In this paper we study a possibility of a decomposition of a bounded operator in a Hilbert space $H$ as a product of a J-unitary and a J-self-adjoint operators, where J is a conjugation (an antilinear involution). This decomposition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompositions in the finite-dimensional case are also obtained. Matrix representations for J-symmetric and J-skew-symmetric operators are studied. Simple basic properties of J-symmetric, J-skew-symmetric and J-isometric operators are obtained.