Generating Ideals of Bloch Mappings via Pietsch’s Quotients
In this paper, we introduce the notion of the normalized Bloch left-hand quotient ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">A</mi><...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-01-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/3/391 |
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| Summary: | In this paper, we introduce the notion of the normalized Bloch left-hand quotient ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><msup><mi mathvariant="script">I</mi><mover accent="true"><mi mathvariant="script">B</mi><mo stretchy="false">^</mo></mover></msup></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> is an operator ideal and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">I</mi><mover accent="true"><mi mathvariant="script">B</mi><mo stretchy="false">^</mo></mover></msup></semantics></math></inline-formula> is a normalized Bloch ideal, as a nonlinear extension of the concept of the left-hand quotient of operator ideals. We show that these quotients constitute a new method for generating normalized Bloch ideals, complementing the existing methods of generation by composition and transposition. In fact, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="script">I</mi><mover accent="true"><mi mathvariant="script">B</mi><mo stretchy="false">^</mo></mover></msup></semantics></math></inline-formula> has the linearization property in a linear operator ideal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">J</mi></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi mathvariant="script">A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><msup><mi mathvariant="script">I</mi><mover accent="true"><mi mathvariant="script">B</mi><mo stretchy="false">^</mo></mover></msup></mrow></semantics></math></inline-formula> is a composition ideal of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><msup><mi mathvariant="script">A</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>∘</mo><mi mathvariant="script">J</mi><mo>)</mo></mrow><mo>∘</mo><msup><mi mathvariant="script">I</mi><mover accent="true"><mi mathvariant="script">B</mi><mo stretchy="false">^</mo></mover></msup></mrow></semantics></math></inline-formula>. We conclude this work by introducing two important subclasses of Bloch maps; these are Bloch maps with the Grothendieck and Rosenthal range. We focus on showing that they form normalized Bloch ideals which can be seen as normalized Bloch left-hand quotients ideals. In addition, we pose an open problem concerning when a Bloch quotient without the linearization property in an operator ideal cannot be related to a normalized Bloch ideal of the composition type, for which we will use the subclass of <i>p</i>-summing Bloch maps. |
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| ISSN: | 2227-7390 |