{"title":"Commuting tuple of multiplication operators homogeneous under the unitary group","authors":"Soumitra Ghara, Surjit Kumar, Gadadhar Misra, Paramita Pramanick","doi":"10.1112/jlms.12890","DOIUrl":null,"url":null,"abstract":"<p>Let <span></span><math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {U}(d)$</annotation>\n </semantics></math> be the group of <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>×</mo>\n <mi>d</mi>\n </mrow>\n <annotation>$d\\times d$</annotation>\n </semantics></math> unitary matrices. We find conditions to ensure that a <span></span><math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {U}(d)$</annotation>\n </semantics></math>-homogeneous <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math>-tuple <span></span><math>\n <semantics>\n <mi>T</mi>\n <annotation>$\\bm{T}$</annotation>\n </semantics></math> is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space <span></span><math>\n <semantics>\n <mrow>\n <msub>\n <mi>H</mi>\n <mi>K</mi>\n </msub>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>B</mi>\n <mi>d</mi>\n </msub>\n <mo>,</mo>\n <msup>\n <mi>C</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n <mo>⊆</mo>\n <mtext>Hol</mtext>\n <mrow>\n <mo>(</mo>\n <msub>\n <mi>B</mi>\n <mi>d</mi>\n </msub>\n <mo>,</mo>\n <msup>\n <mi>C</mi>\n <mi>n</mi>\n </msup>\n <mo>)</mo>\n </mrow>\n </mrow>\n <annotation>$\\mathcal {H}_K(\\mathbb {B}_d, \\mathbb {C}^n) \\subseteq \\mbox{\\rm Hol}(\\mathbb {B}_d, \\mathbb {C}^n)$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>=</mo>\n <mo>dim</mo>\n <msubsup>\n <mo>∩</mo>\n <mrow>\n <mi>j</mi>\n <mo>=</mo>\n <mn>1</mn>\n </mrow>\n <mi>d</mi>\n </msubsup>\n <mo>ker</mo>\n <msubsup>\n <mi>T</mi>\n <mi>j</mi>\n <mo>∗</mo>\n </msubsup>\n </mrow>\n <annotation>$n= \\dim \\cap _{j=1}^d \\ker T^*_{j}$</annotation>\n </semantics></math>. We describe this class of <span></span><math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {U}(d)$</annotation>\n </semantics></math>-homogeneous operators, equivalently, nonnegative kernels <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> quasi-invariant under the action of <span></span><math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {U}(d)$</annotation>\n </semantics></math>. We classify quasi-invariant kernels <span></span><math>\n <semantics>\n <mi>K</mi>\n <annotation>$K$</annotation>\n </semantics></math> transforming under <span></span><math>\n <semantics>\n <mrow>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$\\mathcal {U}(d)$</annotation>\n </semantics></math> with two specific choice of multipliers. A crucial ingredient of the proof is that the group <span></span><math>\n <semantics>\n <mrow>\n <mi>S</mi>\n <mi>U</mi>\n <mo>(</mo>\n <mi>d</mi>\n <mo>)</mo>\n </mrow>\n <annotation>$SU(d)$</annotation>\n </semantics></math> has exactly two inequivalent irreducible unitary representations of dimension <span></span><math>\n <semantics>\n <mi>d</mi>\n <annotation>$d$</annotation>\n </semantics></math> and none in dimensions <span></span><math>\n <semantics>\n <mrow>\n <mn>2</mn>\n <mo>,</mo>\n <mtext>…</mtext>\n <mo>,</mo>\n <mi>d</mi>\n <mo>−</mo>\n <mn>1</mn>\n </mrow>\n <annotation>$2, \\ldots , d-1$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\geqslant 3$</annotation>\n </semantics></math>. We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.12890","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
Let be the group of unitary matrices. We find conditions to ensure that a -homogeneous -tuple is unitarily equivalent to multiplication by the coordinate functions on some reproducing kernel Hilbert space , . We describe this class of -homogeneous operators, equivalently, nonnegative kernels quasi-invariant under the action of . We classify quasi-invariant kernels transforming under with two specific choice of multipliers. A crucial ingredient of the proof is that the group has exactly two inequivalent irreducible unitary representations of dimension and none in dimensions , . We obtain explicit criterion for boundedness, reducibility, and mutual unitary equivalence among these operators.
让 U ( d ) $\mathcal {U}(d)$ 是 d × d $d\times d$ 单元矩阵群。我们要找到一些条件,以确保 U ( d ) $mathcal {U}(d)$ -homogeneous d $d$ -tuple T $\bm{T}$ 与某个重现核 Hilbert 空间 H K ( B d , C n ) 上的坐标函数相乘是单位等价的。 ⊆ Hol ( B d , C n ) $\mathcal {H}_K(\mathbb {B}_d, \mathbb {C}^n) \subseteq \mbox\rm Hol}(\mathbb {B}_d, \mathbb {C}^n)$ 、 n = dim ∩ j = 1 d ker T j∗ $n= \dim \cap _{j=1}^d \ker T^*_{j}$ 。我们描述这一类 U ( d ) $\mathcal {U}(d)$ -同调算子,等价地,非负核 K $K$ 在 U ( d ) $\mathcal {U}(d)$ 作用下准不变。我们将在 U ( d ) $mathcal {U}(d)$ 作用下变换的准不变核 K $K$ 用两种特定的乘数选择进行分类。证明的一个关键要素是群 S U ( d ) $SU(d)$ 在维数为 d $d$ 的情况下有两个不等价的不可还原单元表示,而在维数为 2 , ... , d - 1 $2, \ldots , d-1$ , d ⩾ 3 $d\geqslant 3$ 的情况下没有。我们得到了这些算子的有界性、可还原性和相互单元等价性的明确标准。
期刊介绍:
The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.