A note about maximal almost-invariant subspaces and maximal hyperinvariant subspaces

In this paper, we show that for T∈B(H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T\in B({\mathcal {H}})$$\end{document}, if M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document} is almost-invariant for T, then every maximal almost-invariant subspace of M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document} is of codimension 1 in M\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {M}}$$\end{document}, where H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document} is a separable, infinite-dimensional Hilbert space. We also describe the maximal hyperinvariant subspaces for normal operators with all the dimensions of eigenspaces at most 1 acting on H\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {H}}$$\end{document}. Our result is that for each hyperinvariant subspace, all its maximal hyperinvariant subspaces are also of codimension 1 in it.


Introduction
Let H be a separable, infinite-dimensional Hilbert space and denote by B (  For an operator T ∈ B(H) and an invariant subspace M for T , a T -invariant subspace N is called a maximal invariant subspace of M, if N M and there is no T -invariant subspace L such that N L M. Hedenmalm [6] obtained first the result that every maximal invariant subspace of the Bergman space is of codimension 1. For further generalizions of the Bergman space, we refer the interested readers to [1,16]. Later, Guo et al. [5] extended the result to a much more general situation. Their result is the following. Here S p ( p > 0) denotes the set of Schattenp class operators; and for two subspaces Motivated by the above work, we intend to study the maximal almost-invariant subspaces and maximal hyperinvariant subspaces.
This concept was first introduced in [2]. The minimal dimension of such a subspace F is referred to as the defect of M for T . It is obvious that every finite-dimensional or finite-codimensional subspace is almost-invariant under T . So we only need to consider a half-space, that is, a subspace of H which is infinite-dimensional and infinite-codimensional. For further information about almost-invariant subspaces, we refer the interested readers to [2,[12][13][14].
In a similar way, we give the definition of maximal almost-invariant subspace. There are many unsolved problems in the theory of invariant subspaces, hence these problems need close attention. In this paper, we first deal with hyperinvariant subspaces. For a further discussion about hyperinvariant subspaces, we recommend to the interested readers the recent papers [4,[7][8][9][10][11]15].
We also define maximal hyperinvariant subspaces analogously. Throughout the paper, for a closed subspace E of H, P E denotes the orthogonal projection from H to E and T | E is the operator T restricted to E.

Maximal almost-invariant subspaces
In this section, we give a characterization of maximal almost-invariant subspaces. The main result can be formulated as follows.
The first step in the proof of Theorem 2.1 is the following lemma.
Conversely, we assume that there exists a subspace L 0 with for some finite-dimensional subspace F , that is, L is T -almost invariant. So the assertion of this lemma is proved.
The following lemma, proved by Popov and Tcaciuc in [14], is quite important to get the main result of this section.

Lemma 2.3 Let T be a bounded operator on an infinite-dimensional, reflexive Banach space X . Then X admits an almost-invariant half-space with defect 1.
Using this lemma, we can prove the following result, which is the key idea of the proof of Theorem 2.1. At the end of this section, we want to pose a question to the interested readers. In the proof, the finite-dimensional subspaces making sure that N , L, M are T -almost invariant may not have the same dimension or even be the same subspace. Of course, here we mean such a finite-dimensional subspace with minimal dimension to make sure N , L, M are T -almost invariant. Hence, it is natural to ask the following question:

Maximal hyperinvariant subspaces
In the case when we consider the maximal hyperinvariant subspaces, we focus on the normal operators acting on a separable, infinite-dimensional Hilbert space H. The main result relies on the following lemma, which is proved in a similar way to [5, Lemma 2.3]. Since W T = T W , thus Therefore, we conclude that That is, L is hyperinvariant for T by the arbitrariness of W ∈ {T } , which proves the first assertion of this lemma.
Conversely, note that T is normal, i.e., T * ∈ {T } . Therefore, M, N are both reducing subspaces of T , so is M N . Thus we conclude that S = T | M N . Then the operator T has the corresponding decomposition So the second assertion of the lemma is obtained. The next result can be found in [3].

Lemma 3.3 A normal operator that is not a multiple of the identity has a non-trivial hyperinvariant subspace.
Using the previous lemmas, we are now ready to give the required generalization about maximal hyperinvariant subspaces. Proof Note that the condition that all the dimensions of eigenspaces of T are at most 1 guarantees that P M N T | M N is not a multiple of the identity for each T -hyperinvariant subspace N with N M and dim M N ≥ 2. Then the assertion comes easily from the lemmas above.
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