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<p>We investigate the local preservation of <i>A</i>-orthogonality at a point by <i>A</i>-bounded operators within the semi-Hilbertian framework induced by a positive operator <i>A</i> on a complex Hilbert space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {H}.\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">H</mi> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> We provide complete characterizations of such preservation. Additionally, we explore properties of the <i>A</i>-norm attainment set of an <i>A</i>-bounded operator in light of <i>A</i>-orthogonality preservation. We also study analogous properties for the minimum <i>A</i>-norm attainment set of an <i>A</i>-bounded operator. We then characterize the <i>A</i>-isometries as the <i>A</i>-norm one operators preserving <i>A</i>-orthogonality. Finally, we characterize those subsets of complex Hilbert spaces for which such preservation by an <i>A</i>-norm one operator implies that the operator is an <i>A</i>-isometry.</p>

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On A-orthogonality preservation and Blanco-Koldobsky-Turnšek theorem in semi-Hilbert spaces

  • 期刊论文

  • 发表日期：

    2026年4月6日

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  • Jayanta Manna,
  • Somdatta Barik,
  • Kallol Paul,
  • Debmalya Sain

摘要

We investigate the local preservation of A-orthogonality at a point by A-bounded operators within the semi-Hilbertian framework induced by a positive operator A on a complex Hilbert space \(\mathbb {H}.\) H . We provide complete characterizations of such preservation. Additionally, we explore properties of the A-norm attainment set of an A-bounded operator in light of A-orthogonality preservation. We also study analogous properties for the minimum A-norm attainment set of an A-bounded operator. We then characterize the A-isometries as the A-norm one operators preserving A-orthogonality. Finally, we characterize those subsets of complex Hilbert spaces for which such preservation by an A-norm one operator implies that the operator is an A-isometry.