Boo Rim Choe

Commuting Toeplitz operators on the polydisk

We obtain characterizations of (essentially) commuting Toeplitz operators with pluriharmonic symbols on the Bergman space of the polydisk. We show that commuting and essential commuting properties are the same for dimensions bigger than 2, while they are not for dimensions less than or equal to 2. Also, the corresponding results for semi-commutators are obtained.

Toeplitz operators on harmonic Bergman spaces

We study Toeplitz operators on the harmonic Bergman spaces on bounded smooth domains. Two classes of symbols are considered; one is the class of positive symbols and the other is the class of uniformly continuous symbols. For positive symbols, boundedness, compactness, and membership in the Schatten classes are characterized. For uniformly continuous symbols, the essential spectra are described.

Positive Toeplitz operators between the harmonic Bergman spaces

On the setting of the upper half space we study positive Toeplitz operators between the harmonic Bergman spaces. We give characterizations of bounded and compact positive Toeplitz operators taking a harmonic Bergman space bp into another bq for 1<p<∞, 1<q<∞. The case p=1 or q=1 seems more intriguing and is left open for further investigation. Also, we give criteria for positive Toeplitz operators acting on b2 to be in the Schatten classes. Some applications are also included.

Representations and interpolations of harmonic Bergman functions on half-spaces

On the setting of the half-space of the euclidean n-space, we prove representation theorems and interpolation theorems for harmonic Bergman functions in a constructive way. We also consider the harmonic (little) Bloch spaces as limiting spaces. Our results show that well-known phenomena for holomorphic cases continue to hold. Our proofs of representation theorems also yield a uniqueness theorem for harmonic Bergman functions. As an application of interpolation theorems, we give a distance estimate to the harmonic little Bloch space. In the course of the proofs, pseudohyperbolic balls are used as substitutes for Bergman metric balls in the holomorphic case.

Sums of Toeplitz products with harmonic symbols

On the Bergman space of the unit disk, we consider a class of operators which contain sums of finitely many Toeplitz products with harmonic symbols. We give characterizations of when an operator in that class has finite rank or is compact. Our results provide a unified way of treating several known results.

Positive Toeplitz Operators of Schatten–Herz Type

Motivated by a recent work of Loaiza et al. for the holomorphic case on the disk, we introduce and study the notion of Schatten-Herz type Toeplitz operators acting on the harmonic Bergman space of the ball. We obtain characterizations of positive Toeplitz operators of Schatten-Herz type in terms of averaging functions and Berezin transforms of symbol functions. Our characterization in terms of Berezin transforms settles a question posed by Loaizaet al.

Composition operators acting on holomorphic Sobolev spaces

We study the action of composition operators on Sobolev spaces of analytic functions having fractional derivatives in some weighted Bergman space or Hardy space on the unit disk. Criteria for when such operators are bounded or compact are given. In particular, we find the precise range of orders of fractional derivatives for which all composition operators are bounded on such spaces. Sharp results about boundedness and compactness of a composition operator are also given when the inducing map is polygonal.

Bloch-to-BMOA pullbacks on the disk

For a given holomorphic self map (P of the unit disk, we consider the Bloch-to-BMOA composition property (pullback property) of (p. Our results are (1) (p cannot have the pullback property if (p touches the boundary too smoothly, (2) while (p has the pullback property if (p touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.

On the Boundary Behavior of Functions Holomorphic on the Ball

We obtain an improved analogue for general measures of Aleksandrovs theorem concerning the prescription of the radial limits of functions in the Smirnov class on the ball. Considering general measures, we replace the radial limits by the radial maximal functions. We also give an application concerning the canonical factorization of functions in a certain algebra contained in the Smirnov class on the disk.

Fractional derivatives of Bloch functions, growth rate, and interpolation

Bloch functions on the ball are usually described by means of a restriction on the growth rate of ordinary derivatives of holomorphic functions. In this paper we first give a characterization of Bloch functions in terms of fractional derivatives. Then we show that the growth rate suggested by such a characterization is optimal in a certain sense. Also we prove a result concerning interpolating sequences for fractional derivatives of Bloch functions. 0. Introduction and Results Let B be the unit ball of the complex n-space C with norm |z| = 〈z, z〉 where 〈 , 〉 is the usual Hermitian inner product on C. A holomorphic function f on B is said to be a Bloch function if |∇f(z)|(1 − |z|2) is bounded on B where ∇f denotes the complex gradient of f . The space of Bloch functions endowed with norm ||f ||B = |f(0)|+ sup z∈B |∇f(z)|(1− |z|2) is called the Bloch space and denoted by B(B). If f ∈ B(B) satisfies the additional boundary vanishing condition |∇f(z)|(1 − |z|2) → 0 as |z| ↗ 1, we say f ∈ B0(B), the little Bloch space. In this paper we will investigate some properties of the Bloch space in terms of fractional derivatives. Let f be a function holomorphic on B with homogeneous expansion f = ∑∞ k=0 fk. Following [BB], we define the fractional derivative D f of order α > 0 as follows: D f(z) = ∞ ∑