Joint numerical ranges of operators in semi-Hilbertian spaces

In this paper we aim to investigate the concept of numerical range and maximal numerical range relative to a positive operator of a d-tuple of bounded linear operators on a Hilbert space. Some properties and applications of these sets are studied. Mainly, it is proved that they are convex for d=1, this generalizes the well known Toeplitz-Hausdorff Theorem [24], [16] and Stampfli's result [23]. Moreover, under additional hypotheses, we show that these sets are convex for d≥2.