On square roots of self-adjoint weighted composition operators on H2
In this talk, we characterize square roots of self-adjoint weighted composition operators on the Hardy space H2, i.e., Wg, ψ such that Wg, ψ =Wf, Φ2 is self-adjoint. In particular, we find symbol functions f and Φ in this case. Some of Wf, Φ may be other, nonself-adjoint weighted composition operators. We also investigate several properties of such Wf, Φ. Finally, we give equivalent conditions for such Wf, Φ to be self-adjoint or normal, respectively.
- P. Bourdon and S. K. Narayan, Normal weighted composition operators on the Hardy space H2(D), J. Math. Anal. Appl. 367 (2010), 278-286.
- C. C. Cowen, The commutants of an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), 1-31.
- C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator, J. Functional Analysis 36(2)(1980), 169-184.
- C. C. Cowen, Composition operators on H2, J. Operator Theory 9(1983), 77-106.
- C. C. Cowen, Linear fractional composition operators on H2, Int. Eq. Op. Th. 11(1988), 151-160
- C. C. Cowen, S. Jung, and E. Ko, Normal and cohyponormal weighted composition operators on H2, Operator Theory: Adv. and Appl. 240(2014), 69-85.
- C. C. Cowen and E. Ko, Hermitian weighted composition operators on H2, Trans. Amer. Math. Soc. 362(2010), 5771-5801.
- C. C. Cowen and B. D. MacCluer, Composition operators on spaces of analytic Functions, CRC Press, Boca Raton, 1995.
- J. A. Deddens, Analytic Toeplitz and composition operators, Canad. J. Math. 24(1972), 859-865.
- P. Duren and A. Schuster, Bergman Spaces, Amer. Math. Soc., Providence, 2004.
- F. Forelli, The isometries of Hp, Canad. J. Math. 16(1964), 721-728.
- R. E. Greene and S. G. Krantz, Function theory of one complex variable, Grad. Studies 40, Amer. Math. Soc., 2002.
- H. Hedenmalm, B. Korenblum, and K. Zhu, Theory of Bergman Spaces, Springer-Verlag, New York, 2000.
- S. Jung, Y. Kim, and E. Ko, Characterizations of square roots of self-adjoint weighted composition operators on H2, preprint.
- H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, 1973.
This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2016R1D1A1B03931937).