Logo image
Back
Hyponormality and subnormality of block Toeplitz operators
Journal article   Open access   Peer reviewed

Hyponormality and subnormality of block Toeplitz operators

Raúl E Curto, In Sung Hwang and Woo Young Lee
Advances in mathematics (New York. 1965), Vol.230(4-6), pp.2094-2151
07/2012
DOI: 10.1016/j.aim.2012.04.019
url
https://doi.org/10.1016/j.aim.2012.04.019View
Published (Version of record) Open Access

Abstract

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space HCn2 of the unit circle. First, we establish a tractable and explicit criterion on the hyponormality of block Toeplitz operators having bounded type symbols via the triangularization theorem for compressions of the shift operator. Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos’s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Φ is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator TΦ whose square is also hyponormal must be either normal or analytic. Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following “Toeplitz completion” problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix A≔[U∗??U∗] so that A becomes subnormal, where U is the unilateral shift on H2.
 Block Toeplitz operators Square-hyponormal Trigonometric polynomials Subnormal Bounded type functions Rational functions Subnormal completion problem Hyponormal

Details

Metrics

12 Record Views
37 Times Cited - Web of Science
Logo image