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Abstract toeplitz operatörlerin spektral teorisi

Abstract toeplitz operatörlerin spektral teorisi

##### Dosyalar

##### Tarih

1996

##### Yazarlar

Özdemir, Gülşen

##### Süreli Yayın başlığı

##### Süreli Yayın ISSN

##### Cilt Başlığı

##### Yayınevi

Fen Bilimleri Enstitüsü

##### Özet

Bu çalışmada Abstract Toeplitz operatörleri tanımlanmış, örnekler verilmiş ve bazı özellikleri incelenmiştir. Ayrıca Abstract Toeplitz operatörleri, spektral teorisi ve Banach cebrinin genel teorisi bakımından da incelenmiştir. Çalışmanın son kısmında ise, Riesz sistemlerin denkliği tanımlanarak sonsuz sayıda denk olmayan Riesz sistem olduğu gösterilmiştir.

In this work,Spektral theory of Abstract Toeplitz operators are studied. Toeplitz operators are related to different branches of mathematics such as convergence theory of analytic functions, the integral equation, Wiener- Hopf equation and control theory. For this reason, Toeplitz operators have a great importance in the theory of operators. This work contains four section. The first section is introduce section. The second section contains preliminaires about bounded linear operators on Hubert spaces, Banach and operator algebras. Now let us give some fundamental concepts. The set of all measurable complex function on X is denoted by L2(X, v) which satisfy / |/| < oo. The unit circle is denoted by T and the normal- Jx ized measure on T is denoted by y,. In this condition, L2(T, fj.) denotes the st of all measurable functions in the unit circle which satisfy / |/|2 < oo. Jt The orthonormal basis in L2(T,fi) is given as en(0) = ein$, 0 < 8 < 2tt, n = 0, ±1,.. The inner product on L2(T,fi) is defined by (/, 0 : fz(x ? T : \f(x)\ > c) = 0}. Let us define the H2 space H2 = {fe L2(T, fj.) : / f.endfi = 0, Vn < 0} Jt H2 = {fe L2(T,n) : ± J * f.endd = 0, Vn > 0} The functions in H2 space are called analytic functions and this space is called Hardy space. The linear bounded operators space on Hubert space is denoted by B{H). We give some definitions about operators in B(H), and definition of spektral measure and some theorems about it. In the third section,we have studied spektral theory of Abstract Toeplitz operators. Firstly, we give some definitions. Let H be any infinite dimensional Hubert space and let R be a maxsimal abelian von Neumann algebra of operators acting on R. A closed subspace K t^ {0} Ç H is said to be a weak Riesz subspace for R if each 0 ^ / G K is seperating vector for R. By definition this means that if L ? R and Lf = 0, then L = 0. A proper subspace K C H is a Riesz subspace for R if both K and K1- are weak Riesz subspaces for R. A triple (H, R, K) is called a Riesz system if H is an infinite dimen sional Hubert space, R is a maximal abelian von Neumann algebra on H and K C H is a Riesz subspace for R. If (H, İ2, K) is any Riesz system, the linear operator that projects H onto the subspace K is always denoted by P. Furthermore, every operator L ? R is called Abstract Laurent operator and every operator of form PL acting on the Hubert space K is called a Abstract Topelitz operator. Then, we have given some examples of Riesz systems. First, we have showed that (L2,L°°,H2) is a Riesz system. In this section we suppose as given some fixed but arbitrary Riesz system (H, R, K). Recall that P denotes the projection of H onto K. We first obtain some preliminary results that lead to the spectral inclusion theorem. If M is any linear manifold in H, the closure of M