# Article

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Keywords:
reflexive subspace; transitive subspace; Toeplitz operator; Hardy space; upper half-plane
Summary:
The reflexivity and transitivity of subspaces of Toeplitz operators on the Hardy space on the upper half-plane are investigated. The dichotomic behavior (transitive or reflexive) of these subspaces is shown. It refers to the similar dichotomic behavior for subspaces of Toeplitz operators on the Hardy space on the unit disc. The isomorphism between the Hardy spaces on the unit disc and the upper half-plane is used. To keep weak* homeomorphism between $L^\infty$ spaces on the unit circle and the real line we redefine the classical isomorphism between $L^1$ spaces.
References:
[1] Azoff, E. A., Ptak, M.: A dichotomy for linear spaces of Toeplitz operators. J. Funct. Anal. 156 411-428 (1998). DOI 10.1006/jfan.1998.3275 | MR 1636960 | Zbl 0922.47021
[2] Bercovici, H., Foiaş, C., Pearcy, C.: Dual Algebras with Applications to Invariant Subspaces and Dilation Theory. Reg. Conf. Ser. Math. 56, 1985. DOI 10.1090/cbms/056 | Zbl 0569.47007
[3] Brown, S., Chevreau, B., Pearcy, C.: Contractions with rich spectrum have invariant subspaces. J. Oper. Theory 1 (1979), 123-136. MR 0526294 | Zbl 0449.47003
[4] Conway, J. B.: A Course in Functional Analysis. 2nd ed. Graduate Texts in Mathematics, 96. Springer, New York (1990). MR 1070713 | Zbl 0706.46003
[5] Conway, J. B.: A Course in Operator Theory. Graduate Studies in Mathematics 21, American Mathematical Society, Providence (2000). MR 1721402 | Zbl 0936.47001
[6] Douglas, R. G.: Banach Algebra Techniques in Operator Theory. Pure and Applied Mathematics, 49. Academic Press, New York (1972). MR 0361893 | Zbl 0247.47001
[7] Duren, P. L.: Theory of $H^p$ Spaces. Pure and Applied Mathematics, 38. Academic Press, New York-London (1970). MR 0268655
[8] Hoffman, K.: Banach Spaces of Analytic Functions. Prentice-Hall Series in Modern Analysis, Englewood Cliffs, N.J., Prentice-Hall (1962). MR 0133008 | Zbl 0117.34001
[9] Koosis, P.: Introduction to $H_p$ Spaces. London Mathematical Society Lecture Note Series. 40. Cambridge University Press, Cambridge (1980). MR 0565451 | Zbl 0435.30001
[10] Nikolski, N. K.: Operators, Functions, and Systems: An Easy Reading. Volume I: Hardy, Hankel, and Toeplitz. Transl. from the French by Andreas Hartmann. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence (2002). MR 1864396 | Zbl 1007.47001
[11] Sarason, D.: Invariant subspaces and unstarred operator algebras. Pac. J. Math. 17 (1966), 511-517. DOI 10.2140/pjm.1966.17.511 | MR 0192365 | Zbl 0171.33703

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