Toeplitz operatörlerinin cebirsel özellikleri

Abstract

Bu çalışmada, Toeplitz operatörlerinin cebirsel özellikleri araştırılmış ve bu operatörleri belirleyen koşullar gösterilmiştir. Bütün normal olan Toeplitz operatörler sınıfı bulunmuştur. En sonunda ise, bazı Toeplitz o- peratörleriyle üretilmiş kendine eş olmayan operatörler cebiri araştırılmış ve normal Toeplitz operatörü ve I -birim operatörü ile üretilmiş C* -cebirinin, [0, 1] kapalı aralığında tanımlanmış ve değerleri kompleks sayılar olan bütün sürekli fonksiyonların cebirine izometrik izomorf ol duğu gösterilmiştir.

In this work, the algebraic properties of Toeplitz operators are studied. The theory of Toeplitz operators, which is first introduced and studied by Toeplitz in 1911, has become increasingly important after the paper of Brown-Halmos in 1962. [ 1 ] Toeplitz operators are strongly related to different branches of math ematics such as the convergence theory of analytic functions, the integral equations, Wiener- Hopf equations and control theory. For this reason, Toeplitz operators have a great importance in the theory of operators. This work contains five sections including the introduction. The in troduction part contains a brief explanation of the contents of this work. The second section contains an account of those basic aspects of bounded linear operators and the topologies on Hilbert space. Now let us give some fundamental concepts concerning this and the following sections. Let < X, ç, v > be a measure space and ç be a finite measure. Let L2(X, v) denote the set of all measurable complex functions on X which satisfy fx | / 12 dv < oo. In the special case, let T denote the unit circle, \z G <="" t,="" is a measure space. In that case, L2(T, fi) denotes the set of all Lebesgue measurable functions in the unit circle which satisfies JT \ f |2 dv < oo. The functions, en(6) which are given as en{6) = ein9, 0 < 0 < 2tt, n = 0, ±1, ±2,... constitute an orthonormal basis in L2(T,(j,). The inner product on L2(T, ft) is defined by (/,?) = J f-ğdfi, Vf,geL2(T,fi) v Now, let L (T,fx) denote the essentially bounded, complexed valued functions in the unit circle, that is the functions / for which the set {xeT:\f(x)\>c} has measure zero for c sufficiently large and, let

/

^ denote the smallest such c. We define the H2 space :, H2 = {feL\T,(x): J f.endfi = 0, Vn < 0}. For V contains at least one atom, then there exists a multiplication operator on L2(X, v) such that the multi plicity of the proper value of this multiplication operator is one. Then, we have studied the characterization of Laurent operators in terms of W, where W is the bilateral shift operator and then we have proved that An operator on i2(T, /i) is a Laurent operator if and only if it com mutes with the bilateral shift operator.