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<p>This paper is devoted to the study of propagation phenomena for 2–hyponormal, quadratically hyponormal, and cubically hyponormal operator-valued weighted shifts. First, we show that every <i>quadratically</i> hyponormal matrix-valued weighted shift with two equal weights (<i>excluding the initial weight</i>) is flat. Second, we show that a <i>cubically</i> hyponormal operator-valued weighted shift with two equal weights (<i>possibly including the initial weight</i>) is flat. Next, we introduce a <i>local flatness</i> notion for matrix-valued weighted shifts. We prove that 2–hyponormal (in particular, subnormal) matrix-valued weighted shifts satisfy this stronger propagation phenomenon. As a result, we prove a <i>structural decomposition theorem</i> for 2–hyponormal matrix-valued weighted shifts.</p>

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Propagation Phenomena for Operator-Valued Weighted Shifts

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年12月13日

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  • Raúl Curto,
  • Abderrazzak Ech-charyfy,
  • Hamza El Azhar,
  • El Hassan Zerouali

摘要

This paper is devoted to the study of propagation phenomena for 2–hyponormal, quadratically hyponormal, and cubically hyponormal operator-valued weighted shifts. First, we show that every quadratically hyponormal matrix-valued weighted shift with two equal weights (excluding the initial weight) is flat. Second, we show that a cubically hyponormal operator-valued weighted shift with two equal weights (possibly including the initial weight) is flat. Next, we introduce a local flatness notion for matrix-valued weighted shifts. We prove that 2–hyponormal (in particular, subnormal) matrix-valued weighted shifts satisfy this stronger propagation phenomenon. As a result, we prove a structural decomposition theorem for 2–hyponormal matrix-valued weighted shifts.