Unbounded Antilinear Operators on Hilbert Spaces

Abstract

The paper introduces unbounded antilinear operators on Hilbert spaces and develops their fundamental theory. In particular, we establish a closed range theorem, a polar decomposition theorem, and the convexity of the numerical range for antilinear operators. Furthermore, we present several new results on antilinear normal operators and provide necessary and sufficient conditions for the existence of a minimal antilinear normal extension of an antilinear subnormal operator. We further develop a comprehensive characterization of antilinear block operator matrices with purely antilinear entries, establishing necessary and sufficient criteria for their closability through the framework of Schur and quadratic complements.