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Let $H$ be an infinite dimensional separable Hilbert space.

Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} \} $.

Definitions : An operator $A \in B(H)$ is :

  • Irreducible (Halmos 1968) if its commutant $\{ A\}'$ does not contain projections other than $0$ and $I$ ($A \ne A_{1} \oplus A_{2}$, $A$ generates $B(H)$ as von Neumann algebra : $\{A,A^{*}\}''=B(H)$).

  • Nonnormal if $\{ A\}'$ does not contain $A^{*}$ (i.e. $AA^{*} \ne A^{*}A$).

  • Noncompact commuting if $\{ A\}'$ does not contain a compact operator.

Definition : The spectrum $\sigma(A)$ of $A \in B(H)$ is $\{\lambda \in \mathbb{C} : A - \lambda I \text{ not bijective} \}$.
It decomposes as follows:

  • Point spectrum: $\sigma_{p}(A) = \{\lambda \in \mathbb{C} : A - \lambda I \text{ not injective} \}$
  • Continuous spectrum: $\sigma_{c}(A) = \{\lambda \in \mathbb{C} : A - \lambda I \ \text{injective, dense nonclosed range} \}$
  • Residual spectrum: $\sigma_{r}(A) = \{\lambda \in \mathbb{C} : A - \lambda I \ \text{ injective, nondense range} \}$

The spectrum of $A$ is strictly continuous if $\sigma(A) = \sigma_{c}(A)$.

Examples:

  • Let $S$ be the bilateral shift defined on $H = l^{2}(\mathbb{Z})$ by $S.e_{n} = e_{n+1} $.
    Its spectrum is strictly continuous : $\sigma(S) = \sigma_{c}(S) = \mathbb{S}^{1}$.
    It's also a unitary operator ($SS^{*} = S^{*}S = I$), so a fortiori a normal operator.
    It is noncompact commuting and reducible.
  • Let $T$ be the unilateral shift defined on $H = l^{2}(\mathbb{N})$ by $T.e_{n} = e_{n+1} $.
    Its spectrum is not strictly continuous because $0 \in \sigma_{r}(T)$.
    It's a nonnormal operator because $[T^{*},T].e_{0} = e_{0}$.
    It is noncompact commuting and irreducible.
  • Let $V$ the Volterra operator defined on $H= L^{2}[0,1]$ by $(V.f)(t)=\int_0^tf(x)dx$.
    Its spectrum is strictly continuous $\sigma(V) = \sigma_{c}(V) = \{ 0\}$.
    It is compact, irreducible and nonnormal (see here).
  • Let $p$ be a non-constant polynomial (see here).
    Then $p(V)$ is nonnormal, compact commuting and irreducible.
    Its spectrum is strictly continuous $\sigma(p(V)) = \sigma_{c}(p(V)) = \{ p(0)\}$.
    It's compact commuting, nonnormal and irreducible.
  • Let $S \oplus V$ defined on $l^{2}(\mathbb{Z}) \oplus L^{2}[0,1]$.
    It is reducible, compact commuting, nonnormal and with spectrum strictly continuous.

If you find a mistake, thank you let me know in comment.

The main question: Is there an irreducible, noncompact commuting and nonnormal operator, with spectrum strictly continuous ?

Bonus questions : How classify these operators ?

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  • $\begingroup$ Cross-posted on MO. $\endgroup$ – Julien Jul 21 '13 at 13:41
  • $\begingroup$ Hi Julien, I did not know that I'm "supposed to let people know when you ask the same question on both site", thank you for this information. I hesitated between the two. I have posted it on MO first, and then on MSE. Where do you think is best for it ? $\endgroup$ – Sebastien Palcoux Jul 21 '13 at 14:10
  • $\begingroup$ Honestly, I don't know... $\endgroup$ – Julien Jul 21 '13 at 14:17
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This issue has an answer on mathoverflow, by Ollie Margetts here, by using a weight shift.

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