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<p>In analogy to normal eigenvalues we introduce approximately normal eigenvalues of unbounded operators in Banach spaces. Roughly speaking, those are isolated eigenvalues of the approximate point spectrum of closed range and finite algebraic multiplicity. Thereby we define an approximate version of the discrete spectrum and study its complement in the approximate point spectrum. In the bounded case, this turns out to be a well-known type of essential spectrum introduced in [<CitationRef CitationID="CR39">39</CitationRef>] for which we prove natural, yet, to the best knowledge of the author, new characterisations. We study the connexion of this type of essential spectrum with the essential numerical ranges in Hilbert spaces. As a final result we show that the essential numerical range contains the whole approximate point spectrum except, possibly, some isolated eigenvalues of the approximate point spectrum of finite multiplicity.</p>

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Which Approximate Eigenvalues Are Contained in the Essential Numerical Range? Approximately Normal Eigenvalues of Unbounded Operators in Hilbert and Banach Spaces

  • 期刊论文

  • 发表日期：

    2026年2月18日

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  • Nicolas Hefti

摘要

In analogy to normal eigenvalues we introduce approximately normal eigenvalues of unbounded operators in Banach spaces. Roughly speaking, those are isolated eigenvalues of the approximate point spectrum of closed range and finite algebraic multiplicity. Thereby we define an approximate version of the discrete spectrum and study its complement in the approximate point spectrum. In the bounded case, this turns out to be a well-known type of essential spectrum introduced in [39] for which we prove natural, yet, to the best knowledge of the author, new characterisations. We study the connexion of this type of essential spectrum with the essential numerical ranges in Hilbert spaces. As a final result we show that the essential numerical range contains the whole approximate point spectrum except, possibly, some isolated eigenvalues of the approximate point spectrum of finite multiplicity.