Toeplitz Operators with Vertical Symbols Acting on the Poly-Bergman Spaces of the Upper Half-Plane. II

In this work Toeplitz operators with vertical symbols and acting on the n -poly-Bergman space A n 2 ( Π ) are studied, where Π ⊂ C is the upper half-plane. A vertical symbol is a bounded measurable function on Π depending only on y = Im z and having limit values at y = 0 , + ∞ . We show that the C ∗ -algebra generated by a finite number of Toeplitz operators with vertical symbols is isomorphic and isometric to the C ∗ -algebra consisting of all the matrix-valued functions M ( x ) ∈ M n ( C ) ⊗ C [ 0 , + ∞ ] such that M (0) and M ( + ∞ ) are scalar matrices. Alternatively, the upper half-plane can be endowed with the affine group structure, where the left-invariant Haar measure d μ = ( 1 / y 2 ) d x d y is taken into account. Then the poly-Bergman space A n 2 ( Π ) can be identified with a wavelet subspace A n - 1 + ⊂ L 2 ( Π , d μ ) . Thus, the study of Toeplitz operators on A n 2 ( Π ) , with vertical symbols, can be carried out on the wavelet space A n - 1 + using representation theory and wavelet analysis, as it is shown below. From this point of view, we also study Toeplitz operators on wavelet spaces on the direct product P = Π n , instead of using poly-analytic function spaces in several complex variables.