Factorization of matrices into partial isometries (1989)

In this paper, we characterize complex square matrices which are expressible as products of partial isometries and orthogonal projections. More precisely, we show that a matrix <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> is the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> partial isometries <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis k greater-than-or-equal-to 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>k</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(k \geq 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if and only if <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> is a contraction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-vertical-bar upper T double-vertical-bar less-than-or-equal-to 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>T</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\left \| T \right \| \leq 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 minus upper T Superscript asterisk Baseline upper T right-parenthesis less-than-or-equal-to k dot"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>−<!-- − --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>T</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msup> </mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>≤<!-- ≤ --></mml:mo> <mml:mi>k</mml:mi> <mml:mo>⋅<!-- ⋅ --></mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1 - {T^*}T) \leq k \cdot</mml:annotation> </mml:semantics> </mml:math> </inline-formula> nullity <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>. It follows, as a corollary, that any <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n times n"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n \times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> singular contraction is the product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> partial isometries and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the smallest such number. On the other hand, <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> is the product of finitely many orthogonal projections if and only if <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> is unitarily equivalent to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 circled-plus upper S"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⊕<!-- ⊕ --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \oplus S</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a singular strict contraction <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis double-vertical-bar upper S double-vertical-bar greater-than 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mo symmetric="true">‖</mml:mo> <mml:mi>S</mml:mi> <mml:mo symmetric="true">‖</mml:mo> </mml:mrow> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(\left \| S \right \| &gt; 1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As contrasted to the previous case, the number of factors can be arbitrarily large.