×

A characterization of the invariant subspaces of direct sums of strictly cyclic algebras. (English) Zbl 0615.47005

Let \({\mathcal A}\) be a unicellular strictly cyclic Abelian algebra of operators on a Hilbert space H. As the title indicates, the author has characterized (theorem 1) the invariant subspaces of \({\mathcal A}^{(n)}\) as those subspaces which are spans of at most n invariant graph subspaces with domains invariant of \({\mathcal A}.\)
H\({}^{(n)}\) stands for the sum of n copies of H and \({\mathcal A}^{(n)}:=\{T^{(n)}\), \(T\in {\mathcal A}\}\) where T is a linear operator on H, and \(T^{(n)}\) is defined in the standard way. An invariant subspace M of \({\mathcal A}^{(n)}\) is called an invariant graph subspace on the ith-coordinate if M has the form: \[ M=\{(T_ 1x,...,T_{i- 1}x,x,T_{i+1}x,...,T_ nx),\quad x\in D\} \] for some linear manifold D of H, and linear transformations \(T_ j\) with domain D and range in H.
Leaning on theorem 1 the author also claims that the invariant subspaces of \((T^*)^{(n)}\) are precisely the spans of finite dimensional invariant subspaces of \((T^*)^{(n)}\), whenever T has a unipunctual spectrum and the weakly closed algebra generated by T is hereditarily strictly cyclic.
The result of theorem 1 is also generalized to the \(\ell_ 2\)-sum of denumerably many copies of H, with some stronger assumptions on the operator algebra \({\mathcal A}\).
Reviewer: E.Martin-Peinador

MSC:

47A15 Invariant subspaces of linear operators
47L30 Abstract operator algebras on Hilbert spaces
15A04 Linear transformations, semilinear transformations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] William B. Arveson, A density theorem for operator algebras, Duke Math. J. 34 (1967), 635 – 647. · Zbl 0183.42403
[2] W. F. Donoghue Jr., The lattice of invariant subspaces of a completely continuous quasi-nilpotent transformation, Pacific J. Math. 7 (1957), 1031 – 1035. · Zbl 0078.29504
[3] Mary R. Embry, Maximal invariant subspaces of strictly cyclic operator algebras, Pacific J. Math. 49 (1973), 45 – 50. · Zbl 0236.46069
[4] Ciprian Foiaş and J. P. Williams, Some remarks on the Volterra operator, Proc. Amer. Math. Soc. 31 (1972), 177 – 184. · Zbl 0246.47012
[5] Paul R. Halmos, Finite-dimensional vector spaces, The University Series in Undergraduate Mathematics, D. Van Nostrand Co., Inc., Princeton-Toronto-New York-London, 1958. 2nd ed. · Zbl 1368.15001
[6] James H. Hedlund, Strongly strictly cyclic weighted shifts, Proc. Amer. Math. Soc. 57 (1976), no. 1, 119 – 121. · Zbl 0342.47017
[7] Domingo A. Herrero, Transitive algebras of operators that contain a subalgebra of finite strict multiplicity, Rev. Un. Mat. Argentina 26 (1972/73), 77 – 84 (Spanish). · Zbl 0253.46124
[8] Domingo A. Herrero, Operator algebras of finite strict multiplicity, Indiana Univ. Math. J. 22 (1972/73), 13 – 24. · Zbl 0235.46098 · doi:10.1512/iumj.1972.22.22003
[9] Domingo A. Herrero and Alan Lambert, On strictly cyclic algebras, \?-algebras and reflexive operators, Trans. Amer. Math. Soc. 185 (1973), 229 – 235. · Zbl 0253.46127
[10] Edward Kerlin and Alan Lambert, Strictly cyclic shifts on \?_{\?}, Acta Sci. Math. (Szeged) 35 (1973), 87 – 94. · Zbl 0266.47021
[11] A. L. Lambert, Strictly cyclic operator algebras, Dissertation, Univ. of Michigan, 1970. · Zbl 0213.40701
[12] Alan Lambert, Strictly cyclic weighted shifts, Proc. Amer. Math. Soc. 29 (1971), 331 – 336. · Zbl 0214.14201
[13] Alan Lambert, Strictly cyclic operator algebras, Pacific J. Math. 39 (1971), 717 – 726. · Zbl 0213.40701
[14] A. L. Lambert and T. R. Turner, The double commutant of invertibly weighted shifts, Duke Math. J. 39 (1972), 385 – 389. · Zbl 0241.46062
[15] N. K. Nikol\(^{\prime}\)skiĭ, Invariant subspaces of weighted shift operators, Mat. Sb. (N.S.) 74 (116) (1967), 172 – 190 (Russian).
[16] Eric A. Nordgren, Closed operators commuting with a weighted shift, Proc. Amer. Math. Soc. 24 (1970), 424 – 428. · Zbl 0188.44201
[17] Eric A. Nordgren, Invariant subspaces of a direct sum of weighted shifts, Pacific J. Math. 27 (1968), 587 – 598. · Zbl 0172.16804
[18] Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77. · Zbl 0269.47003
[19] Erik J. Rosenthal, Power bounded strictly cyclic operators, Proc. Amer. Math. Soc. 72 (1978), no. 2, 276 – 280. · Zbl 0408.47015
[20] Erik J. Rosenthal, A Jordan form for certain infinite-dimensional operators, Acta Sci. Math. (Szeged) 41 (1979), no. 3-4, 365 – 374. · Zbl 0441.47018
[21] Allen L. Shields, Weighted shift operators and analytic function theory, Topics in operator theory, Amer. Math. Soc., Providence, R.I., 1974, pp. 49 – 128. Math. Surveys, No. 13. · Zbl 0303.47021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.