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Öteleme operatörlerinin invaryant altuzayları, devirsel vektörleri ve bazı uygulamaları

Başlık çevirisi mevcut değil.

  1. Tez No: 75549
  2. Yazar: TOLGA KÜRKÇÜOĞLU
  3. Danışmanlar: PROF. DR. NAZIM SADIKOV
  4. Tez Türü: Yüksek Lisans
  5. Konular: Matematik, Mathematics
  6. Anahtar Kelimeler: Belirtilmemiş.
  7. Yıl: 1998
  8. Dil: Türkçe
  9. Üniversite: İstanbul Teknik Üniversitesi
  10. Enstitü: Fen Bilimleri Enstitüsü
  11. Ana Bilim Dalı: Matematik Mühendisliği Ana Bilim Dalı
  12. Bilim Dalı: Belirtilmemiş.
  13. Sayfa Sayısı: 74

Özet

Bu çalışmada öteleme operatörünün ve ağırlıklı öteleme operatörünün in- varyant altuzayları, devirsel vektörleri incelenmiştir. Öteleme operatörü Hardy uzayında bağımsız değişkenle çarpım operatörü gibi düşünülerek, bu operatörün invaryant altuzayları fonksiyonel analiz yöntemleri ile incelenmiş ve elde edilen bazı sonuçlar, kompleks analizin bazı teoremlerini fonksiyonel analiz yöntemleri ile ispatlamanın mümkün olduğunu ve sonsuz boyutlu uzayda tek hücreli ope ratörün mevcut olduğunu göstermiştir.

Özet (Çeviri)

