Abstract
The basic theory of Toeplitz and Hankel operators acting on the Paley-Weiner space is developed. This includes criteria for boundedness, compactness, being of finite rank, and membership in the Schatten-von Neumann ideals. Similar questions are considered for the related operators formed by commuting the discrete Hilbert transform with a multiplication operator.
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[AR] J. M. Anderson and R. Rochberg, Toeplitz operators associated with subalgebras of the disk algebra, Indiana U. Math. J. 30 (1981), 813–820.
[BC] C. A. Berger and L. A. Coburn, Toeplitz operators and quantum mechanics, to appear, J. Funct. Anal.
[B] R. Boas, Entire Functions, Academic Press, New York, 1954.
[Bu] J. Burbea, Trace ideal criteria for Hankel operators on the ball of ℂn, preprint, 1986.
[CR] R. R. Coifman and R. Rochberg, Representation theorems for holomorphic and harmonic functions in Lp, Asterisque, 1980.
[CW] R. R. Coifman and R. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.
[Do] J. R. Dorronsoro, Mean oscillation and Besov spaces, Canad. Math. Bull. 28 (1985), 474–488.
[D] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, New York and London, 1972.
[F] M. Feldman.
[FR] R. Frankfurt and J. Rovnyak, Recent results and unsolved problems on finite convolution operators, Linear Spaces and Approximation, P. L. Butzer and B. Sz.-Nagy eds, Birkhauser Verlag, Basel, Stuttgart, 1978.
[G] V. Guilliman, Toeplitz operators in n-dimensions, Integral Equations and Operator Theory 7 (1984), 145–205.
[H] J. R. Higgins, Five short stories about the cardinal series, Bull. Amer. Math. Soc. 12 (1985), 45–90.
[J] S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat. 16 (1978), 263–270.
[JPR] S. Janson, J. Peetre, and R. Rochberg, Hankel forms on the Fock space, preprint, 1986.
[JPS] S. Janson, J. Peetre, and S. Semmes, On the action of Hankel and Toeplitz operators on some function spaces, Duke Math. J 51 (1984), 937–958.
[L] D. Luecking, Trace ideal criteria for Toeplitz operators, preprint, 1985.
[McD S] G. McDonald and C. Sundberg, Toeplitz operators on the disc, Indiana U. Math. J. 28 (1979), 595–611.
[N] N. K. Nikol'skii, Two applications of Hankel operators, Operator Theory: Advances and Applications, Vol. 14, Birkhauser Verlag, Basel, 1981, 239–246.
[Pe 1] J. Peetre, New thoughts on Besov Spaces, Duke University Press, Durham, 1976.
[Pe 2] J. Peetre, Invariant function spaces connected with the holomorphic discrete series, Aniversary Volume on Approximation Theory and Functional Analysis, Oberwohlfach 1983, pp 119–134, Birkhauser.
[Pel 1] V. V. Peller, Vectorial Hankel operators, commutators and related operators of the Schatten-Von Neumann class γp Integral Equations and Operator Theory, 5 (1982), 244–272.
[Pel 2] —, Nuclear Hankel operators acting between HP spaces, Operator Theory: Advances and Applications, Vol. 14, Birkhauser Verlag, Basel, 1984 213–220.
[Pel 3] V. V. Peller, Wiener-Hopf operators on a finite interval and Schatten-von Neumann classes, Uppsala Univ. Dept of Math. Report 1986: 9
[PH] V. V. Peller and S. V. Hruschev, Hankel operators, best approximation, and stationary Gaussian processes I, II, III, Russian Math Surveys 37, (1982), 61–144.
[Po 1] S. Power, Hankel operators on Hilbert space, Bull. Lond. Math. Soc., 12, (1980), 422–442.
[Po 2] —, Hankel operators on Hilbert space, Pitmann Books LTD., London, 1982.
[RT] F. Ricci and M. Taibleson, Boundary values of harmonic functions in mixed norm spaces and their atomic structure, Ann Sc. Norm. Super. Pisa, Cl. Sci., IV, Ser. 10 (1983) 1–54.
[R 1] R. Rochberg, Decomposition theorems for Bergman spaces and their applications, Operators and Function Theory, S. C. Power ed., Reidel, Dordrecth, 1985, 225–278.
[R 2] —, Trace ideal criteria for Hankel operators and commutators, Ind. U. Math. J. 31 (1982), 913–925.
[RS] R Rochberg and S. Semmes, A decomposition theorem for BMO and applications, J. Funct. Anal., 67 (1986), 228–263.
[S] D. Sarason, Generalized interpolation in H∞, Trans. Amer. Math. Soc. 127 (1967), 179–203.
[T] H. Triebel, Theory of Function Spaces, Birkhauser Verlag, Boston, 1983.
[Un] A. Unterberger, Symbolic calculi and the duality of homogeneous spaces, Contemporary Mathematics 27 (1984) 237–252, Amer. Math. Soc., Providence RI.
[Up] H. Upmeier, Topeplitz C*-algebras on bounded symmetric domains, Ann. Math 119, (1984), 549–576.
[W] H. Widom, Asymptotic behavior of the eigenvalues of certain integral equations. II, Arch. Rat. Mech. Anal. 17 (1964), 215–229.
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Supported in part by a grant from the National Science Foundation.
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Rochberg, R. Toeplitz and Hankel operators on the Paley-Wiener space. Integr equ oper theory 10, 187–235 (1987). https://doi.org/10.1007/BF01199078
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DOI: https://doi.org/10.1007/BF01199078