is denoted by [M]. Lemma 3.2: E £ is Abstract Laurent operator and [LH\ ^ if, then [LS] = [LK] = [LK-*-]. vi Here, the following result is taken: Corollary 3.3: H is infinite dimensional Hubert space, R is maximal abelian von Neumann algebra on H, then (H,R,K) is a Riesz system if and only if EH = [EK] = [.Elf-1-] for every nonzero projection E < 1 in R. The following proposition is proved by using the above corollary. Proposition 3.5: The maximal abelian von Neumann algebra R contains no minimal projections and the subspaces K and K1- are both infinite di mensional. If A is any operator, we denoted by cr(A) the spektrum of A and by tt(A) the approximate point spektrum of A. If L 6 R (L is Abstract Laurent operator), then the operator PL (PL is Abstract Toeplitz operator) on K is denoted by T&. Teorem 3.6: E L ? R, then a(L) C tt(Tl) C (t(Tl). Then, the spektral radius of an operator A is denoted by r(A). Let (if, R, K) be Riesz system. The collection of Abstract Toeplitz operators is denoted by GT. It is obvious that GT is closed subset of B(K). Let $ : R-^GT be defined by $(L) = TL. Corollary 3.7: The mapping %j} is Hnear,one-to-one,isometric-isomorph and preserves adjoints. To prove Corollary 3.7 is used Teorem 3.6. Corollary 3.8: If L ? R that is not scalar, then L and Tl have no proper value in common. We obtain a corollary about compact Abstract Toeplitz operator. Corollary 3.11: The only compact Abstract Toeplitz operator is zero. The proving of Corollary 3.11 is used Theorem 3.6 and the following Lemma. Lemma 3.9: If a ? B(H) is compact operator, A ^ 0 and A 6 cr(A), then A is proper value of A. Abstract Laurent operator L and its associated Abstract Topelitz opera tor are said to be analytic if LK C K, and to be co-analytic if LK1- C K1-. It follows L is analytic if and only if L* is co-analytic. vu In this section, it is showed when Tl is isometry operator. Theorem 3.14: A Tl is an isometry if and only if L is analytic unitary operator. Let A ? B(H). A is called quasinilpotent, if cr{A) = 0. Let H be algebra. The radical of H is denned by Radii = {A ? H '? (I - AB)-1 ? ft, V.B ? 11, exists }. If Radii = {0}, then H is called semisimple. Lemma 3.15: The collection of analytic Abstract Toeplitz operators with any Riesz system form a semisimple commutative Banach algebra under the operator norm. The collection of analytic Abstract Toeplitz operators is denoted by ACT. Let o~agt(Tl) denote the spektrum of an analytic Abstract Toeplitz operator Tl regarded as an element of AQT. Lemma 3.16: VTL ? AQT, of H onto H\ such that {K) = üTı and such that (frRcf)"1 = R\. Proposition 4.1: If two Riesz system (H,R,K) and (H\, R\, K\ ) are equivalent, then the spaces GT and GT\ are isometrically isomorphic which is given by FTL = T^-i. Let F{n) be any finite subset of the non-negative integer and the sub- space Mp(n) C K spaned by the orthonormal vectors {em0 : n 6 N/F(n) }. MF(n) = V{eine : n ? N/F(n) }. The Riesz system (H,R,MF{n)) is de noted by Rp(n). Lemma 4.2: Hermitian Abstract Toeplitz operator associated with the Riesz system Rf(ti) has a null space whose dimension is at most n and there exists a hermitian Abstract Toeplitz operator whose null space dimension is n. vm Following theorem is proved by the using of Lemma 4.2. It shows that there are a lot of Riesz systems which can be differ. Theorem 4.3: If m and n are distinct possitive integers. The Riesz systems Rp(n) arLd i?F(m) are not equivalent. Furthermore no Riesz system Rf(ti) is equivalent to (L2, L°°,H2).

