Abstract
We will discuss methods for proving invariant space results which were first introduced by Scott Brown for subnormal operators. These methods are now related with the idea of a dual algebra.
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References
J. Agler, An invariant subspace theorem, J. Funct. Anal. 38 (1980), 315–323.
J. Agler and J.E. McCarthy, Operators that dominate normal operators, J. Operator Theory 40 (1998), 385–407.
E. Albrecht and B. Chevreau, Invariant subspaces for l p-operators having Bishop’s property (β) on a large part of their spectrum, J. Operator Theory 18 (1987), 339–372.
E. Albrecht and J. Eschmeier, Analytic functional models and local spectral theory, Proc. London Math. Soc. (3) 75 (1997), 323–348.
A. Aleman, H. Hedenmalm and S. Richter, Recent progress and open problems in the Bergman space, Oper. Theory Adv. Appl. 156 (2005), 27–59.
A. Aleman, S. Richter and C. Sundberg, Beurling’s theorem for the Bergman space, Acta Math. 177 (1996), 275–310.
S. Richter and C. Sundberg, Analytic contractions, nontangential limits, and the index of invariant subspaces, Trans. Amer. Math. Soc. 359 (2007), 3369–3407.
C. Ambrozie and V. Müller, Invariant subspaces for polynomially bounded operators, J. Funct. Anal. 213 (2004), 321–345.
V. Müller, Invariant subspaces for n-tuples of contractions with dominating Taylor spectrum, J. Operator Theory 61 (2009), 63–73.
C. Apostol, Spectral decompositions and functional calculus, Rev. Roumaine Math. Pures Appl. 13 (1968), 1481–1528.
C. Apostol, Ultraweakly closed operator algebras, J. Operator Theory 2 (1979), 49–61.
C. Apostol, Functional calculus and invariant subspaces, J. Operator Theory 4 (1980), 159–190.
C. Apostol The spectral flavour of Scott Brown’s techniques, J. Operator Theory 6 (1981), 3–12.
C. Apostol, H. Bercovici, C. Foias and C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. I, J. Funct. Anal. 63 (1985), 369–404.
H. Bercovici, C. Foias and C. Pearcy, Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra. II, Indiana Univ. Math. J. 34 (1985), 845–855.
A. Arias and G. Popescu, Factorization and reflexivity on Fock spaces, Integral Equations Operator theory 23 (1995), 268–286.
W.B. Arveson, Interpolation problems in nest algebras, J. Funct. Anal. 20 (1975), 208–233.
H. Bercovici, C. Foias, J. Langsam and C. Pearcy, (BCP)-operators are reflexive, Michigan Math. J. 29 (1982), 371–379.
H. Bercovici, Factorization theorems for integrable functions, Analysis at Urbana. II (ed. E.R. Berkson et al.), Cambridge University Press, Cambridge, 1989.
H. Bercovici, Factorization theorems and the structure of operators on Hilbert space, Ann. of Math. (2) 128 (1988), 399–413.
H. Bercovici, A factorization theorem with applications to invariant subspaces and the reflexivity of isometries, Math. Research Letters, 1 (1994), 511–518.
H. Bercovici, Hyper-reflexivity and the factorization of linear functionals, J. Funct Anal. 158 (1998), 242–252.
H. Bercovici, B. Chevreau, C. Foias and C. Pearcy, Dilation theory and systems of simultaneous equations in the predual of an operator algebra. II, Math. Z. 187 (1984), 97–103.
H. Bercoviciand J.B. Conway, A note on the algebra generated by a subnormal operator, Operator Theory: Advances and Applications 32 (1988), 53–56.
H. Bercovici, C. Foias, J. Langsam and C. Pearcy, (BCP)-operators are reflexive, Michigan Math. J. 29 (1982), 371–379.
H. Bercovici, C. Foias. and C. Pearcy, Dual algebras with applications to invariant subspaces and dilation theory, CBMS Regional Conf. Ser. in Math., No. 56, Amer. Math. Soc, Providence, R.I., 1985.
C. Foias. and C. Pearcy, Dilation theory and systems of simultaneous equations in the predual of an operator algebra. I, Michigan Math. J. 30 (1983), 335–354.
C. Foias. and C. Pearcy, On the reflexivity of algebras and linear spaces of operators, Michigan Math. J. 33 (1986), 119–126.
C. Foias. and C. Pearcy, Two Banach space methods and dual operator algebras, J. Funct. Anal. 78 (1988), 306–345.
H. Bercovici and W.S. Li, A near-factorization theorem for integrable functions, Integral Equations Operator Theory 17 (1993), 440–442.
H. Bercovici, V. Paulsen and C. Hernandez, Universal compressions of representations of H∞(G), Math. Ann. 281 (1988), 177–191.
H. Bercovici and B. Prunaru, An improved factorization theorem with applications to subnormal operators, Acta Sci. Math. (Szeged) 63 (1997), 647–655.
