On Invariant Subspaces for the Shift Operator
In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace <i>M</i> in the Hardy space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi&g...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2019-06-01
|
| Series: | Symmetry |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2073-8994/11/6/743 |
| _version_ | 1798006055579418624 |
|---|---|
| author | Junfeng Liu |
| author_facet | Junfeng Liu |
| author_sort | Junfeng Liu |
| collection | DOAJ |
| description | In this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace <i>M</i> in the Hardy space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is invariant under the shift operator <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula> on <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> if and only if it is hyperinvariant under <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula>, and that a closed linear subspace <i>M</i> in the Lebesgue space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is reducing under the shift operator <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> on <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> if and only if it is hyperinvariant under <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula>. At the same time, we show that there are two large classes of invariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> that are not hyperinvariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> and are also not reducing subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula>. Moreover, we still show that there is a large class of hyperinvariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula> that are not reducing subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula>. Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, which are the analogue of the formula of the reproducing function in the Bergman space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>A</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces. |
| first_indexed | 2024-04-11T12:49:20Z |
| format | Article |
| id | doaj.art-baf36af847ff455ba90026f607283c6d |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T12:49:20Z |
| publishDate | 2019-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-baf36af847ff455ba90026f607283c6d2022-12-22T04:23:15ZengMDPI AGSymmetry2073-89942019-06-0111674310.3390/sym11060743sym11060743On Invariant Subspaces for the Shift OperatorJunfeng Liu0Faculty of Information Technology, Macau University of Science and Technology, Avenida Wai Long, Taipa, Macau 999078, ChinaIn this paper, we improve two known invariant subspace theorems. More specifically, we show that a closed linear subspace <i>M</i> in the Hardy space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo><</mo> <mo>∞</mo> <mo stretchy="false">)</mo> </mrow> </semantics> </math> </inline-formula> is invariant under the shift operator <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula> on <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mi>p</mi> </msup> <mrow> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> if and only if it is hyperinvariant under <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula>, and that a closed linear subspace <i>M</i> in the Lebesgue space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is reducing under the shift operator <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> on <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mo>∂</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> if and only if it is hyperinvariant under <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula>. At the same time, we show that there are two large classes of invariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> that are not hyperinvariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula> and are also not reducing subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <msup> <mi>e</mi> <mrow> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </msub> </semantics> </math> </inline-formula>. Moreover, we still show that there is a large class of hyperinvariant subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula> that are not reducing subspaces for <inline-formula> <math display="inline"> <semantics> <msub> <mi>M</mi> <mi>z</mi> </msub> </semantics> </math> </inline-formula>. Furthermore, we gave two new versions of the formula of the reproducing function in the Hardy space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>H</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, which are the analogue of the formula of the reproducing function in the Bergman space <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mi>A</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">D</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In addition, the conclusions in this paper are interesting now, or later if they are written into the literature of invariant subspaces and function spaces.https://www.mdpi.com/2073-8994/11/6/743invariant subspacehyperinvariant subspacereducing subspaceshift operatorhardy spacelebesgue spaceBergman space |
| spellingShingle | Junfeng Liu On Invariant Subspaces for the Shift Operator Symmetry invariant subspace hyperinvariant subspace reducing subspace shift operator hardy space lebesgue space Bergman space |
| title | On Invariant Subspaces for the Shift Operator |
| title_full | On Invariant Subspaces for the Shift Operator |
| title_fullStr | On Invariant Subspaces for the Shift Operator |
| title_full_unstemmed | On Invariant Subspaces for the Shift Operator |
| title_short | On Invariant Subspaces for the Shift Operator |
| title_sort | on invariant subspaces for the shift operator |
| topic | invariant subspace hyperinvariant subspace reducing subspace shift operator hardy space lebesgue space Bergman space |
| url | https://www.mdpi.com/2073-8994/11/6/743 |
| work_keys_str_mv | AT junfengliu oninvariantsubspacesfortheshiftoperator |