Uniqueness of commuting compact approximations (1975)

Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an infinite dimensional complex Hilbert space, and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {B}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (resp. <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) be the algebra of all bounded (resp. compact) linear operators on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It is well known that every <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T element-of script upper B left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T \in \mathcal {B}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has a best approximation from the subspace <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper C left-parenthesis upper H right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">C</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>H</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {C}(H)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The purpose of this paper is to study the uniqueness problem concerning the best approximation of a bounded linear operator by compact operators. Our criterion for selecting a unique representative from the set of best approximants is that the representative should commute with <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>. In particular, many familiar operators are shown to have zero as a unique commuting best approximant.