A note on some operator theory in certain semi-inner-product spaces.

Abstract

Let <italic>X</italic> be a smooth uniformly convex Banach space and let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket dot comma dot right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo>,</mml:mo> <mml:mo>⋅<!-- ⋅ --></mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[\cdot ,\cdot ]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the unique semi-inner-product generating the norm of <italic>X</italic>. If <italic>A</italic> is a bounded linear operator on <italic>X</italic>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript dagger"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†<!-- † --></mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">{A^\dagger }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> mapping <italic>X</italic> to <italic>X</italic> is called the generalized adjoint of <italic>A</italic> if and only if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-bracket upper A left-parenthesis x right-parenthesis comma y right-bracket equals left-bracket x comma upper A Superscript dagger Baseline left-parenthesis y right-parenthesis right-bracket"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">]</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†<!-- † --></mml:mo> </mml:msup> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">]</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">[A(x),y] = [x,{A^\dagger }(y)]</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for all <italic>x</italic> and <italic>y</italic> in <italic>X</italic>. In this setting adjoint abelian (iso abelian) operators [<bold>5</bold>] are characterized as those operators <italic>A</italic> for which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper A Superscript dagger Baseline equals upper A left-parenthesis upper A Superscript dagger Baseline equals upper A Superscript negative 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†<!-- † --></mml:mo> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mo>†<!-- † --></mml:mo> </mml:msup> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>A</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{A^\dagger } = A({A^\dagger } = {A^{ - 1}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, i.e. the invertible isometries). It is also shown that the compression spectrum of an operator is contained in its numerical range.