Roots of Toeplitz Operators on the Bergman space

2010/04/01 by Issam Louhichi, Louhichi, Issam, N. V. Rao +1
Mathematics · #Holomorphic and Operator Theory #Algebraic and Geometric Analysis #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1004.0121

Abstract

One of the major questions in the theory of Toeplitz operators on the Bergman space over the unit disk \mathbb D in the complex plane \mathbb C is a complete description of the commutant of a given Toeplitz operator, that is the set of all Toeplitz operators that commute with it. In \citel, the first author obtained a complete description of the commutant of Toeplitz operator T with any quasihomogeneous symbol ϕ(r)e^ipθ, p>0 in case it has a Toeplitz p-th root S with symbol ψ(r)e^iθ, namely, commutant of T is the closure of the linear space generated by powers Sⁿ which are Toeplitz. But the existence of p-th root was known until now only when ϕ(r)=r^m,m ≥ 0. In this paper we will show the existence of p-th roots for a much larger class of symbols, for example, it includes such symbols for which ϕ(r)=∑_i=1^kr^a_i(ln r)^b_i,0≤ a_i, b_i for all 1≤ i≤ k .

Related

• Commutants of Toeplitz Operators with Harmonic Symbols
• Commutants of a certain class of Toeplitz operators
• Commutants of quasihomogeneous Toeplitz operators on the harmonic Bergman space
• Commutant of sum of two quasihomogeneous Toeplitz operators
• Commutants of the sum of two quasihomogeneous Toeplitz operators