Universal Elements for Non-Linear Operators and Their Applications
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The Rational and Jordan Forms Linear Algebra Notes
The Rational and Jordan Forms Linear Algebra Notes Satya Mandal November 5, 2005 1 Cyclic Subspaces In a given context, a "cyclic thing" is an one generated "thing". For example, a cyclic groups is a one generated group. Likewise, a module M over a ring R is said to be a cyclic module if M is one generated or M = Rm for some m 2 M: We do not use the expression "cyclic vector spaces" because one generated vector spaces are zero or one dimensional vector spaces. 1.1 (De¯nition and Facts) Suppose V is a vector space over a ¯eld F; with ¯nite dim V = n: Fix a linear operator T 2 L(V; V ): 1. Write R = F[T ] = ff(T ) : f(X) 2 F[X]g L(V; V )g: Then R = F[T ]g is a commutative ring. (We did considered this ring in last chapter in the proof of Caley-Hamilton Theorem.) 2. Now V acquires R¡module structure with scalar multiplication as fol- lows: Define f(T )v = f(T )(v) 2 V 8 f(T ) 2 F[T ]; v 2 V: 3. For an element v 2 V de¯ne Z(v; T ) = F[T ]v = ff(T )v : f(T ) 2 Rg: 1 Note that Z(v; T ) is the cyclic R¡submodule generated by v: (I like the notation F[T ]v, the textbook uses the notation Z(v; T ).) We say, Z(v; T ) is the T ¡cyclic subspace generated by v: 4. If V = Z(v; T ) = F[T ]v; we say that that V is a T ¡cyclic space, and v is called the T ¡cyclic generator of V: (Here, I di®er a little from the textbook.) 5. -
Linear Algebra Midterm Exam, April 5, 2007 Write Clearly, with Complete
Mathematics 110 Name: GSI Name: Linear Algebra Midterm Exam, April 5, 2007 Write clearly, with complete sentences, explaining your work. You will be graded on clarity, style, and brevity. If you add false statements to a correct argument, you will lose points. Be sure to put your name and your GSI’s name on every page. 1. Let V and W be finite dimensional vector spaces over a field F . (a) What is the definition of the transpose of a linear transformation T : V → W ? What is the relationship between the rank of T and the rank of its transpose? (No proof is required here.) Answer: The transpose of T is the linear transformation T t: W ∗ → V ∗ sending a functional φ ∈ W ∗ to φ ◦ T ∈ V ∗. It has the same rank as T . (b) Let T be a linear operator on V . What is the definition of a T - invariant subspace of V ? What is the definition of the T -cyclic subspace generated by an element of V ? Answer: A T -invariant subspace of V is a linear subspace W such that T w ∈ W whenever w ∈ W . The T -cyclic subspace of V generated by v is the subspace of V spanned by the set of all T nv for n a natural number. (c) Let F := R, let V be the space of polynomials with coefficients in R, and let T : V → V be the operator sending f to xf 0, where f 0 is the derivative of f. Let W be the T -cyclic subspace of V generated by x2 + 1. -
Cyclic Vectors and Invariant Subspaces for the Backward Shift Operator Annales De L’Institut Fourier, Tome 20, No 1 (1970), P
ANNALES DE L’INSTITUT FOURIER R. G. DOUGLAS H. S. SHAPIRO A. L. SHIELDS Cyclic vectors and invariant subspaces for the backward shift operator Annales de l’institut Fourier, tome 20, no 1 (1970), p. 37-76 <http://www.numdam.org/item?id=AIF_1970__20_1_37_0> © Annales de l’institut Fourier, 1970, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une in- fraction pénale. Toute copie ou impression de ce fichier doit conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Ann. Inst. Fourier, Grenoble 20,1 (1970), 37-76 CYCLIC VECTORS AND INVARIANT SUBSPACES FOR THE BACKWARD SHIFT OPERATOR (i) by R. G. DOUGLAS (2), H. S. SHAPIRO and A.L. SHIELDS 1. Introduction. Let T denote the unit circle and D the open unit disk in the complex plane. In [3] Beurling studied the closed invariant subspaces for the operator U which consists of multiplication by the coordinate function on the Hilbert space H2 = H^D). The operator U is called the forward (or right) shift, because the action of U is to transform a given function into one whose sequence of Taylor coefficients is shifted one unit to the right, that is, its action on sequences is U : (flo,^,^,...) ——>(0,flo,fli ,...). Strictly speaking, of course, the multiplication and the right shift operate on the distinct (isometric) Hilbert spaces H2 and /2. -
