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We characterize wandering subspace property of the shift operator on zero-based invariant subspaces of reproducing kernel Hilbert spaces by using the kernels of Toeplitz operators. As applications, we prove that the wandering subspace property of the shift operator holds on the weighted Bergman space() for zero-based invariant subspaces when the zeros lie in a small disk centred at the origin.

Finite zero-based invariant subspaces of the shift operator on reproducing kernel spaces