On the Berezin symbols and Toeplitz operators

Abstract

The present paper mainly gives some new applications of Berezin symbols. In particular, the Berezin symbol is used in approximation problem for 𝐻∞-functions. We study also asymptotic multiplicativity of the Berezin symbols. Moreover, we study the solvability of some Riccati operator equations of the form XAX + XB- CX = D on the Toeplitz algebra 𝜏 , which is the C*-subalgebra of the operator algebra 𝛣(L² ₐ) generated by the Toeplitz operators {T𝑔 : 𝑔 ϵ H∞} on the Bergman space (L² ₐ) (𝔻). We characterizecompactness of truncated Toeplitz operators Aᵩ = PₖɵT ǁ, Kɵ, 𝜑ϵL∞(𝕋), in terms of Berezin symbols. The spectrum of model operators 𝜑(Mɵ), 𝜑 ϵ H∞, is localized in terms of the so-called Berezin set by proving that σ(𝜑(Mѳ)) ⊂ closBer(𝜑(Mѳ)). Reducing subspaces of n-tuple of invertible operators on the Hilbert space H are described.