Narrow Results

On the setting of the unit ball U the author considers Toeplitz operators on Bergman space. The Bergman space B^p(U)(1 ≤ p < ∞) is the closed subspace of the usual Lebesgue space L^p(U) consisting of holomorphic functions. For a function β ∈ L²(U), the Toeplitz operator T_β with symbol β is defined by T_β f = ℘(β f) for function f ∈ B²(U). Here ℘ is the Bergman projection from L²(U) onto B²(U). Two bounded linear operators S, T on the Hilbert H are said to be essentially commuting on H if the commutator ST-TS is compact on H. In this paper, a criterion of essentially Toeplitz operators with the vanishing property is obtained.

In this paper, the following composite analytic functions $F(z)=f(\mathrm{\Phi }(z))$ and $H(z)=G(\mathrm{\Phi }(z),\dots ,\mathrm{\Phi }(z))$ are considered, where $f:ℂ\to ℂ,$$\mathrm{\Phi }:{𝔹}^n\to ℂ,$$G:{ℂ}^m\to ℂ.$ We established conditions which provide equivalence of boundedness of the $l$-index of the function $f$ and boundedness of the $L$-index in joint variables of the function $F,$ where $l:ℂ\to {ℝ}_{+}$ is a continuous function, $L(z)=(l(\mathrm{\Phi }(z))|\frac{\partial \mathrm{\Phi }(z)}{\partial z_1}|,\dots ,$$l(\mathrm{\Phi }(z))\left|\frac{\partial \mathrm{\Phi }(z)}{\partial z_n}\right|).$ For the function $H$ with additional restrictions, the function $L$ is constructed such that $H$ has bounded $L$-index in joint variables in the case when the function $G$ has bounded $L$-index in the direction $1=(1,\dots ,1)$, where $L:{ℂ}^n\to {ℝ}_{+}^n$ is a positive continuous function. Our proofs are based on the application of analog of Hayman’s theorem for these classes of functions.