Toeplitz Symbol Calculus On The Unit Sphere

This thesis deals mainly with Berezin transform and Toeplitz operators on the unit sphere S in Cⁿ. In particular, Toeplitz symbol calculus is mainly investigated. Given an operator T in I(L²(S)), Englis constructed its symbol by means of Berezin transform. The main aim of this paper is to generalize Englis's constructive version in Hardy space H²(S).Chapter 1 is concerned with the definitions and the basic properties of reproducing kernel, Cauchy-Szeg(?) kernel (transform), Poisson kernel (transform), Toeplitz operator, Berezin transform, commutator ideal and semicommutator ideal, etc. What's more, we introduce the background of Toeplitz symbol calculus.In Chapter 2, we investigate the properties of Berezin transform and Toeplitz operators on the unit sphere S in Cⁿ. We obtain the symbol of an operator T in the C^*-algebra I(L^∞(S)) generated by {T_φ, φ∈ L^∞(S)} as the nontangential boundary value of a certain function (?) (called the Berezin transform of T) on the unit ball B.In Chapter 3, we obtain interesting generalizations of the classical Toeplitz symbol calculus. One approach for extending the Toeplitz calculus is accomplished by adjoining the set of operators F := {H ∈ (?)(H²(S)) : ||Hk_z|| and ||H^*k_z||→ 0 radially} to the kernel of the symbol map. Then the map η : A₁ → L^∞(S) still enjoys the required properties by denoting A₁ := {T_φ + H : φ ∈ L^∞(S), H ∈ F}. Another approach is constructed by enlarging the class in which the symbols are taken. More precisely, another extended symbol calculus is obtained for the algebra A :={T ∈ B(H²(S)) : ||Tk_z||² - |(Tk_z, k_z)|² and ||T^*k_z||² - |(Tk_z, k_z)|²â†’0 radially} by employing "symbols" from the factor algebra BC(B)/V_?. Here BC(B) stands...