Commentary on Thurston's Work on Foliations
COMMENTARY ON FOLIATIONS* Quoting Thurston's definition of foliation [F11]. \Given a large supply of some sort of fabric, what kinds of manifolds can be made from it, in a way that the patterns match up along the seams? This is a very general question, which has been studied by diverse means in differential topology and differential geometry. ... A foliation is a manifold made out of striped fabric - with infintely thin stripes, having no space between them. The complete stripes, or leaves, of the foliation are submanifolds; if the leaves have codimension k, the foliation is called a codimension k foliation. In order that a manifold admit a codimension- k foliation, it must have a plane field of dimension (n − k)." Such a foliation is called an (n − k)-dimensional foliation. The first definitive result in the subject, the so called Frobenius integrability theorem [Fr], concerns a necessary and sufficient condition for a plane field to be the tangent field of a foliation. See [Spi] Chapter 6 for a modern treatment. As Frobenius himself notes [Sa], a first proof was given by Deahna [De]. While this work was published in 1840, it took another hundred years before a geometric/topological theory of foliations was introduced. This was pioneered by Ehresmann and Reeb in a series of Comptes Rendus papers starting with [ER] that was quickly followed by Reeb's foundational 1948 thesis [Re1]. See Haefliger [Ha4] for a detailed account of developments in this period. Reeb [Re1] himself notes that the 1-dimensional theory had already undergone considerable development through the work of Poincare [P], Bendixson [Be], Kaplan [Ka] and others. -
Fm
proceedings OF the AMERICAN MATHEMATICAL SOCIETY Volume 78, Number 1, January 1980 THE INACCESSIBLE INVARIANT SUBSPACES OF CERTAIN C0 OPERATORS JOHN DAUGHTRY Abstract. We extend the Douglas-Pearcy characterization of the inaccessi- ble invariant subspaces of an operator on a finite-dimensional Hubert space to the cases of algebraic operators and certain C0 operators on any Hubert space. This characterization shows that the inaccessible invariant subspaces for such an operator form a lattice. In contrast to D. Herrero's recent result on hyperinvariant subspaces, we show that quasisimilar operators in the classes under consideration have isomorphic lattices of inaccessible in- variant subspaces. Let H be a complex Hubert space. For T in B(H) (the space of bounded linear operators on H) the set of invariant subspaces for T is given the metric dist(M, N) = ||PM — PN|| where PM (PN) is the orthogonal projection on M (N) and "|| ||" denotes the norm in B(H). An invariant subspace M for T is "cyclic" if there exists x in M such that { T"x) spans M. R. G. Douglas and Carl Pearcy [3] have characterized the isolated invariant subspaces for T in the case of finite-dimensional H (see [9, Chapters 6 and 7], for the linear algebra used in this article): An invariant subspace M for T is isolated if and only if M n M, = {0} or M, for every noncyclic summand M, in the primary decomposition for T. In [1] we showed how to view this result as a sharpening of the previously known conditions for the isolation of a solution to a quadratic matrix equation. -
Morse Theory for Codimension-One Foliations
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 298. Number 1. November 1986 MORSE THEORY FOR CODIMENSION-ONE FOLIATIONS STEVEN C. FERRY AND ARTHUR G. WASSERMAN ABSTRACT. It is shown that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every smooth real- valued function on the manifold has a cusp singularity. Introduction. Morse theory is a method of obtaining information about a smooth manifold by analyzing the singularities of a generic real-valued function on the manifold. If the manifold has the additional structure of a foliation, one can then consider as well the singularities of the function restricted to each leaf. The resulting singular set becomes more complicated, higher dimensional, and