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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 decompo- sition shows an inner structure of a bounded operator in a Hilbert space. Some decompositions of J-unitary and unitary operators which generalize decompo- sitions 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.

On a J-polar decomposition of a bounded operator and matrices of J-symmetric and J-skew-symmetric operators