荒牧数学談話会 / 第6案内

第6回荒牧数学談話会の開催について

群馬大学荒牧キャンパスの教室での対面開催です。
講演内容は以下の通りです。

開催日時・内容

日時：2026年1月27日（火）13:00開始
場所: 群馬大学共同教育学部（荒牧キャンパス） 6号館1階 C103

講演者:
大坂 博幸 (立命館大学)
タイトル:
可算次元ヒルベルト空間上の有界線形作用素における不変部分空間問題の最 近の進展について
(Recent progress in the invariant subspace problem for bounded linear operators on countably dimensional Hilbert spaces)
概要:
The invariant subspace problem is formally stated as follows:
For any complex Banach space \(\mathcal{B}\) of dimension \(> 1\) does every bounded linear operator \(T\colon B \to B\) have a non-trivial closed subspace of \(W\) of \( \mathcal{B}\), which is different from \(\{0\}\) and from \( \mathcal{B}\), such that \(T(W) \subset W\)?

The first example of an operator on a Banach space with no non-trivial invariant subspaces was found by Per Enflo in 1976 [1] which was out line (the detailed version was submitted in 1981 [2]) and very complicated. In 1985 B. Beauzamy simpified this example [3]. Note that if \(H\) is a uncountourble Hilbert space, then if you consider a closed subapce \(M\) generated by \(\{T^n x\mathrel{;} n ∈ \mathbb{N}\}\), \(M \ne \{ 0 \}\), \(M \ne H\), and \(TM \subset M\). So, we may consider a separable Hilbert space H. Until now, the invariant subspace problem for a separable Hilbert space is opened. (I believe)

A bounded linear operator \(T\colon H_1 \to H_2\), where \(H_1, H_2\) are Hilbert spaces, is said to be norm attaining if there exists a unit vector \(x \in H \) such that \(|\!|T x |\!| = |\!| T |\!|\) and absolutely norm attaining (or \(\mathcal{AN}\)-operator) if \( T\vert_M \colon M \to H_2\) is norm attaining for every closed subspace \(M\) of \(H\).

Let \(\mathcal{R}_T\) denote the set of all reducing subspaces of \(T\). Define \[\beta(H) :=\{ T \in \mathcal{B}(H) \mathrel{;} T\vert_M \colon M → M\ \text{is norm attaining}\ \forall\, M \in \mathcal{R}_T \}.\] In this talk we present the outline of the History of Invariant subspace problem and we introduce a structure theorem for positive operators in \(\beta(H)\) and compare our results with those of absolutely norm attaining operators. Then, we characterize all operators in this new class. Lastly, we present the denseness of \(\beta(H)\) in \(\mathcal{B}(H) \), that is, for any \(T \in \mathcal{B}(H)\) and \(\varepsilon > 0\) there exists \(S \in \beta(H)\) such thar \(|\!|T −S |\!| < \varepsilon \) and \(S\) has non-trivial invariant subspace. I recommend a book by Rdjavi and Rosenthal [8] for the good introduction for an invariant subspace problem.

References

  1. P. Enflo, On the invariant subspace problem in Banach spaces, S ́eminaire Maurey–Schwartz (1975–1976) Espaces \(L_p\), applications radonifiantes et gomtrie des espaces de Banach, Exp. Nos. 14-15. Centre Math., cole Polytech., Palaiseau. p. 7.
  2. P. Enflo, On the invariant subspace problem for Banach spaces, Acta Mathematica. 158 (3): 213313. doi:10.1007/BF02392260.
  3. B. Beauzmy, Un oprateur sans sous-espace invariant: simplification de l’exemple de P. Enflo An operator with no invariant subspace: simplification of the example of P. Enflo, Integral Equations Operator Theory 8 (1985), no. 3, 314384.
  4. R. Golla and H. Osaka, On a subclass of norm attaining operators, Acta. Sci. Math. (Szegrd), 87 (2021), 247-263.
  5. R. Golla and H. Osaka, On operators which attain their norm on every reducing subspace, Ann. Funct. Annal. 13(19) (2022).
  6. R. Golla, H. Osaka, and S. S. Sequeira, On the denseness of \(\beta(H)\), to appear in J. O. T.
  7. H. Osaka and T. Yamazaki, Limits of iteration of the induced Althuge transformations of centered operators, Trans. A.M.S. 30 (2025), 6857-6884.
  8. R. Radjavi and R. Rosenthal, Invariant subspaces, Second edition, Dover Publications, INC, 2003.