Vertical Toeplitz Operators on the Upper Half-Plane and Very Slowly Oscillating Functions

C*-Algebra Generated by Angular Toeplitz Operators on the Weighted Bergman Spaces Over the Upper Half-Plane

“…The following theorem gives a criterion for an operator to be angular and is analogous to the Zorboska result [20] for radial operators and Herrera Yañez, Maximenko, Vasilevski result [12] for vertical operators. For the unweighted Bergman space, this criterion was established in [7].…”

“…(4.1) Therefore, if f ∈ VSO(R), then f | R+ ∈ VSO(R + ), where the class of functions VSO(R + ) was defined in [12] and mentioned in the Introduction. Furthermore, Herrera Yañez, Hutník, and Maximenko [11] have shown that for every σ ∈ VSO(R + ) and ε > 0 there exists b ∈ L ∞ (R + ) such that…”

“…The papers [11,12] continue this program and give a description of the commutative algebra generated by Toeplitz operators with bounded vertical symbols. The result states that their spectral functions form a dense subset in the C * -algebra VSO(R + ) of very slowly oscillating functions on R + , i.e.…”

Toeplitz Operators Defined by Sesquilinear Forms: Bergman Space Case

“…Vertical operators Recall (see [9]) that the vertical operators S (acting on A 2 (Π)) are those that commute with the horizontal translation operators (…”

“…The C * -algebra generated by vertical Toeplitz operators with bounded symbols is isomorphic [9] to the algebra VSO(R + ), which consists of those functions in L ∞ (R + ) that are uniformly continuous with respect to the logarithmic metric ρ(x, y) = |ln x − ln y| .…”