| 要約: | We investigate a novel operator seminorm, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced open="∥" close="∥"><mi>Q</mi></mfenced><mrow><mi>A</mi><mo>,</mo><msub><mi mathvariant="fraktur">m</mi><mi>λ</mi></msub><mo>,</mo><mi>f</mi></mrow></msub></semantics></math></inline-formula>, for an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula>-bounded operator <i>Q</i>, 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 a positive operator on a complex Hilbert space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi mathvariant="script">K</mi><mo>,</mo><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo><mo>)</mo></mrow></semantics></math></inline-formula>. This seminorm is defined using a continuous increasing and bijective function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup><mo>⟶</mo><msup><mi mathvariant="double-struck">R</mi><mo>+</mo></msup></mrow></semantics></math></inline-formula> and an interpolational path <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="fraktur">m</mi><mi>λ</mi></msub></semantics></math></inline-formula> of the symmetric mean <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">m</mi></semantics></math></inline-formula>. Specifically, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mfenced open="∥" close="∥"><mi>Q</mi></mfenced><mrow><mi mathvariant="script">A</mi><mo>,</mo><msub><mi mathvariant="fraktur">m</mi><mi>λ</mi></msub><mo>,</mo><mi>f</mi></mrow></msub><mo>=</mo><mi>sup</mi><mfenced separators="" open="{" close="}"><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced separators="" open="(" close=")"><mi>f</mi><mfenced open="(" close=")"><mfenced separators="" open="|" close="|"><msub><mfenced separators="" open="⟨" close="⟩"><mi>Q</mi><mi>y</mi><mo>,</mo><mi>y</mi></mfenced><mi mathvariant="script">A</mi></msub></mfenced></mfenced><msub><mi mathvariant="fraktur">m</mi><mi>λ</mi></msub><mi>f</mi><mfenced open="(" close=")"><msub><mfenced separators="" open="∥" close="∥"><mi>Q</mi><mi>y</mi></mfenced><mi mathvariant="script">A</mi></msub></mfenced></mfenced><mo>:</mo><mi>y</mi><mo>∈</mo><mi mathvariant="script">K</mi><mo>,</mo><msub><mfenced open="∥" close="∥"><mi>y</mi></mfenced><mi mathvariant="script">A</mi></msub><mo>=</mo><mn>1</mn></mfenced><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>f</mi><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> represents the reciprocal function of <i>f</i>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>⟨</mo><mo>·</mo><mo>,</mo><mo>·</mo><mo>⟩</mo></mrow><mi mathvariant="script">A</mi></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mfenced open="∥" close="∥"><mo>·</mo></mfenced><mi mathvariant="script">A</mi></msub></semantics></math></inline-formula> denote the semi-inner product and seminorm, respectively, induced by <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">A</mi></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">K</mi></semantics></math></inline-formula>. We explore various bounds and relationships associated with this new concept, establishing connections with existing literature.
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