Abstract
We define and study Toeplitz operators in the space of Herglotz solutions of the Helmholtz equation in \({\mathbb R}^d\). Since the most traditional definition of Toeplitz operators via Bergman-type projection is not available here, we use the approach based upon the reproducing kernel nature of the Herglotz space and sesquilinear forms, which results in a meaningful theory. For two important patterns of sesquilinear forms we discuss a number of properties, including the uniqueness of determining the symbols from the operator, the finite rank property, the conditions for boundedness and compactness, spectral properties, certain algebraic relations.
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Agmon, S.: A representation theorem for solutions of the Helmholtz equation and resolvent estimates for the Laplacian. Analysis, et cetera. Research Papers Published in Honor of Jurgen Moser’s 60th Birthday, Academic Press, pp. 39–76 (1990)
Agmon, S.: Representation theorems for solutions of the Helmholtz equation on \({\mathbb{R}}^n\). Differential operators and spectral theory, Amer. Math. Soc. Transl. Ser. 2, 189, Amer. Math. Soc., Providence, RI, pp. 27-43 (1999)
Agmon, S., Hörmander, L.: Asymptotic properties of solutions of differential equations with simple characteristics. J. Analyse Math. 30, 1–38 (1976)
Alexandrov, A., Rozenblum, G.: Finite rank Toeplitz operators: some extensions of D. Luecking’s theorem. J. Funct. Anal. 256, 2291–2303 (2009)
Alvarez, J., Folch-Gabayet, M., Pérez-Esteva, S.: Banach spaces of solutions of the Helmholtz equation in the plane. J. Fourier Anal. Appl. 7(1), 49–62 (2001)
Atkinson, K., Han, W.: Spherical Harmonics and Approximations on the Unit Sphere: An Introduction. Lecture Notes in Mathematics, vol. 2044. Springer-Verlag, Berlin Heidelberg (2012)
Barcelo, J., Folch-Gabayet, M., Pérez-Esteva, S., Ruiz, A.: Toeplitz operators on Herglotz wave functions. J. Math. Anal. Appl. 358, 364–379 (2009)
Bauer, W., Herrera Yañez, C., Vasilevski, N.: Eigenvalue characterization of radial operators on weighted Bergman spaces over the unit ball. Integr. Equ. Oper. Theory 78(2), 271–300 (2014)
Bauer, W., Le, T.: Algebraic properties and the finite rank problem for Toeplitz operators on the Segal-Bargmann space. J. Funct. Anal. 261(9), 2617–2640 (2011)
Colton, D., Kress, R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd Edition, Springer. Appl. Math. Sci. 93 (1998)
Gelfand, I.M., Shilov, G.E.: Generalized functions, II. Spaces of fundamental and generalized functions. Academic Press, Cambridge (1968)
Gradshtein, I.S., Ryzhik, I.M.: Tables of integrals, series and products. Acad. Press, Cambridge (2007)
Grudsky, S., Vasilevski, N.: Toeplitz operators on the Fock space: radial component effect. Integr. Equ. Oper. Theory 44(1), 10–37 (2002)
Guo, K.: A uniform \(L^p\) estimate of Bessel functions and distributions supported on \(S^{n-1}\). Proc. AMS 125(5), 1329–1340 (1997)
Helgason, S.: Topics in harmonic analysis on homogeneous spaces. Birkhäuser, Boston (1981)
Koosis, P.: The Logarithmic Integral. I. Cambridge University Press, Cambridge (1988)
Lorentz, G., von Golitschek, M., Makovoz, Y.: Constructive approximation. Advanced problems. Grundlehren d. Mathematischen Wissenschaften. Springer, New York (1996)
Luecking, D.: Finite rank Toeplitz operators on the Bergman space. Proc. Am. Math. Soc. 136(5), 1717–1723 (2008)
Maz’ya, V., Verbitsky, I.: The Schrodinger operator on the energy space: boundedness and compactness criteria. Acta Math. 188(2), 263–302 (2002)
Paley, R., Wiener, N.: Fourier transforms in the complex domain. AMS (1934)
Pérez-Esteva, S., Valenzuela-Diaz, S.: Reproducing kernel for the Herglotz functions in \({\mathbb{R}}^n\) and solutions of the Helmholtz equation. To appear
Rodberg, L., Thaler, R.: Introduction to the quantum theory of scattering. Academic Press, Cambridge (1965)
Rozenblum, G., Shirokov, N.: Some weighted estimates for the \(\bar{\partial }\)-equation and a finite rank theorem for Toeplitz operators in the Fock space. Proc. Lond. Math. Soc. (3) 109(5), 1281–1303 (2014)
Rozenblum, G.: Finite rank Toeplitz operators in Bergman spaces, In: Around the Research of Vladimir Maz’ya. III: Analysis and Applications.”, Springer, pp. 331–358 (2010) arXiv:0904.0171
Rozenblum, G., Vasilevski, N.: Toeplitz operators defined by sesquilinear forms: Fock space case. J. Funct. Anal. 267(11), 4399–4430 (2014)
Rozenblum, G., Vasilevski, N.: Toeplitz operators defined by sesquilinear forms: Bergman space case. J. Math. Sci. (Springer) 213(4), 582–609 (2016)
Taskinen, J., Virtanen, J.: Toeplitz operators on Bergman spaces with locally integrable symbols. Rev. Mat. Iberoam. 26(2), 693–706 (2010)
Vasilevski, N.: Commutative algebras of Toeplitz operators on the Bergman space. Birkhäuser (2008)
Yafaev, D.: Scattering theory; some old and new problems. Lecture Notes in Math., 1735, Springer (2000)
Zorboska, N.: Toeplitz operators with BMO symbols and the Berezin transform. Int. J. Math. Sci. 46, 2929–2945 (2003)
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Grigori Rozenblum thanks CINVESTAV, Mexico for hospitality and financial support during an essential part of the work with the paper.
Nikolai Vasilevski has been partially supported by CONACYT Project 238630, México.
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Rozenblum, G., Vasilevski, N. Toeplitz Operators in the Herglotz Space. Integr. Equ. Oper. Theory 86, 409–438 (2016). https://doi.org/10.1007/s00020-016-2331-0
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DOI: https://doi.org/10.1007/s00020-016-2331-0