An extension and refinement of the theorems of Douglas and Sebestyén for unbounded operators

For a closed densely defined operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> from a Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper H"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">H</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{\mathfrak {H}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to a Hilbert space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , necessary and sufficient conditions are established for the factorization of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with a bounded nonnegative operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X"> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding="application/x-tex">X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German upper K"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {K}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . This result yields a new extension and a refinement of a well-known theorem of R.G. Douglas [Proc. Amer. Math. Soc. 17 (1966), pp. 413–415], which shows that the operator inequality <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript asterisk Baseline upper A less-than-or-equal-to lamda squared upper B Superscript asterisk Baseline upper B comma lamda greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>A</mml:mi> <mml:mo> ≤ </mml:mo> <mml:msup> <mml:mi> λ </mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>B</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>B</mml:mi> <mml:mo>,</mml:mo> <mml:mi> λ </mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">A^*A\leq \lambda ^2 B^*B, \lambda \geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , is equivalent to the factorization <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A equals upper C upper B"> <mml:semantics> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:mi>B</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">A=CB</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar upper C double-vertical-bar less-than-or-equal-to lamda"> <mml:semantics> <mml:mrow> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mi>C</mml:mi> <mml:mo fence="false" stretchy="false"> ‖ </mml:mo> <mml:mo> ≤ </mml:mo> <mml:mi> λ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">\|C\|\leq \lambda</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The main results give necessary and sufficient conditions for the existence of an intermediate selfadjoint operator <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H greater-than-or-equal-to 0"> <mml:semantics> <mml:mrow> <mml:mi>H</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">H\geq 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript asterisk Baseline upper A less-than-or-equal-to lamda upper H less-than-or-equal-to lamda squared upper B Superscript asterisk Baseline upper B"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mo> ∗ </mml:mo> </mml:msup> <mml:mi>A</mml:mi> <mml:mo> ≤ </mml:mo> <mml:mi> λ </mml:mi> <mml:mi>H</mml:mi> <mml:mo> ≤ </mml:mo> <mml:msup>