Analytic Function Spaces On Multi-connected Domain And Those Operators

Since Brown-Halmos re-expressed Toeplitz matrix in terms of function theory, close attention has been paid to the study of Toeplitz operator. Probably, because of its close and important connection with control theory, function theory and other branches of science, its has been spread in many different directions, including the simple variable Bergman space, Hardy space and Bergman space on several variable complex space, etc. In the situation of Hardy space and Bergman space on unite disc, Hardy space and Bergman space on unite ball of several variable complex space, the study concerning Toeplitz operators greatly benefit from the theory of reproductive kernel in these spaces. However, in other spaces, such as the situation in Bergman space of some general domains, it is very difficult to write out the concrete expression , Therefore, the application of reproductive kernel is greatly limited. In fact, in general domain and manifold, Toeplitz operator theory is much less rich than unite disc and unite ball. Particularly, there merely is little study of Toeplitz operator of general domain including annulus, and furthermore, the systematic theory has not been developed.One of other important operator in the function spaces is composition operator. Because this kind of operator has natural relation to function theory, and what's more, many problems concerning function theory can be changed into corresponding problems concerning operator theory by means of composition operator, importance has gradually been attached to it. The studies concerning this kind of operator roughly can be classified as boundedness, compactness, spectrum, algebraic properties (such as normality and subnormality) and so on. However, compared with Toeplitz operator and Hankel operator, composition operator theory is far from maturity, still with many problems to solve.In this paper, we will study Toeplitz operator and composition operator as follow. 1. Toeplitz operator and composition operator on general domain and Riemannsurface.(1) Mihaila asked in [28]: Does the invertibility of the composition operator induced by a map between open Riemann surfaces imply the invertibility of the map? In this paper, the problem was negatively answered firstly. Furthermore., in some particular Riemann surfaces, we got positive answer for the problem. We proved as follow, (a) Suppose M and N are open Riemann surfaces, if M is a simply connected hyperbolic Riemann surface, p is an analytic map from M to N, then p is invertible if and only if the composition operator induced by p is invertible. (b) Suppose p is an analytic map from an annulus to an open Riemann surface, then p is invertible if and only if the composition operator induced by p is invertible. (c) Suppose p is an analytic map form the disc removed infinite holes to a Riemann surface, the composition operator induced by p is Fredholm operator, then p is a injection.(2) The properties of Toeplitz operators on the space of square-integrable analytic 1-forms on Riemann surface is studied. We prove as follow, (a) The algebra that is generated by Toeplitz operators with essential bounded symbol of Bergman space on annulus is dense with respect to the weak star topology in the space of all bounded linear operators on the Bergman space, (b) If the dimension of the space of square-integrable analytic 1-forms on Riemann surface is not less than 1, then the spectrum of the Toeplitz operator with analytic symbol is equal to the close hull of the range of the symbol function, (c) If the dimension of the space of square-integrable analytic 1-forms on Riemann surface is equal to infinite, and value of the symbol function is enough small outside the compact sets, then the Toeplitz operator is compact operator, (d) If the Toeplitz operator with continuous symbol of the space of square-integrable analytic 1-forms on the disc removed infinite holes is Fredholm operator, then the index of the Toeplitz operator is the topology degree of symbol function with respect to origin, (e) Suppose G is a boun...