$$H^{\infty }$$ calculus for submarkovian semigroups on weighted $$L^2$$ spaces

Let \((T_t)_{t \geqslant 0}\) be a markovian (resp. submarkovian) semigroup on some \(\sigma \)-finite measure space \((\Omega ,\mu )\). We prove that its negative generator A has bounded \(H^{\infty }(\Sigma _\theta )\) calculus the weighted \(L^2(\Omega ,wd\mu as long weight \(w : \Omega \rightarrow (0,\infty finite characteristic defined by \(Q^A_2(w) = \sup _{t > 0} \left\| T_t(w) T_t \left( w^{-1} \right) \right\| _{L^\infty (\Omega )}\) variant for submarkovian semigroups). Some additional technical conditions have to imposed and their validity in examples is discussed. Any angle \(\theta \frac{\pi }{2}\) admissible above }\) calculus, semigroups also certain \theta _w < depending size of \(Q^A_2(w)\). The norm linear \(Q^A_2\) }{2}\). discuss results angles Namely we show there probability w without Hörmander functional ,w d\mu