Berezin range and norm of Volterra-type integral operators on the Fock space

In this article, we investigate several properties of the Volterra-type integral operator V g Vg upper V Subscript g on the Fock space. We give an explicit description of its Berezin range, prove that this range is path connected, and establish several results concerning its convexity. We compute the operator norm, essential norm, Berezin norm, and Berezin number of V g Vg upper V Subscript g . We also derive an exact coefficient representation of its numerical range and obtain a spectral characterization of its numerical radius. In particular, we prove that the operator norm, Berezin norm, Berezin number, and numerical radius coincide if and only if the polynomial expansion of the symbol g contains no linear term. This extremal identity holds even though the operator is non-normal.