Antilinear Normal Operators on Hilbert Space

Abstract An operator A on a complex Hilbert space $${\mathcal {H}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> is called antilinear if $$A(x+y)=Ax+Ay$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> </mml:math> and $$A(\lambda x)=\overline{\lambda } Ax$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mover> <mml:mi>λ</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> <mml:mi>A</mml:mi> <mml:mi>x</mml:mi> </mml:mrow> </mml:math> for $$x,y\in {\mathcal {D}}(A)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$\lambda \in \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> . We investigate some classes of densely defined antilinear unbounded operators, especially antilinear normal operators. We give various characterizations of antilinear normal operators and study a class of such operators in detail. Our main result is a structure theorem for unbounded antilinear normal operators.