Coupled-channel Omnès matrix for the D-wave isoscalar <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mi>π</mml:mi> <mml:mo linebreak="goodbreak">/</mml:mo> <mml:mi>K</mml:mi> <mml:mover a

In this work, we construct the D -wave isoscalar π π / K K ¯ coupled-channel Omnès matrix, formulated to satisfy unitarity, analyticity, and the appropriate asymptotic behavior. We employ a two-channel K -matrix model containing poles associated with the f 2 (1270) and f 2 ′ ( 1525 ) resonances. The resulting unitary scattering matrix, which reproduces the experimental ππ → ππ and π π → K K ¯ data and PDG information, serves as input to the homogeneous two-channel Muskhelishvili-Omnès equation. We compare our Omnès matrix with previous constructions based on π π → K K ¯ phases extracted from sums of Breit-Wigner amplitudes. The Omnès matrix developed here provides a reliable dispersive input for form-factor calculations and resonance studies in the tensor-meson sector. As an application, we show that it enables a simultaneous and accurate description of the BESIII J / ψ → π 0 π 0 γ and J / ψ → K S K S γ spectra in the J = 2 electric-dipole (E1) partial wave.