On the Equality <i>A</i> = <i>A</i><sub>1</sub><i>A</i><sub>2</sub> for Linear Relations
Assume that <i>A</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mn>1</mn></msub></semantics></math></inline-formula>, and <in...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-03-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/4/239 |
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| Summary: | Assume that <i>A</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mn>1</mn></msub></semantics></math></inline-formula>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mn>2</mn></msub></semantics></math></inline-formula> are three selfadjoint linear relations (multi-valued linear operators) in a certain complex Hilbert space. In this study, conditions are presented for the multi-valued operator equality <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> to hold when the inclusion <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>⊂</mo><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> is assumed to be satisfied. The present study is strongly motivated by the invalidity of a classical result from A. Devinatz, A. E. Nussbaum, and J. von Neumann in the general case of selfadjoint linear relations. Two types of conditions for the aforementioned equality to hold are presented. Firstly, a condition is given in terms of the resolvent sets of the involved objects, which does not depend on the product structure of the right-hand side, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mn>1</mn></msub><msub><mi>A</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula>. Secondly, a condition is also presented where the structure of the right-hand side is taken into account. This one is based on the notion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">L</mi></semantics></math></inline-formula>-stability of a linear operator under linear subspaces. It should be mentioned that the classical Devinatz–Nussbaum–von Neumann theorem is obtained as a particular case of one of the main results. |
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| ISSN: | 2075-1680 |