In this paper we consider the subnormality of block Toeplitz operators TΦ, where Φ is an n × n matrix-valued function on the unit circle <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11854_2025_358_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="12" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb{T}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">T</mi> </mrow> </math></EquationSource> </InlineEquation> of the form<Equation ID="Equa"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="11854_2025_358_Article_Equa.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="371" /> </MediaObject> <EquationSource Format="TEX">$\Phi=Q\Phi^{\ast}\;\;\;\;(Q\;\text{is}\; \text{a}\; \text{finite}\; \text{Blaschke-Potapov}\; \text{product})$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>Q</mi> <msup> <mi mathvariant="normal">Φ</mi> <mrow> <mo>∗</mo> </mrow> </msup> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>Q</mi> <mspace width="thickmathspace" /> <mtext>is</mtext> <mspace width="thickmathspace" /> <mtext>a</mtext> <mspace width="thickmathspace" /> <mtext>finite</mtext> <mspace width="thickmathspace" /> <mtext>Blaschke-Potapov</mtext> <mspace width="thickmathspace" /> <mtext>product</mtext> <mo stretchy="false">)</mo> </math></EquationSource> </Equation>This is related to a matrix-valued version of Halmos’ Problem 5 and the Nakazi–Takahashi Theorem. We ask whether TΦ is either normal or analytic if TΦ is subnormal, where Φ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when Φ* is not of bounded type.

Subnormal block Toeplitz operators

期刊论文
发表日期：

2025年1月21日

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Mankunikuzhiyil Abhinand,
Raúl E. Curto,
In Sung Hwang,
Woo Young Lee,
Thankarajan Prasad

摘要

In this paper we consider the subnormality of block Toeplitz operators T_Φ, where Φ is an n × n matrix-valued function on the unit circle $\mathbb{T}$ $T$ of the form

$\Phi=Q\Phi^{\ast}\;\;\;\;(Q\;\text{is}\; \text{a}\; \text{finite}\; \text{Blaschke-Potapov}\; \text{product})$ $Φ = Q Φ^{*} (Q is a finite Blaschke-Potapov product)$

This is related to a matrix-valued version of Halmos’ Problem 5 and the Nakazi–Takahashi Theorem. We ask whether T_Φ is either normal or analytic if T_Φ is subnormal, where Φ is of the above form. We give answers to this problem for different cases of the symbol. Moreover, we provide a sufficient condition for the answer to be affirmative when Φ* is not of bounded type.