L. Brown, A. Shields and K. Zeller, On absolutely convergent exponential sums, Trans. Amer. Math. Soc. 96 (1960), 162–183.
S.W. Brown, Some invariant subspaces for subnormal operators, Integral Equations Operator Theory 1 (1978), 310–333.
S.W. Brown, Hyponormal operators with thick spectra have invariant subspaces, Ann. of Math. (2) 125 (1987), 93–103.
S.W. Brown, Contractions with spectral boundary, Integral Equations Operator Theory 11 (1988), 49–63.
S.W. Brown and B. Chevreau, Toute contraction à calcul fonctionnel isométrique est réflexive, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 185–188.
S.W. Brown, B. Chevreau and C. Pearcy, Contractions with rich spectrum have invariant subspaces, J. Operator Theory 1 (1979), 123–136.
B. Chevreau and C. Pearcy, On the structure of contraction operators. II, J. Funct. Anal. 76 (1988), 30–55.
S.W. Brown and E. Ko, Operators of Putinar type, Oper. Theory Adv. Appl. 104 (1998), 49–57.
G. Cassier, Un exemple d’opérateur pour lequel les topologies faible et ultrafaible ne coïncident pas sur l’algèbre duale, J. Operator Theory 16 (1986), 325–333.
E. Ko, Algèbres duales uniformes d’opérateurs sur l’espace de Hilbert, Studia Math. 95 (1989), 17–32.
G. Cassier, I. Chalendar and B. Chevreau, A mapping theorem for the boundary set X t of an absolutely continuous contraction T, J. Operator Theory 50 (2003), 331–343.
I. Chalendar and J. Esterle, L 1-factorization for C oo-contractions with isometric functional calculus, J. Funct. Anal. 154 (1998), 174–194.
I. Chalendar, J.R. Partington and R.C. Smith, L 1 factorizations, moment problems and invariant subspaces, Studia Math. 167 (2005), 183–194.
B. Chevreau, Sur les contractions à calcul fonctionnel isométrique. II, J. Operator Theory 20 (1988), 269–293.
B. Chevreau, G. Exner and C. Pearcy, On the structure of contraction operators. Ill, Michigan Math. J. 36 (1989), 29–62.
G. Exner and C. Pearcy, Boundary sets for a contraction, J. Operator Theory 34 (1995), 347–380.
B. Chevreau and W.S. Li, On certain representations of H ∞ (G) and the reflexivity of associated operator algebras, J. Funct. Anal. 128 (1995), 341–373.
B. Chevreau and C. Pearcy, On the structure of contraction operators. I, J. Funct. Anal. 76 (1988), 1–29.
C. Pearcy, On Sheung’s theorem in the theory of dual operator algebras, Oper. Theory Adv. Appl. 28 (1988), 43–49.
I. Colojoară and C. Foias, Theory of generalized spectral operators, Gordon and Breach, New York, 1968.
J.B. Conway and M. Ptak, The harmonic functional calculus and hyperreflexivity, Pacific J. Math. 204 (2002), 19–29.
K.R. Davidson, The distance to the analytic Toeplitz operators, Illinois J. Math. 31 (1987), 265–273.
K.R. Davidson and R. Levene, 1-hyperreflexivity and complete hyperreflexivity, J. Funct. Anal. 235 (2006), 666–701.
K.R. Davidson and D.R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (1999), 401–430.
M. Didas, Invariant subspaces for commuting pairs with normal boundary dilation and dominating Taylor spectrum, J. Operator Theory 54 (2005), 169–187.
J. Eschmeier, Operators with rich invariant subspace lattices, J. Reine Angew. Math. 396 (1989), 41–69.
J. Eschmeier, Representations of H ∞ (G) and invariant subspaces, Math. Ann. 298 (1994), 167–186.
J. Eschmeier, Algebras of subnormal operators on the unit ball, J. Operator Theory 42 (1999), 37–76.
J. Eschmeier, On the reflexivity of multivariable isometries, Proc. Amer. Math. Soc. 134 (2006), 1783–1789
J. Eschmeier and B. Prunaru, Invariant subspaces for operators with Bishop’s property (β) and thick spectrum, J. Funct. Anal. 94 (1990), 196–222.
B. Prunaru, Invariant subspaces and localizable spectrum, Integral Equations Operator Theory 42 (2002), 461–471.
G. Exner, Y.S. Jo and LB. Jung, Representations of H∞(∞N), J. Operator Theory 45 (2001), 233–249.
D.W. Hadwin, Compressions, graphs, and hyperreflexivity, J. Funct. Anal. 145 (1997), 1–23.
P.R. Halmos, Normal dilations and extensions of operators, Summa Brasil. Math. 2 (1950). 125–134.
P.R. Halmos, Ten problems in Hilbert space, Bull. Amer. Math. Soc. 76 (1970), 887–933.
