Mixed Bram-Halmos and Agler-Embry conditions

The subnormality of a Hilbert space operator may be characterized either by the Bram-Halmos conditions (positivity of certain operator matrices) or the Agler-Embry conditions (positivity of certain operator differences). We define and consider mixed conditions involving matrices of operator differences, thus yielding conditions whose extremes are the Bram-Halmos and Agler-Embry conditions. We study these conditions for weighted shifts, showing that they reduce to matrices of differences of the moments of the shifts, and examine these conditions under the perturbation of a single weight of a subnormal shift.