less trivially placed. In exchange, one obtains information about the. foliation. In this paper we consider codimension-one smooth foliations and show that, in this case, the occurrence of one type of singularity, the cusp, is related to the existence of compact leaves. Specifically, we show that a smooth codimension-one foliation on a compact simply-connected manifold has a compact leaf if and only if every real-valued function on M has a cusp. The singularities of a function on a foliated manifold are very similar to the singularities of a map to the plane. See [F, H-W, L 1, T 1, Wa, W]. However, Levine [L 1] has shown that the only obstruction to finding a map of a compact manifold to the plane without a cusp singularity is the mod 2 Euler characteristic of the manifold. -
1 Symplectic Vector Spaces
Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U. -
SYMPLECTIC GEOMETRY Lecture Notes, University of Toronto
SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes, University of Toronto These are lecture notes for two courses, taught at the University of Toronto in Spring 1998 and in Fall 2000. Our main sources have been the books “Symplectic Techniques” by Guillemin-Sternberg and “Introduction to Symplectic Topology” by McDuff-Salamon, and the paper “Stratified symplectic spaces and reduction”, Ann. of Math. 134 (1991) by Sjamaar-Lerman. Contents Chapter 1. Linear symplectic algebra 5 1. Symplectic vector spaces 5 2. Subspaces of a symplectic vector space 6 3. Symplectic bases 7 4. Compatible complex structures 7 5. The group Sp(E) of linear symplectomorphisms 9 6. Polar decomposition of symplectomorphisms 11 7. Maslov indices and the Lagrangian Grassmannian 12 8. The index of a Lagrangian triple 14 9. Linear Reduction 18 Chapter 2. Review of Differential Geometry 21 1. Vector fields 21 2. Differential forms 23 Chapter 3. Foundations of symplectic geometry 27 1. Definition of symplectic manifolds 27 2. Examples 27 3. Basic properties of symplectic manifolds 34 Chapter 4. Normal Form Theorems 43 1. Moser’s trick 43 2. Homotopy operators 44 3. Darboux-Weinstein theorems 45 Chapter 5. Lagrangian fibrations and action-angle variables 49 1. Lagrangian fibrations 49 2. Action-angle coordinates 53 3. Integrable systems 55 4. The spherical pendulum 56 Chapter 6. Symplectic group actions and moment maps 59 1. Background on Lie groups 59 2. Generating vector fields for group actions 60 3. Hamiltonian group actions 61 4. Examples of Hamiltonian G-spaces 63 3 4 CONTENTS 5. Symplectic Reduction 72 6. Normal forms and the Duistermaat-Heckman theorem 78 7. -
1. Overview Over the Basic Notions in Symplectic Geometry Definition 1.1
AN INTRODUCTION TO SYMPLECTIC GEOMETRY MAIK PICKL 1. Overview over the basic notions in symplectic geometry Definition 1.1. Let V be a m-dimensional R vector space and let Ω : V × V ! R be a bilinear map. We say Ω is skew-symmetric if Ω(u; v) = −Ω(v; u), for all u; v 2 V . Theorem 1.1. Let Ω be a skew-symmetric bilinear map on V . Then there is a basis u1; : : : ; uk; e1 : : : ; en; f1; : : : ; fn of V s.t. Ω(ui; v) = 0; 8i and 8v 2 V Ω(ei; ej) = 0 = Ω(fi; fj); 8i; j Ω(ei; fj) = δij; 8i; j: In matrix notation with respect to this basis we have 00 0 0 1 T Ω(u; v) = u @0 0 idA v: 0 −id 0 In particular we have dim(V ) = m = k + 2n. Proof. The proof is a skew-symmetric version of the Gram-Schmidt process. First let U := fu 2 V jΩ(u; v) = 0 for all v 2 V g; and choose a basis u1; : : : ; uk for U. Now choose a complementary space W s.t. V = U ⊕ W . Let e1 2 W , then there is a f1 2 W s.t. Ω(e1; f1) 6= 0 and we can assume that Ω(e1; f1) = 1. Let W1 = span(e1; f1) Ω W1 = fw 2 W jΩ(w; v) = 0 for all v 2 W1g: Ω Ω We claim that W = W1 ⊕ W1 : Suppose that v = ae1 + bf1 2 W1 \ W1 . We get that 0 = Ω(v; e1) = −b 0 = Ω(v; f1) = a and therefore v = 0. -
The Asymptotic Behavior of the Codimension Sequence of Affine G-Graded Algebras
