<i>m</i>-Isometric Operators with Null Symbol and Elementary Operator Entries

<i>m</i>-Isometric Operators with Null Symbol and Elementary Operator Entries

A pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false"&g...

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Bibliographic Details
Main Author:	Bhagwati Prashad Duggal
Format:	Article
Language:	English
Published:	MDPI AG 2025-06-01
Series:	Axioms
Subjects:	Banach space m-isometric left/right multiplication operator null symbol strict m-isometric operator
Online Access:	https://www.mdpi.com/2075-1680/14/7/503
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Summary:	A pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> of Banach space operators is strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-isometric for a Banach space operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi mathvariant="script">X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and a positive integer <i>m</i> if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>▵</mo><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><mi>m</mi></msubsup><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mfenced separators="" open="(" close=")"><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>0</mn></mrow><mi>m</mi></munderover></mstyle><mfenced separators="" open="(" close=")"><mtable><mtr><mtd><mi>m</mi></mtd></mtr><mtr><mtd><mi>j</mi></mtd></mtr></mtable></mfenced><msubsup><mi>L</mi><mi>A</mi><mi>j</mi></msubsup><msubsup><mi>R</mi><mi>B</mi><mi>j</mi></msubsup></mfenced><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mo>▵</mo><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msubsup><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><mo>≠</mo><mn>0</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>A</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>B</mi></msub><mo>∈</mo><mi>B</mi><mrow><mo stretchy="false">(</mo><mi>B</mi><mrow><mo stretchy="false">(</mo><mi mathvariant="script">X</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> are, respectively, the operators of left multiplication by <i>A</i> and right multiplication by <i>B</i>. Define operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>E</mi><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">E</mi><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>E</mi><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mo>=</mo><msub><mi>L</mi><mi>A</mi></msub><msub><mi>R</mi><mi>B</mi></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mfenced separators="" open="(" close=")"><msub><mi mathvariant="script">E</mi><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mfenced><mi>n</mi></msup><mo>=</mo><msubsup><mi>E</mi><mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow><mi>n</mi></msubsup><mrow><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> for all non-negative integers <i>n</i>. Using little more than an algebraic argument, the following generalised version of a result relating <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-isometric properties of pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> to pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="script">E</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi mathvariant="script">E</mi><mrow><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></msub><mo>,</mo><msub><mi>E</mi><mrow><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is proved: if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>,</mo><msub><mi>B</mi><mi>i</mi></msub><mo>,</mo><msub><mi>S</mi><mi>i</mi></msub><mo>,</mo><mi>X</mi></mrow></semantics></math></inline-formula> are operators in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><mo stretchy="false">(</mo><mi mathvariant="script">X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <i>X</i> a quasi-affinity, then the pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi mathvariant="script">E</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow><mo>,</mo><msub><mi mathvariant="script">E</mi><mrow><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> (resp., the pair <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi><mrow><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></msub><mo>,</mo><msub><mi>E</mi><mrow><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>) is strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>m</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-isometric for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>B</mi><mo stretchy="false">(</mo><mi mathvariant="script">X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> if and only if there exist positive integers <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>m</mi><mi>i</mi></msub><mo>≤</mo><mi>m</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mn>2</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mo>=</mo><msub><mi>m</mi><mn>1</mn></msub><mo>+</mo><msub><mi>m</mi><mn>2</mn></msub><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, and a non-zero scalar <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> such that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>−</mo><msub><mi mathvariant="script">E</mi><mrow><mi>β</mi><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is (strict) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mn>1</mn></msub></semantics></math></inline-formula>-nilpotent and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>I</mi><mo>−</mo><msub><mi mathvariant="script">E</mi><mrow><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>β</mi></mfrac></mstyle><msub><mi>B</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>2</mn></msub></mrow></msub><mrow><mo stretchy="false">(</mo><msub><mi>S</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> is (strict) <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>m</mi><mn>2</mn></msub></semantics></math></inline-formula>-nilpotent (resp., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mi>β</mi><msub><mi>A</mi><mn>1</mn></msub><mo>,</mo><msub><mi>B</mi><mn>1</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>m</mi><mn>1</mn></msub><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-isometric and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mn>1</mn><mi>β</mi></mfrac></mstyle><msub><mi>B</mi><mn>2</mn></msub><mo>,</mo><msub><mi>A</mi><mn>2</mn></msub><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> is strict <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>m</mi><mn>2</mn></msub><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula>-isometric).
ISSN:	2075-1680