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<p>We solve a completion problem of a <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(3\times 3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>×</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> upper triangular operator matrix acting on a direct sum of Banach spaces and hence generalize the famous result of Han et&#xa0;al. (Proc. Am. Math. Soc. 128(1):119–123, 2000) to a greater dimension of a matrix. Our main tools are Harte’s ghost of an index theorem and Banach spaces embeddings. We overcome the lack of orthogonality in Banach spaces by exploiting decomposition properties of inner regular operators. We provide some illustrative examples, and we comment on further generalizations to matrix dimensions <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n&gt;3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&gt;</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Completion Problem of Upper Triangular \(3\times 3\) Operator Matrices on Arbitrary Banach Spaces

  • 期刊论文

  • 发表日期：

    2025年11月25日

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  • Nikola Sarajlija,
  • Dragan S. Djordjević

摘要

We solve a completion problem of a \(3\times 3\) 3 × 3 upper triangular operator matrix acting on a direct sum of Banach spaces and hence generalize the famous result of Han et al. (Proc. Am. Math. Soc. 128(1):119–123, 2000) to a greater dimension of a matrix. Our main tools are Harte’s ghost of an index theorem and Banach spaces embeddings. We overcome the lack of orthogonality in Banach spaces by exploiting decomposition properties of inner regular operators. We provide some illustrative examples, and we comment on further generalizations to matrix dimensions \(n>3\) n > 3 .