A class of complex symmetric Toeplitz operators on Hardy and Bergman spaces

An operator T∈B(H) is complex symmetric if there exists a conjugation C on H so that CTC=T⁎. In this paper, we characterize the complex symmetric Toeplitz operator on the Hardy and Bergman space. In particular, we show that the Toeplitz operator induced by the Berezin transform of a complex symmetric operator on Hardy space is also complex symmetric with the same conjugation. However, we see that this is not true for Bergman space by providing some examples.