Generalized Halmos conjectures and constrained unitary dilations

In this paper, we investigate a problem of Halmos on various generalizations of the numerical range. We generalize a finite-dimensional result of Gau, Li and Wu, by showing that for k ∈ N , the closure of the rank- k numerical range of a contraction T acting on a separable Hilbert space H is the intersection of the closure of the rank- k numerical ranges of all unitary dilations of T to H ⊕ H . The same is true for k = ∞ provided the rank- ∞ numerical range of T is non-empty. We also show that when both defect indices of a contraction are equal and finite ( = N ), one may restrict the intersection to a smaller family consisting of all unitary N -dilations. We also investigate this problem in the matricial range and the C -numerical range. We obtain a few interesting results and conclude the answers in negative.