Dilation and Birkhoff-James orthogonality

We study the interaction between unitary $\rho$-dilations of a pair of Hilbert space operators and Birkhoff-James orthogonality. We prove that for two orthogonal operators $T,A$ if $\mathrm{\parallel }T\mathrm{\parallel }=\rho$, then $U_T{\perp }_B{U}_A$ for any unitary $\rho$-dilations $U_T$ of $T$ and $U_A$ of $A$ acting on a common space. We characterize $\epsilon$-approximate Birkhoff-James orthogonality for complex Hilbert space operators. Then find a sharp bound on $\epsilon$ such that $T{\perp }_{B}A$ implies that $U_T{\perp }_B^{\epsilon }{U}_A$ for any unitary $\rho$-dilations $U_T,U_A$ of $T$ and $A$ respectively. The Schäffer unitary dilations of a pair of contractions $T,A$ are not orthogonal in general. We construct Schäffer-type unitary dilations for contractions which are pairwise orthogonal.