Backward shift invariant subspaces in reproducing kernel Hilbert spaces
DOI:
https://doi.org/10.7146/math.scand.a-119120Abstract
In this note, we describe the backward shift invariant subspaces for an abstract class of reproducing kernel Hilbert spaces. Our main result is inspired by a result of Sarason concerning de Branges-Rovnyak spaces (the non-extreme case). Furthermore, we give new applications in the context of the range space of co-analytic Toeplitz operators and sub-Bergman spaces.
References
Aleksandrov, A. B., Invariant subspaces of the backward shift operator in the space $H^p$ $(p\in (0,\,1))$: Investigations on linear operators and the theory of functions, IX, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 92 (1979), 7–29, 318.
Aleman, A. and Malman, B., Hilbert spaces of analytic functions with a contractive backward shift, J. Funct. Anal. 277 (2019), no. 1, 157–199. https://doi.org/10.1016/j.jfa.2018.08.019
Aleman, A., Richter, S., and Sundberg, C., Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm, in “Bergman spaces and related topics in complex analysis”, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, 2006, pp. 1–25. https://doi.org/10.1090/conm/404/07631
Bolotnikov, V. and Rodman, L., Finite dimensional backward shift invariant subspaces of Arveson spaces, Linear Algebra Appl. 349 (2002), 265–282. https://doi.org/10.1016/S0024-3795(02)00251-3
Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. https://doi.org/10.2307/2035178
Douglas, R. G., Shapiro, H. S., and Shields, A. L., Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20, fasc. 1 (1970), 37–76.
Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
Fricain, E. and Mashreghi, J., The theory of $\mathcal H(b)$ spaces. Vol. $1$, New Mathematical Monographs, vol. 20, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781139226752
Fricain, E. and Mashreghi, J., The theory of $\mathcal H(b)$ spaces. Vol. $2$, New Mathematical Monographs, vol. 21, Cambridge University Press, Cambridge, 2016.
Girela, D. and González, C., Division by inner functions, in “Progress in analysis, Vol. I, II (Berlin, 2001)”, World Sci. Publ., River Edge, NJ, 2003, pp. 215–220.
Girela, D., González, C., and Peláez, J. Á., Multiplication and division by inner functions in the space of Bloch functions, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1309–1314. https://doi.org/10.1090/S0002-9939-05-08049-4
Girela, D., González, C., and Peláez, J. Á., Toeplitz operators and division by inner functions, in “Proceedings of the First Advanced Course in Operator Theory and Complex Analysis”, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 85–103.
Havin, V. P., The factorization of analytic functions that are smooth up to the boundary, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 202–205.
Izuchi, K. and Nakazi, T., Backward shift invariant subspaces in the bidisc, Hokkaido Math. J. 33 (2004), no. 1, 247–254. https://doi.org/10.14492/hokmj/1285766003
Koosis, P., Introduction to $H_p$ spaces, second ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998.
Lax, P. D., Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2002.
Mashreghi, J., Representation theorems in Hardy spaces, London Mathematical Society Student Texts, vol. 74, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511814525
Paulsen, V. I. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics, vol. 152, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316219232
Sarason, D., Doubly shift-invariant spaces in $H^2$, J. Operator Theory 16 (1986), no. 1, 75–97.
Sarason, D., Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994.
Shirokov, N. A., Analytic functions smooth up to the boundary, Lecture Notes in Mathematics, vol. 1312, Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/BFb0082810
Suárez, D., Backward shift invariant spaces in $H^2$, Indiana Univ. Math. J. 46 (1997), no. 2, 593–619. https://doi.org/10.1512/iumj.1997.46.877
Sultanic, S., Sub-Bergman Hilbert spaces, J. Math. Anal. Appl. 324 (2006), no. 1, 639–649. https://doi.org/10.1016/j.jmaa.2005.12.035
Sz.-Nagy, B., Foias, C., Bercovici, H., and Kérchy, L., Harmonic analysis of operators on Hilbert space, second ed., Universitext, Springer, New York, 2010. https://doi.org/10.1007/978-1-4419-6094-8
Zhu, K., Sub-Bergman Hilbert spaces on the unit disk, Indiana Univ. Math. J. 45 (1996), no. 1, 165–176. https://doi.org/10.1512/iumj.1996.45.1097
Zhu, K., Sub-Bergman Hilbert spaces in the unit disk. II, J. Funct. Anal. 202 (2003), no. 2, 327–341. https://doi.org/10.1016/S0022-1236(02)00086-1
Zhu, K., Operator theory in function spaces, second ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138
Aleman, A. and Malman, B., Hilbert spaces of analytic functions with a contractive backward shift, J. Funct. Anal. 277 (2019), no. 1, 157–199. https://doi.org/10.1016/j.jfa.2018.08.019
Aleman, A., Richter, S., and Sundberg, C., Invariant subspaces for the backward shift on Hilbert spaces of analytic functions with regular norm, in “Bergman spaces and related topics in complex analysis”, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, 2006, pp. 1–25. https://doi.org/10.1090/conm/404/07631
Bolotnikov, V. and Rodman, L., Finite dimensional backward shift invariant subspaces of Arveson spaces, Linear Algebra Appl. 349 (2002), 265–282. https://doi.org/10.1016/S0024-3795(02)00251-3
Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Amer. Math. Soc. 17 (1966), 413–415. https://doi.org/10.2307/2035178
Douglas, R. G., Shapiro, H. S., and Shields, A. L., Cyclic vectors and invariant subspaces for the backward shift operator, Ann. Inst. Fourier (Grenoble) 20, fasc. 1 (1970), 37–76.
Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970.
Fricain, E. and Mashreghi, J., The theory of $\mathcal H(b)$ spaces. Vol. $1$, New Mathematical Monographs, vol. 20, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781139226752
Fricain, E. and Mashreghi, J., The theory of $\mathcal H(b)$ spaces. Vol. $2$, New Mathematical Monographs, vol. 21, Cambridge University Press, Cambridge, 2016.
Girela, D. and González, C., Division by inner functions, in “Progress in analysis, Vol. I, II (Berlin, 2001)”, World Sci. Publ., River Edge, NJ, 2003, pp. 215–220.
Girela, D., González, C., and Peláez, J. Á., Multiplication and division by inner functions in the space of Bloch functions, Proc. Amer. Math. Soc. 134 (2006), no. 5, 1309–1314. https://doi.org/10.1090/S0002-9939-05-08049-4
Girela, D., González, C., and Peláez, J. Á., Toeplitz operators and division by inner functions, in “Proceedings of the First Advanced Course in Operator Theory and Complex Analysis”, Univ. Sevilla Secr. Publ., Seville, 2006, pp. 85–103.
Havin, V. P., The factorization of analytic functions that are smooth up to the boundary, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 22 (1971), 202–205.
Izuchi, K. and Nakazi, T., Backward shift invariant subspaces in the bidisc, Hokkaido Math. J. 33 (2004), no. 1, 247–254. https://doi.org/10.14492/hokmj/1285766003
Koosis, P., Introduction to $H_p$ spaces, second ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998.
Lax, P. D., Functional analysis, Pure and Applied Mathematics (New York), Wiley-Interscience, New York, 2002.
Mashreghi, J., Representation theorems in Hardy spaces, London Mathematical Society Student Texts, vol. 74, Cambridge University Press, Cambridge, 2009. https://doi.org/10.1017/CBO9780511814525
Paulsen, V. I. and Raghupathi, M., An introduction to the theory of reproducing kernel Hilbert spaces, Cambridge Studies in Advanced Mathematics, vol. 152, Cambridge University Press, Cambridge, 2016. https://doi.org/10.1017/CBO9781316219232
Sarason, D., Doubly shift-invariant spaces in $H^2$, J. Operator Theory 16 (1986), no. 1, 75–97.
Sarason, D., Sub-Hardy Hilbert spaces in the unit disk, University of Arkansas Lecture Notes in the Mathematical Sciences, vol. 10, John Wiley & Sons, Inc., New York, 1994.
Shirokov, N. A., Analytic functions smooth up to the boundary, Lecture Notes in Mathematics, vol. 1312, Springer-Verlag, Berlin, 1988. https://doi.org/10.1007/BFb0082810
Suárez, D., Backward shift invariant spaces in $H^2$, Indiana Univ. Math. J. 46 (1997), no. 2, 593–619. https://doi.org/10.1512/iumj.1997.46.877
Sultanic, S., Sub-Bergman Hilbert spaces, J. Math. Anal. Appl. 324 (2006), no. 1, 639–649. https://doi.org/10.1016/j.jmaa.2005.12.035
Sz.-Nagy, B., Foias, C., Bercovici, H., and Kérchy, L., Harmonic analysis of operators on Hilbert space, second ed., Universitext, Springer, New York, 2010. https://doi.org/10.1007/978-1-4419-6094-8
Zhu, K., Sub-Bergman Hilbert spaces on the unit disk, Indiana Univ. Math. J. 45 (1996), no. 1, 165–176. https://doi.org/10.1512/iumj.1996.45.1097
Zhu, K., Sub-Bergman Hilbert spaces in the unit disk. II, J. Funct. Anal. 202 (2003), no. 2, 327–341. https://doi.org/10.1016/S0022-1236(02)00086-1
Zhu, K., Operator theory in function spaces, second ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138
Downloads
Published
2020-03-29
How to Cite
Fricain, E., Mashreghi, J., & Rupam, R. (2020). Backward shift invariant subspaces in reproducing kernel Hilbert spaces. MATHEMATICA SCANDINAVICA, 126(1), 142–160. https://doi.org/10.7146/math.scand.a-119120
Issue
Section
Articles