H. Hedenmalm, An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993), 1–9.
H. Hedenmalm, B. Korenblum and K. Zhu, Beurling type invariant subspaces of the Bergman spaces, J. London Math. Soc. (2) 53 (1996), 601–614.
H. Hedenmalm, S. Richter and K. Seip, Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math. 477 (1996), 13–30.
K. Horák, and V. Müller, On commuting isometries, Czechoslovak Math. J. 43(118) (1993), 373–382.
F. Jaëck and S.C. Power, Hyper-reflexÃvÃty of free semigroupoid algebras, Proc. Amer. Math. Soc. 134 (2006), 2027–2035.
B. Korenblum and K. Zhu, Complemented invariant subspaces in Bergman spaces, J. London Math. Soc. (2) 71 (2005), 467–480.
M. Kosiek and A. Octavio, On common invariant subspaces for commuting contractions with rich spectrum, Indiana Univ. Math. J. 53 (2004), 823–844.
W.S. Li, On polynomially bounded operators. I, Houston J. Math. 18 (1992), 73–96.
A.I. Loginov and V.S. Shulmán, Hereditary and intermediate reflexivity of W*-algebras, Akad. Nauk SSSR Ser. Mat. 39 (1975), 1260–1273.
M. Marsalli, Systems of equations in the predual of a von Neumann algebra, Proc. Amer. Math. Soc. 111 (1991), 517–522.
V. Müller and M. Ptak, Hyperreflexivity of finite-dimensional subspaces, J. Funct. Anal. 218 (2005), 395–408.
R.F. Olin and J.E. Thomson, Algebras of subnormal operators, J. Funct. Anal. 37 (1980), 271–301.
J.E. Thomson, Algebras generated by a subnormal operator, Trans. Amer. Math. Soc. 271 (1982), 299–311.
V. Pata, K.X. Zheng and A. Zucchi, On the reflexivity of operator algebras with isometric functional calculus, J. London Math. Soc. (2) 61 (2000), 604–616.
K.X. Zheng and A. Zucchi, Cellular-indecomposable subnormal operators. III, Integral Equations Operator Theory 29 (1997), 116–121.
G. Popescu, A generalization of Beurling’s theorem and a class of reflexive algebras, J. Operator Theory 41 (1999), 391–320.
B. Prunaru, A structure theorem for singly generated dual uniform algebras, Indiana Univ. Math. J. 43 (1994), 729–736.
B. Prunaru, Approximate factorization in generalized Hardy spaces, Integral Equations Operator Theory 61 (2008), 121–145.
M. Putinar, Hyponormal operators are subscalar, J. Operator Theory 12 (1984), 385–395.
G. F. Robel, On the structure of (BCP)-operators and related algebras, I, J. Operator Theory 12 (1984), 23–45.
D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math. 17 (1966), 511–517.
D. Sarason, Weak-star density of polynomials, J. Reine Angew. Math. 252 (1972), 1–15.
J. Sheung, On the preduals of certain operator algebras, Ph.D. Thesis, Univ. Hawaii, 1983.
I. Singer, Bases in Banach spaces. II. Editura Academiei, Bucharest, 1981.
B. Sz.-Nagy and C. Foias, Harmonic analysis of operators on Hilbert space, North-Holland, Amsterdam, 1970.
J.G. Stampfli, An extension of Scott Brown’s invariant subspace theorem: K-spectral sets, J. Operator Theory 3 (1980), 3–21.
P. Sullivan, Dilations and subnormal operators with rich spectrum, J. Operator Theory 29 (1993), 29–42.
J.E. Thomson, Invariant subspaces for algebras of subnormal operators, Proc. Amer. Math. Soc. 96 (1986), 462–464.
J.E. Thomson, Approximation in the mean by polynomials, Ann. of Math. (2) 133 (1991), 477–507.
J. Wermer, On invariant subspaces of normal operators, Proc. Amer. Math. Soc. 3 (1952), 270–277.
D.J. Westwood, On C oo -contractions with dominating spectrum, J. Funct. Anal. 66 (1986), 96–104.
D.J. Westwood, Weak operator and weak* topologies on singly generated algebras, J. Operator Theory 15 (1986), 267–280.
O. Yavuz, Invariant subspaces for Banach space operators with a multiply connected spectrum, Integral Equations Operator Theory 58 (2007), 433–446.
C. Zenger, On convexity properties of the Bauer field of values of a matrix, Numer. Math. 12 (1968), 96–105.
A. Zygmund, Trigonometric series, 2nd ed., Cambridge University Press, Cambridge, 1968.
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In memory of Paul Halmos, who knew how to ask the right questions
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Bercovici, H. (2010). Dual Algebras and Invariant Subspaces. In: Axler, S., Rosenthal, P., Sarason, D. (eds) A Glimpse at Hilbert Space Operators. Operator Theory Advances and Applications, vol 207. Springer, Basel. https://doi.org/10.1007/978-3-0346-0347-8_10
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DOI: https://doi.org/10.1007/978-3-0346-0347-8_10
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