On the Symbols of Strictly <i>m</i>-Null Elementary Operators

On the Symbols of Strictly <i>m</i>-Null Elementary Operators

This paper extends the previous work by the author on <i>m</i>-null pairs of operators in Hilbert space. If an elementary operator <i>L</i> has elementary symbols <i>A</i> and <i>B</i> that are <i>p</i>-null and <i>q</i>-null, r...

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Bibliographic Details
Main Author:	Isabel Marrero
Format:	Article
Language:	English
Published:	MDPI AG 2025-06-01
Series:	Mathematics
Subjects:	elementary operator <i>m</i>-isometry <i>m</i>-null operators (<i>m</i>, T)-null operators operator arithmetic progression
Online Access:	https://www.mdpi.com/2227-7390/13/12/2026
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Summary:	This paper extends the previous work by the author on <i>m</i>-null pairs of operators in Hilbert space. If an elementary operator <i>L</i> has elementary symbols <i>A</i> and <i>B</i> that are <i>p</i>-null and <i>q</i>-null, respectively, then <i>L</i> is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-null. Here, we prove the converse under strictness conditions, modulo some nonzero multiplicative constant—if <i>L</i> is strictly <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>-null, then a scalar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula> exists such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>λ</mi><mi>A</mi></mrow></semantics></math></inline-formula> is strictly <i>p</i>-null and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>λ</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>B</mi></mrow></semantics></math></inline-formula> is strictly <i>q</i>-null. Our constructive argument relies essentially on algebraic and combinatorial methods. Thus, the result obtained by Gu on <i>m</i>-isometries is recovered without resorting to spectral analysis. For several operator classes that generalize <i>m</i>-isometries and are subsumed by <i>m</i>-null operators, the result is new.
ISSN:	2227-7390