Strong uniqueness for stochastic evolution equations with unbounded measurable drift term

We consider stochastic evolution equations in Hilbert spaces with merely measurable and locally bounded drift term and cylindrical Wiener noise. We prove pathwise (hence strong) uniqueness in the class of global solutions. This paper extends our previous paper (Da Prato et al. in Ann Probab 41:3306-3344, 2013) which generalized Veretennikov's fundamental result to infinite dimensions assuming boundedness of the drift term. As in Da Prato et al. (Ann Probab 41:3306-3344, 2013), pathwise uniqueness holds for a large class, but not for every initial condition. We also include an application of our result to prove existence of strong solutions when the drift is assumed only to be measurable and bounded and grow more than linearly.