Fractional Integrodifferentiation and Toeplitz Operators with Vertical Symbols - Karapetyants, Alexey (2020)
We consider the so-called vertical Toeplitz operators on the weighted Bergman space over the half plane. The terminology “vertical” is motivated by the fact that if a is a symbol of such Toeplitz operator, then a(z) depends only on y = ℑz, where z = x + iy. The main question raised in this paper can be formulated as follows: given two bounded vertical Toeplitz operators Tλa and Tλb, under which conditions is there a symbol h such that TλaTλb=Tλh? It turns out that this problem has a very nice connection with fractional calculus! We shall formulate our main results using the well-known theory of Riemann–Liouville fractional integrodifferentiation.