Invariant subspaces, reflexivity, hyperreflexivity

Marek Ptak

Abstrakt


The problem of existence of a nontrivial invariant subspace for an arbitrary
operator in a complex Hilbert space is one of the most challenging unsolved
problem in the whole operator theory. The problem and partial results are
presented. The notions of the reflexivity and the hyperreflexivity are closely
related to the main one.

Bibliografia


C. Ambrozie, V. Müller, "Invariant subspaces for polynomially bounded operators", J. Funct. Anal., 213(2004), 321–345, MR 2078629, Zbl 1056.47006.

N. Aronszajn, K. T. Smith, "Invariant subspaces of completely continuous opertors", Ann. Math., 60(1954), 345–350, MR 65807, Zbl 0056.11302.

W. T. Arveson, "Interpolation problems in nest algebras", J. Funct. Anal. 20 (1975), 208–233, MR 383098, Zbl 0309.46053.

E. A. Azoff, "On finite rank operators and preannihilators", Mem. Amer. Math. Soc., No. 357, 1986, MR 858467, Zbl 0606.47042.

E. A. Azoff, W. S. Li, M. Mbektha, M. Ptak, "On Consistent Operators and Reflexivity", Integr. Equ. Oper. Theory 71 (2011), 1–12, MR 2822424, Zbl 1231.47070.

A. Beurling, "On two problems concerning linear transformations in Hilbert space", Acta Math., 81(1949), 239–255, MR 27954, Zbl 0033.37701.

S. Brown, "Some invariant subspaces for subnormal operators", Integr. Equat. Oper. Th., 1(1978), 310–333, MR 511974, Zbl 0416.47009.

S. Brown, B. Chevreau, C. Pearcy, "On the structure of contraction operators", II, J. Funct. Anal., 76(1988), 30–55, MR 923043, Zbl 0641.47013.

J. B. Conway, "A Course in Functional Analysis", Springer–Verlag, New York, 1990, MR 1070713, Zbl 0706.46003.

J. B. Conway, "A Course in Operator Theory", American Mathematical Society, Providence, 2000, MR 1721402, Zbl 0936.47001.

K. Davidson, "The distance the analytic Toeplitz operators", Illinois J. Math. 31 (1987), 265–273, MR 882114, Zbl 0599.47034.

J. A. Deddens, P. A. Fillmore, "Reflexive linear transformations", Lin. Alg. Appl. 10 (1975), 89–93, MR 358390, Zbl 0301.15011.

P. L. Duren, "Hp spaces, Academic Press", New York, 1970, MR 268655, Zbl 0215.20203.

P. Enflo, "On invariant subspace problem in Banach spaces", Acta Math., 158(1987), 213–313, MR 892591, Zbl 0663.47003.

E. Helly, "Über Mengen Konvexer Körper mit gemeinschaftlichen Punkten", Jahressbericht der Deutschen Mathematiker–Vereinigung, 32(1923), 175–176, JFM 49.0534.02.

R. Horn, Ch. Jonhson, "Matrix analysis", Cambridge University Press, 1989, pp. 656, Zbl 0734.15002.

K. Klis, M. Ptak, "Quasinormal operators are hyperreflexive", Banach Center Publications, 67 (2005), 241–244, MR 2143929, Zbl 1068.47087.

J. Kraus, D. Larson, "Reflexivity and distance formulae", Proc. London Math. Soc. 53 (1986), 340–356, MR 850224, Zbl 0623.47046.

V. L. Lomonosov, "Invariant subspaces for operators commuting with compact operator", Funct. Anal. Appl., 115(1973), 213–214.

W. Mlak, "Wstep do Teorii Przestrzeni Hilberta", PWN, Biblioteka Matematyczna t. 35, Warszawa, 1970.

V. Müller, M. Ptak, "Hyperreflexivity of finite-dimenssional subspaces", J. Funct. Anal. 218 (2005), 395–408, MR 2108117, Zbl 1074.47032.

K. Piwowarczyk, M. Ptak, "On the hyperreflexivity of power partial isometries", Linear Algebra Appl., 437 (2012), 623–629, MR 2921722, Zbl 1315.47080.

H. Radjavi, P. Rosenthal, "Invariant Subspaces", Springer, Berlin, 1973, MR 367682, Zbl 0269.47003.

C. J. Read, "A solution to the invariant subspace problem", Bull. London Math. Soc., 16(1984), 337–401, MR 749447, Zbl 0566.47003.

C. J. Read, "A solution to the invariant subspace problem on the space" `1, Bull. London Math. Soc., 17(1985), 305–317, MR 806634, Zbl 0574.47006.

D. Sarason, "Invariant subspaces and unstarred operator algebras", Pacific J. Math 17 (1966), 511–517, MR 192365, Zbl 0171.33703.

M. Unser, P. D. Tafti, "An Introduction to Sparse Stochastic Processes", Cambridge University Press, 2014, MR 3495257, Zbl 1329.60002.

A. D.Wentzell, "Wykłady z teorii procesów stochastycznych", PWNWarszawa, 1980.


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