Inner Products on the Hilbert Space $$S_2$$ of Hilbert–Schmidt Operators

Abstract This work presents a rigorous characterization of continuous inner products on the Hilbert space $$S_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> of Hilbert–Schmidt operators. We begin by addressing the general framework of continuous sesquilinear forms on a Hilbert space $$\mathcal {H}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> and provide a characterization of all continuous inner products, by means of positive operators in $$\mathcal {B(H)}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:mi>H</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Next, we establish necessary and sufficient conditions for an operator in $$\mathcal {B}(S_2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>B</mml:mi> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> to be positive. Identifying a continuous inner product with a positive operator enables us to rigorously describe inner products on $$S_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> .