THE ASYMPTOTIC BEHAVIOR OF THE CODIMENSION SEQUENCE OF AFFINE G-GRADED ALGEBRAS YUVAL SHPIGELMAN Abstract. Let W be an affine PI algebra over a field of characteristic zero graded 1 by a finite group G: We show that there exist α1; α2 2 R, β 2 2 Z, and l 2 N such β n G β n that α1n l ≤ cn (W ) ≤ α2n l . Furthermore, if W has a unit then the asymptotic G β n 1 behavior of cn (W ) is αn l where α 2 R, β 2 2 Z, l 2 N. Introduction Throughout this article F is an algebraically closed field of characteristic zero and W is an affine associative F - algebra graded by a finite group G. We also assume that W is a PI-algebra; i.e., it satisfies an ordinary polynomial identity. In this article we study G-graded polynomial identities of W , and in particular the corresponding G-graded codimension sequence. Let us briefly recall the basic setup. Let XG be a countable G G set of variables fxi;g : g 2 G; i 2 Ng and let F X be the free algebra on the set X . Given a polynomial in F XG we say that it is a G-graded identity of W if it vanishes upon any admissible evaluation on W . That is, a variable xi;g assumes values only from the corresponding homogeneous component Wg. The set of all G-graded identities is an ideal in the free algebra F XG which we denote by IdG(W ). -
Resolving Wild Embeddings of Codimension-One Manifolds in Manifolds of Dimensions Greater Than 3*
Topology and its Applications 24 (1986) 13-40 13 North-Holland RESOLVING WILD EMBEDDINGS OF CODIMENSION-ONE MANIFOLDS IN MANIFOLDS OF DIMENSIONS GREATER THAN 3* Fredric D. ANCEL Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA Received 25 June 1985 Revised 10 January 1986 For ns4, every embedding of an (n-1)-manifold in an n-manifold has a S-resolution for each S > 0. Consequently, for n z 4, every embedding of an (n - I)-manifold in an n-manifold can be approximated by tame embeddings. AMS (MOS) Subj. Class.: Primary 57N45; generalized manifold with boundary 1. Introduction The Codimension-One Tame Approximation Theorem in dimension n states that every embedding of an (n - 1)-manifold in an n-manifold can be approximated by tame embeddings. Bing proved this theorem in dimension 3 in [4], and went on to exploit it to great effect in his study of 3-manifolds. The theorem was established in dimensions 25 by Ancel and Cannon in [3]. It is proved in the remaining dimension, 4, in the present paper. The Codimension-One Tame Approximation Theorem is contained in a more comprehensive proposition which is the principal result of this paper: a Resolution Theorem for Wild Codimension-One Embeddings. The latter theorem is founded on the notion of a b-resolution of a wild embedding. For 6 > 0, a &resolution of a wild embedding e : M + N of a manifold M in a manifold N is, roughly speaking, a cell-like map G : N + N which moves no point of N farther than S and to which is associated a tame embedding f: M + N such that G of= e. -
Locally Flat Imbeddings of Topological Manifolds in Codimension Three
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 157, June 1971 LOCALLY FLAT IMBEDDINGS OF TOPOLOGICAL MANIFOLDS IN CODIMENSION THREE BY GLENN P. WELLERO) Abstract. This paper presents an imbedding theorem for one topological mani- fold M" in another topological manifold Q", provided that the codimension (q —n) is at least three. The result holds even if the manifolds are of the recently discovered non-piecewise-linear type. Denote the boundaries of M and Q by M and Q respec- tively. Suppose that M is 2n—q connected and Q is 2n —q + \ connected. It is then proved that any map /: (M, M) —>(ß, Q) such that f\M is a locally flat imbedding is homotopic relative to M to a proper locally flat imbedding g: M —>Q. It is also shown that if Mis closed and 2n—q + \ connected and Q is 2«—q + 2 connected, then any two homotopic locally flat imbeddings are locally flatly concordant. Introduction. The category of topological manifolds has recently been shown to include manifolds possessing no piecewise linear (P.L.) structure [8]. These non-P.L. manifolds must be investigated in their own right. One approach to the study of topological manifolds is to obtain results analogous to those already known for differentiable and P.L. manifolds. In the study of imbeddings of manifolds, one of the principal results in the P.L. category is the following: Irwin's Imbedding Theorem [7]. If n¿¡q-3, Mn is a compact, 2n-q con- nected P.L. manifold, Qf is a 2n—q+\ connected P.L.