In this work, the invariant subspaces and the cyclic vectors of shifts and weighted shifts has been studied. Shift operators, possesing a profound significance in the operator theory, have a wide range of applications into the approximation theory, the study of stochastic processes, the theory of functions and the study of the properties of operators that are not self-adjoint. First introduced by A. I. Plesner [1] in 1939, shift operators are used as a basic tool in the theory of operators. Shift operators have so interesting algebraic and spectral properties that with shift operators, giving much shorter and clearer proofs for many theorems than the classical approach does is possible. Thus, solving some unanswered problems of the general theory of operators specifically for shift operators is by all means, useful for both the inquiry into a general solution and the improvement of the general theory. The problem of invariant spaces is one of the most important unsolved problems in the theory of operators although it has an equally simple statement. For a Hubert space, the statement of the question is“Does every continuous linear operator have a non-trivial invariant subspace in a separable Hilbert space?”,“non-trivial”meaning different from both {0} and the whole space while“in variant”meaning that the operator maps it into itself. The problem has been solved for Banach spaces and construction of an operator that has no invariant subspaces in a Banach space has been prooved to be possible [16, sf. 317-344]. The studies on the invariant subspace problem have been concerned with which operators have invariant subspaces and what the structure of invariant sub- spaces of an operator is if it has any. The first question has been subjected to -a lot of inquisitions [2], [3]. The answer for the latter has been searched most extensively and into depth for the shift operators. Examination of this problem for the unilateral shift operator requires a functional model to be constructed. In this way, a close interaction between complex analysis and functional analy sis has been developed [10]. Using the techniques and methods of one of these branches in the other and vice versa, has provided new results for both branches and has made some much clearer and shorter, yet elegant proofs for known facts possible. This approach was first practiced by A. Beurling in 1949 in his most celebrated work [4] that specified the structure of all the possible invariant sub- spaces of the shift operator, regarding it as the multiplication operator (with the independent variable) in H2 Hardy space. Helson and Lowsdenslager [5] exhibited the same approach and they were followed by many other. In addi tion to this, W. Donoghue [6] and N. Nikolski [7] have examined all possible invariant subspaces of a class of unilateral weighted shifts. Another concept, still important for the study of invariant subspaces, is the cyclic vectors of the operator under examination. By definiton, if an operator has cyclic vectors, these vectors may not lie in an invariant subspace of the operator. For this reason, identification of the cyclic vectors of an operator helps to determine the structure of the invariant subspaces. In this work, the invariant subspaces has been studied with the methods of functional analysis, regarding it as the multiplication operator with the inde pendent variable and some of the acquired results have shown that proving some theorems of complex analysis with the methods of functional analysis is possible and that a unicellular operator exists in an infinite dimensional space. This work consists of an nine sections. The first section, as a brief introduction, includes the historical background and the motivation of this work are stated. In the second section, the definitions needed throughout the work are given, some basic concepts have been reminded of and some propositions and some theorems, important due to how this work handles its subject, are proved. Some of the keyword definitions are stated below. Adjoint of an Operator: Let A ? B{H). The operator denoted by A* satisfying the condition V/,«7?i? (Af,g) = (f,A*g) -is called the adjoint operator of A. Unilateral Shift Operator: If {en : n = 0, 1, 2,...} is an orthonormal base in a Hilbert space H then the operator U defined as Uen = en+i n = 0,l,2,... is called the unilateral shift. Bilateral Shift Operator: If {en : n G 2} is an orthonormal base in a Hubert space H then the operator W defined as Wen = en+i Mn? 2 is called the bilateral shift. Weighted Shift Operator: If {en ; n G 2} is an orthonormal base in a Hilbert space H and {an : n 6 2} is a bounded set of complex numbers then the operator A defined as Aen = anen+ı Vn G 2 is called the bilateral weighted shift. In the similar way, taking {en : n = 0, 1,2,...} and {an : n = 0, 1, 2,...}, the unilateral weighted shift is defined as Ben = a“en+ı n = 0,1,2,... oo Monotone I2 Shift: Let {an} be a monotone sequence such that ŞD a2n < oo, ocn > an+i and for n = 0, 1, 2, 3,... an > 0 The unilateral weighted shift with weight sequence {an} is called a monotone I2 shift. Invariant Subspace: Let H be a Hilbert space, K be a closed subspace of J?, A G B{H) be an operator. If AK C K, i.e. for every k e K, Ak £ K then if is called an invariant subspace of A.Cyclic vector: Let H be a Hilbert space, A ? B(H) be an operator. If the vectors /, A/, A2 f,... span H then the vector / is called a cyclic vector of A. In the third section, the Beurling-Helson theorem whose statement is given below is proved and making use of this theorem, the uniqueness theorem for the class H2 of F. and M. Riesz brothers is deduced as a corollary. Besides, the other corollaries deduced via factorization theorem establish relations between the boundary values of analytic functions and their values in the domain in which they are analytic. 3.1. Beurling-Helson Theorem: Let W be the bilateral shift on L2{T). The neccessary and sufficient condition for a subspace E of L2 (T) be invariant with respect to the operator W is a) If WE = E then there exists a Lebesgue measurable e C T such that E = XeL2 = { ? L2 : 3/ £ 1? - Xef} where Xe is the characteristic function of e. b) If WE -^ E then there exists a measurable function 6 on T with \8\ = 1 almost everywhere such that E = M$H2. 3.8. Corollary: Let U be the unilateral shift on H2(T). If E C #2(T), E ^ {0} and UE C E then E = MeH2 where 9 e H2 \6\ = 1 almost everywhere onT. 3.9. Corollary: If / ? H2(T) and m{C : /(C) = 0, (gT}>0 then / = 0. 3.10. Factorization Theorem: If / G H2 and / 7^ 0 then / can uniquely (up to multiplacative constants) be written as the product / = fifd of an inner factor fi and an outer factor fd. Moreover, Ef = \J{Wnf : n > 0} = MfiH2. 3.17. Corollary: Let / G H2. The following statements are equivalent. a) / is an outer function. b) g e H2 and j e L2 =» j ? H2 3.19. Proposition: If / and g are outer functions with |/| = \g\ almost everywhere then f - Xg with A ? T. 3.20. Proposition: If / 6 L2 then there exists an outer function g with the same modulus | = |/| if and only if W~xEf (JL Ef. In the fourth section, after giving some definitions, the conditions under which a given function / is not cyclic for the adjoint operator of the unilateral shiftare studied and making use of this, some cyclic vectors for the adjoint of the unilateral shift are determined. 4.2. Proposition: Suppose / is meromorphic on A and analytically continu- able across all points of an open arc a on its boundary with the exception of an isolated branch point on a. Then / is not pseudocontinuable across a into any contiguous domain. 4.9. Theorem: Let / Ç. H2(A). The neccessary and sufficient condition for / be non-cyclic for U* is that there exists a meromorphic function / of bounded (Nevanlinna) type in Ae = {z : 1 < \z\ which is the pseudocontinuation of / in Ae across T. 4.17. Corollary: If / £ IT2 (A) and / is analytically continuable across all points of an arc a of T with the exception of an isolated branch point on a, then / is cyclic. In the fifth section, the sufficient conditions for a weighted shift to be unicellular are examined. 5.9. Donoghue's Theorem: Let A be a monotone I2 shift. Then A is unicel lular and any invariant subspace M of A has one of the following structures: I3k£Af M = V{ei}£* II.M = {0} III.M = H The sixth section is a short one in which some of the divisibility properties of inner functions are described. 6.1. Proposition: The necessary and sufficient condition that an inner func tion 4>i divide another inner function 2 is that M^H2 C M^H2. The seventh section includes two theorems and some related corollaries to in vestigate some further details of the relation between the cyclic vectors of the adjoint of the unilateral shift and inner functions. 7.1. Theorem: A necessary and sufficient condition that a function / ? H2 be non-cyclic for U* is that there exist a pair of inner functions (j> ve V> such that -x-almost everywhere on T. 7.8. Theorem: A function / is non-cyclic for U* if and only if there exists g ? H2 and an inner function such that fie”) = e-rt0(c“)*(ert) (7.9.) Moreover, if we require be normalized and relatively prime to the inner factor of g then and g in (7.9.) are uniquely determined. In this case the closed subspace generated by {U*nf : n > 0} is precisely (M^H2)-1. The eigth section in which a weighted Bergman space is under consideration, reveals the structure of the invariant subspaces of the adjoint of the unilateral shift. 8.31. Theorem: Let a > - 1 and na = min{rc GA/”U {0},n > a] and let S denote the set of functions satisfying the condition 0(».+D(l _ |*|*) logt-^-) e L2((l - \z\2)adA) (8.32.) 1 - \z\ in A?a. Assume that the set M ^ A2a is invariant under U* such that M1- = [S](mz,a2)- Then M consists precisely of functions f ? A2a with the properties: i..fae-H^A) for all^eS ii. f/ has a pseudocontinuation to iV+(Ae) that vanishes at infinity where is the greatest inner divisor of S. In the last section, an example of a space in which the effect of the derivative operator can be represented with the adjoint of a weighted unilateral shift is given and the representation is established. Throughout this work, we made use of some concepts of and some information on Hubert and Hardy spaces, theory of operators, real ve complex analysis.

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