We consider an abstract symmetric Fock space and the Hilbert space generated symmetric polynomials on the space of absolutely summing sequences. In particular, the properties of the symmetric shift operator, such as linearity and unboundedness are investigated. Also, we find eigenvectors of this operator.  We construct an operator in the symmetric Fock space, which is similar to the symmetric shift operator.  In the paper we note that this operator is not hypercyclic.

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