Let \(\big \{A_{n}\big \}_{n\ge 1}\) be a sequence of bounded linear operators on a complex Hilbert space \({\mathcal {H}}.\) Then a bounded operator B on a Hilbert space \({\mathcal {K}} \supseteq {\mathcal {H}}\) is said to be a dilation of this sequence if \(\begin{aligned} A_{n} = P_{{\mathcal {H}}}B^{n}|_{{\mathcal {H}}} \quad \text {for all}\ n\ge 1, \end{aligned}\) where \(P_{{\mathcal {H}}}\) is the projection of \({\mathcal {K}}\) onto \({\mathcal {H}}.\) The question of the existence of dilation is a generalization of the classical moment problem. We obtain the necessary and sufficient conditions for the existence of self-adjoint dilation. Then we provide explicit block operator representations for these dilations. For example, the self-adjoint dilations can be put in block tri-diagonal form. Given a positive invertible operator A, an operator T is said to be in the \({\mathcal {C}}_{A}\) -class if the sequence \(\{ A^{-\frac{1}{2}}T^nA^{-\frac{1}{2}}\}_{n\ge 1}\) admits a unitary dilation. We identify a tractable collection of \({\mathcal {C}}_A\) -class operators for which isometric and unitary dilations can be written explicitly in block operator form. This includes the well-known \({\mathcal {C}}_{\rho }\) -class for positive scalars \(\rho .\) Here the special cases \(\rho =1\) and \(\rho =2\) correspond to Schäffer representation for contractions and Ando’s representation for operators with a numerical radius not more than one respectively.
