Hyponormality and subnormality of block Toeplitz operators

作者：Curto Raul E^*; Hwang In Sung; Lee Woo Young

来源：Advances in Mathematics, 2012, 230(4-6): 2094-2151.

DOI：10.1016/j.aim.2012.04.019

• Block Toeplitz operators
• Hyponormal
• Square-hyponormal
• Subnormal
• Bounded type functions
• Rational function
• Trigonometric polynomials
• Subnormal completion problem

In this paper, we are concerned with hyponormality and subnormality of block Toeplitz operators acting on the vector-valued Hardy space H-Cn(2) of the unit circle. %26lt;br%26gt;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. %26lt;br%26gt;Second, we consider the gap between hyponormality and subnormality for block Toeplitz operators. This is closely related to Halmos%26apos;s Problem 5: Is every subnormal Toeplitz operator either normal or analytic? We show that if Phi is a matrix-valued rational function whose co-analytic part has a coprime factorization then every hyponormal Toeplitz operator T-Phi whose square is also hyponormal must be either normal or analytic. %26lt;br%26gt;Third, using the subnormal theory of block Toeplitz operators, we give an answer to the following %26quot;Toeplitz completion%26quot; problem: find the unspecified Toeplitz entries of the partial block Toeplitz matrix %26lt;br%26gt;A = [(U*)(?) (?)(U*)] %26lt;br%26gt;so that A becomes subnormal, where U is the unilateral shift on H-2.

• 出版日期2012-8