In this work,Spektral theory of Abstract Toeplitz operators are studied. Toeplitz operators are related to different branches of mathematics such as convergence theory of analytic functions, the integral equation, Wiener- Hopf equation and control theory. For this reason, Toeplitz operators have a great importance in the theory of operators. This work contains four section. The first section is introduce section. The second section contains preliminaires about bounded linear operators on Hubert spaces, Banach and operator algebras. Now let us give some fundamental concepts. The set of all measurable complex function on X is denoted by L2(X, v) which satisfy / |/| < oo. The unit circle is denoted by T and the normal- Jx ized measure on T is denoted by y,. In this condition, L2(T, fj.) denotes the st of all measurable functions in the unit circle which satisfy / |/|2 < oo. Jt The orthonormal basis in L2(T,fi) is given as en(0) = ein$, 0 < 8 < 2tt, n = 0, ±1,.. The inner product on L2(T,fi) is defined by (/, 0 : fz(x ? T : \f(x)\ > c) = 0}. Let us define the H2 space H2 = {fe L2(T, fj.) : / f.endfi = 0, Vn < 0} Jt H2 = {fe L2(T,n) : ± J * f.endd = 0, Vn > 0} The functions in H2 space are called analytic functions and this space is called Hardy space. The linear bounded operators space on Hubert space is denoted by B{H). We give some definitions about operators in B(H), and definition of spektral measure and some theorems about it. In the third section,we have studied spektral theory of Abstract Toeplitz operators. Firstly, we give some definitions. Let H be any infinite dimensional Hubert space and let R be a maxsimal abelian von Neumann algebra of operators acting on R. A closed subspace K t^ {0} Ç H is said to be a weak Riesz subspace for R if each 0 ^ / G K is seperating vector for R. By definition this means that if L ? R and Lf = 0, then L = 0. A proper subspace K C H is a Riesz subspace for R if both K and K1- are weak Riesz subspaces for R. A triple (H, R, K) is called a Riesz system if H is an infinite dimen sional Hubert space, R is a maximal abelian von Neumann algebra on H and K C H is a Riesz subspace for R. If (H, İ2, K) is any Riesz system, the linear operator that projects H onto the subspace K is always denoted by P. Furthermore, every operator L ? R is called Abstract Laurent operator and every operator of form PL acting on the Hubert space K is called a Abstract Topelitz operator. Then, we have given some examples of Riesz systems. First, we have showed that (L2,L°°,H2) is a Riesz system. In this section we suppose as given some fixed but arbitrary Riesz system (H, R, K). Recall that P denotes the projection of H onto K. We first obtain some preliminary results that lead to the spectral inclusion theorem. If M is any linear manifold in H, the closure of M is denoted by [M]. Lemma 3.2: E £ is Abstract Laurent operator and [LH\ ^ if, then [LS] = [LK] = [LK-*-]. vi Here, the following result is taken: Corollary 3.3: H is infinite dimensional Hubert space, R is maximal abelian von Neumann algebra on H, then (H,R,K) is a Riesz system if and only if EH = [EK] = [.Elf-1-] for every nonzero projection E < 1 in R. The following proposition is proved by using the above corollary. Proposition 3.5: The maximal abelian von Neumann algebra R contains no minimal projections and the subspaces K and K1- are both infinite di mensional. If A is any operator, we denoted by cr(A) the spektrum of A and by tt(A) the approximate point spektrum of A. If L 6 R (L is Abstract Laurent operator), then the operator PL (PL is Abstract Toeplitz operator) on K is denoted by T&. Teorem 3.6: E L ? R, then a(L) C tt(Tl) C (t(Tl). Then, the spektral radius of an operator A is denoted by r(A). Let (if, R, K) be Riesz system. The collection of Abstract Toeplitz operators is denoted by GT. It is obvious that GT is closed subset of B(K). Let $ : R-^GT be defined by $(L) = TL. Corollary 3.7: The mapping %j} is Hnear,one-to-one,isometric-isomorph and preserves adjoints. To prove Corollary 3.7 is used Teorem 3.6. Corollary 3.8: If L ? R that is not scalar, then L and Tl have no proper value in common. We obtain a corollary about compact Abstract Toeplitz operator. Corollary 3.11: The only compact Abstract Toeplitz operator is zero. The proving of Corollary 3.11 is used Theorem 3.6 and the following Lemma. Lemma 3.9: If a ? B(H) is compact operator, A ^ 0 and A 6 cr(A), then A is proper value of A. Abstract Laurent operator L and its associated Abstract Topelitz opera tor are said to be analytic if LK C K, and to be co-analytic if LK1- C K1-. It follows L is analytic if and only if L* is co-analytic. vu In this section, it is showed when Tl is isometry operator. Theorem 3.14: A Tl is an isometry if and only if L is analytic unitary operator. Let A ? B(H). A is called quasinilpotent, if cr{A) = 0. Let H be algebra. The radical of H is denned by Radii = {A ? H '? (I - AB)-1 ? ft, V.B ? 11, exists }. If Radii = {0}, then H is called semisimple. Lemma 3.15: The collection of analytic Abstract Toeplitz operators with any Riesz system form a semisimple commutative Banach algebra under the operator norm. The collection of analytic Abstract Toeplitz operators is denoted by ACT. Let o~agt(Tl) denote the spektrum of an analytic Abstract Toeplitz operator Tl regarded as an element of AQT. Lemma 3.16: VTL ? AQT, of H onto H\ such that {K) = üTı and such that (frRcf)"1 = R\. Proposition 4.1: If two Riesz system (H,R,K) and (H\, R\, K\ ) are equivalent, then the spaces GT and GT\ are isometrically isomorphic which is given by FTL = T^-i. Let F{n) be any finite subset of the non-negative integer and the sub- space Mp(n) C K spaned by the orthonormal vectors {em0 : n 6 N/F(n) }. MF(n) = V{eine : n ? N/F(n) }. The Riesz system (H,R,MF{n)) is de noted by Rp(n). Lemma 4.2: Hermitian Abstract Toeplitz operator associated with the Riesz system Rf(ti) has a null space whose dimension is at most n and there exists a hermitian Abstract Toeplitz operator whose null space dimension is n. vm Following theorem is proved by the using of Lemma 4.2. It shows that there are a lot of Riesz systems which can be differ. Theorem 4.3: If m and n are distinct possitive integers. The Riesz systems Rp(n) arLd i?F(m) are not equivalent. Furthermore no Riesz system Rf(ti) is equivalent to (L2, L°°,H2).

##### Açıklama

Tez (Yüksek Lisans) -- İstanbul Teknik Üniversitesi, Fen Bilimleri Enstitüsü, 1996

##### Anahtar kelimeler

Matematik,
Operatörler,
Spektrum analizi,
Mathematics,
Operators,
Spectrum analysis