Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be the algebra of all bounded linear operators on an infinite dimensional complex separable Hilbert space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation>. We prove that all operators T in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> satisfying the following two conditions constitute a co-meager subset of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {B(H)}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">B</mi> <mo stretchy="false">(</mo> <mi mathvariant="script">H</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>: (i) For every <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$n\in \{\pm 1,\pm 2,\pm 3,\cdots ,\pm \infty \}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>∈</mo> <mo stretchy="false">{</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> <mo>±</mo> <mn>2</mn> <mo>,</mo> <mo>±</mo> <mn>3</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mo>±</mo> <mi>∞</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, there exists a complex number <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$T-\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>T</mi> <mo>-</mo> <mi>λ</mi> </mrow> </math></EquationSource> </InlineEquation> is semi-Fredholm with index n. (ii) The Wolf spectrum of T has an empty interior. As an application, we show that a typical operator on <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {H}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> has a thick point spectrum, a thick residual spectrum and a rare continuous spectrum. This solves a problem raised by T. Eisner and T. Mátrai in the affirmative, and complements a recent result of M. Scherer. Also the spectra of typical operators in certain special classes of operators are studied.

Spectral pictures of typical Hilbert space operators

期刊论文
发表日期：

2026年6月17日

查看全文

Cun Wang,
Sen Zhu

摘要

Let $\mathcal {B(H)}$ $B (H)$ be the algebra of all bounded linear operators on an infinite dimensional complex separable Hilbert space $\mathcal {H}$ $H$ . We prove that all operators T in $\mathcal {B(H)}$ $B (H)$ satisfying the following two conditions constitute a co-meager subset of $\mathcal {B(H)}$ $B (H)$ : (i) For every $n\in \{\pm 1,\pm 2,\pm 3,\cdots ,\pm \infty \}$ $n \in {\pm 1, \pm 2, \pm 3, \dots, \pm \infty}$ , there exists a complex number $\lambda $ $λ$ such that $T-\lambda $ $T - λ$ is semi-Fredholm with index n. (ii) The Wolf spectrum of T has an empty interior. As an application, we show that a typical operator on $\mathcal {H}$ $H$ has a thick point spectrum, a thick residual spectrum and a rare continuous spectrum. This solves a problem raised by T. Eisner and T. Mátrai in the affirmative, and complements a recent result of M. Scherer. Also the spectra of typical operators in certain special classes of operators are studied.