Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

Raúl E Curto; In Sung Hwang; Dong-O Kang; Woo Young Lee

doi:10.1016/j.aim.2014.01.008

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Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

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Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols

Raúl E Curto, In Sung Hwang, Dong-O Kang and Woo Young Lee

Advances in mathematics (New York. 1965), Vol.255, pp.562-585

04/01/2014

DOI: 10.1016/j.aim.2014.01.008

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Abstract

In this paper we deal with the subnormality and the quasinormality of Toeplitz operators with matrix-valued rational symbols. In particular, in view of Halmos's Problem 5, we focus on the question: Which subnormal Toeplitz operators are normal or analytic? We first prove: Let Φ∈LMn∞ be a matrix-valued rational function having a “matrix pole”, i.e., there exists α∈D for which kerHΦ⊆(z−α)HCn2, where HΦ denotes the Hankel operator with symbol Φ. If(i)TΦ is hyponormal;(ii)ker[TΦ⁎,TΦ] is invariant for TΦ, then TΦ is normal. Hence in particular, if TΦ is subnormal then TΦ is normal. Next, we show that every pure quasinormal Toeplitz operator with a matrix-valued rational symbol is unitarily equivalent to an analytic Toeplitz operator.

Abrahamse's Theorem

Amemiya, Ito and Wong's Theorem

Subnormal

Hyponormal

Toeplitz operators

Quasinormal

Matrix-valued rational functions

Details

Title: Subtitle: Subnormal and quasinormal Toeplitz operators with matrix-valued rational symbols
Creators: Raúl E Curto - Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
In Sung Hwang - Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea
Dong-O Kang - Department of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea
Woo Young Lee - Department of Mathematics, Seoul National University, Seoul 151-742, Republic of Korea
Resource Type: Journal article
Publication Details: Advances in mathematics (New York. 1965), Vol.255, pp.562-585
DOI: 10.1016/j.aim.2014.01.008
ISSN: 0001-8708
eISSN: 1090-2082
Publisher: Elsevier Inc
Grant note: 2009-0083521 / Korea government (MSIP) DMS-0801168 / NSF 2011-0022577 / Ministry of Education, Science and Technology
Language: English
Date published: 04/01/2014
Academic Unit: Mathematics
Record Identifier: 9983985848002771

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