Toeplitz operatörlerinin cebirsel özellikleri
Algebraic properties of toeplitz operators
- Tez No: 46161
- Danışmanlar: PROF.DR. NAZIM SADIKOV
- Tez Türü: Yüksek Lisans
- Konular: Matematik, Mathematics
- Anahtar Kelimeler: Toeplitz operatörleri, Toeplitz operators
- Yıl: 1995
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: Belirtilmemiş.
Özet
ÖZET 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. iv
Özet (Çeviri)
ALGEBRAIC PROPERTIES OF TOEPLITZ OPERATORS SUMMARY 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 <D: \z\ = 1} in the complex plane ; let a denote the collection of Lebesgue measurable sets in the unit circle ; and let fj, denote the normalized Lebesgue measure, then < T, <x, \i > 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) vNow, 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</? £ L (T, /i), the Laurent operator is defined by Lvf = <p.f, V/?L2(λ and denoted by L^. Let P be the orthogonal projection of L2(T,/j.) onto H2. For \/<p ? L (T, (j,), the Toeplitz operator is defined by Ttpf = PLvf = P{?.f), v/eff2 and denoted by Tv. T^ is the compression of Lv to H2. In the third section, we have studied the theory of Laurent operators. We have given the necessary and sufficient conditions on an operator which is defined on an abstract Hubert space in order that it be equivalent to some Laurent operator Lv on the concrete Hilbert space L2(X, v) and we have proved that A normal operator on an K0 dimensional Hilbert space is equivalent to a Laurent operator if and only if it has no proper values of finite multiplicity. In order to prove this theorem, we made use of the following lemma and its corollaries. The dimension of L2(X, u) is equal to 1 if and only if X is itself an atom. If the dimension of L2(X, u) is infinite i) X can be non-atomic; viii) X can consist of infinitely many atoms; and iii) X can be the union of a non-atomic piece and a set of finitely or infinitely many atoms. If the measure space, < X, ç, 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. In the third section, we finally introduced that the matrix of Lv with respect to the orthonormal basis in L2(T,fj.) (em/9,0 < 6 < 2ir,n = 0, ±1, ±2,...) has a simple form related to <p. In order to describe this relation, we define the Laurent matrix as a bilaterally infinite matrix a,ij such that ai+ij+i =oy, Vi,j (=0,±1,±2,...) Then we have proved the following theorem. An operator on L2(T, p.) is a Laurent operator if and only if its matrix with respect to the orthonormal basis (emö,0 < 6 < 2w,n = 0,±1,±2,...) is a Laurent matrix. If that condition is satisfied, then a,ij = «;_,-, where <p = ]T^ otiei is the Fourier expansion of <p. The main reason of studying Laurent operators was to establish the necessary background in order to study Toeplitz operators. In section 4, we have studied the algebraic, metric and spectral properties of Toeplitz operators by using the concepts which we have studied in section 3. Some of these properties are If cp = 1, then T^ = I. Tv depends linearly on ip. T: <p -> Ty, T(<p) = Tv is (1-1). It is determined that the Toeplitz operator Tv is self-adjoint if and only if tp is real. We have proved that it is also true that Tv is positive if and only if <p is positive. Then we have studied that the matrix of T^ with respect to the or thonormal basis {en : n = 0, 1, 2,...} has a simple form related to ip as Ly has. In order to describe this relation, we define the Toeplitz matrix as a unilaterally infinite matrix g.,j such that ûi+ı.i+1 =ay, Vi',.7 (=0,1,2,...) viiThen we have proved the following theorem. An operator on H2 is a Toeplitz operator if and only if its matrix with respect to the orthonormal basis {ere : n = 0, 1, 2,...} is a Toeplitz matrix. We have studied the problems concerning the analytical properties of Toeplitz operators and the multiplicative properties of Toeplitz operators, and we have indicated that An operator A on H2 is a Toeplitz operator if and only if U*AU = A. ( where U is the unilateral shift operator.) By this theorem, we have determined that the condition U*AU = A characterizes Toeplitz operators. Then we have studied for the condition AU = UA. (The chief point is that U is not unitary). To formulate this, the Toeplitz operator Tv is defined to be analytic (or co-analytic) according as p (or <p ) is analytic and we have proved that An operator on H2 is an analytic ( or co-analytic) Toeplitz operator if and only if it commute with the unilateral shift. ( or the adjoint of the unilateral shift.) Then the questions about when the product of two Toeplitz operators is a Toeplitz operator and when the two Toeplitz operators commute are studied and the following theorems are studied. The product TVT^ of two Toeplitz operators is that either <p be ana lytic or ip be analytic ; if the condition is satisfied, then TVT^ = Tv^ The two Toeplitz operators commute if and only if either is both analytic, or both co-analytic, or one is a linear function of the other. As a corollary of this theorem, we have determined the class of all normal Toeplitz operators and we have indicated that The only normal Toeplitz operators are linear functions of the self- adjoint ones. In this section, finally, the spectrum of the Toeplitz operator Tv is studied and it is proved that If tp is a real- valued essentially bounded measurable function vm(tf ? L (T, /i) ), and if a and /? are respectively the essential lower and essential upper bounds of <p, then a(Ttp) = [a,fl After having studied the properties of Toeplitz operators, in section 5 we have determined the operator algebras which are generated by some Toeplitz operators. Firstly, we have given the fundamental concepts con cerning the properties of Banach algebras. Then we have proved that The operator algebra which is generated by Tz operator and I - identity operator, (p(Tz,I)) is isometrically isomorphic to the disc al gebra, (A(A)). Finally, we have given the structure of the C* -algebra which is generated by a normal Toeplitz operator and I -identity operator, and then we have proved that The C* -algebra which is generated by a normal Toeplitz operator and I -identity operator is isometrically isomorphic to the algebra C{0, 1] of continuous complexed valued functions on [0, 1]. In this work we have made use of the concepts concerning the theory of Hubert spaces, real analysis (i.e. measure, atom), the convergence theory of analytic functions in complex analysis, the theory of Hardy spaces, spectral theory and finally the theory of Banach and C* -algebras. IX
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Başlık çevirisi yok
AYŞE NUR DAYIOĞLU
Yüksek Lisans
Türkçe
1997
MatematikSüleyman Demirel ÜniversitesiMatematik Ana Bilim Dalı
PROF. DR. SERPİL PEHLİVAN