argentina (to appear).
  • [6] D. A. Herrero and N. Salinas, Analytically invariant and bi-invariant subspaces (to appear). MR 0312294 (47:856)
  • [7] Alan Lambert, Strictly cyclic operator algebras, Dissertation, University of Michigan, Ann Arbor, Mich., 1970. MR 0310664 (46:9762)
  • [8] -, Strictly cyclic weighted shifts, Proc. Amer. Math. Soc. 29 (1971), 331-336. MR 43 #970. MR 0275213 (43:970)
  • [9] -, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717-726. MR 0310664 (46:9762)
  • [10] C. E. Rickart, General theory of Banach algebras, University Series in Higher Math., Van Nostrand, Princeton, N.J., 1960. MR 22 #5903. MR 0115101 (22:5903)
  • [11] D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511-517. MR 33 #590. MR 0192365 (33:590)
  • [12] A. L. Shields and L. J. Wallen, The commutant of certain Hilbert space operators, Indiana Univ. Math. J. 20 (1971), 777-788. MR 0287352 (44:4558)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46L20, 47A15

Retrieve articles in all journals with MSC: 46L20, 47A15

Additional Information

Keywords: Banach algebras, algebras of operators, $ \mathcal{P}$-algebras, strictly cyclic algebras, reflexive algebra, weighted shifts, invariant subspaces
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society