Apr 22, 2018 - Many problems of engineering, physics, and mathematics lead to problems about vector subspaces of a fixed vector space V, where the vector ...
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Vector subspaces depending on variables First draft posted on Research Gate on April 22, 2018 Most recent update: August 4, 2018 In the real world, it is exceptional that what we are observing is constant. Most of time it depends on variables. Many problems of engineering, physics, and mathematics lead to problems about vector subspaces of a fixed vector space V, where the vector subspaces depend on a variable t moving on a curve, or a surface, or more generally, on a manifold M. Such vector subspaces are Vector-Subspace Valued Functions (VSVF) defined on the manifold M. In many cases, such subspaces are generated by a set of vectors depending on the variable t, that is to say, by a set of Vector-Valued Functions (VVF) of t. These VVFs can be continuous, or differentiable, or continuously differentiable, or twice differentiable, or …, infinitely many times differentiable, or analytic, or holomorphic. I call this the Regularity of the VVFs, and it is also the regularity of the VSVF, and the regularity of the manifold M. When the VVFs are represented by a vector-column containing the coordinates of the VVF in a given basis of the fixed vector space V, the regularity of every VVF is also the regularity of every entry in its representation by a vector column. When a VSVF of constant dimension d is defined on a manifold M and is generated by a set of VVFs of a certain regularity R, an important problem is to find a set of d VVFs of the same regularity R such that at every point t of M, they form a basis of the VSF at t. Such a set of d VVFs is a Basis-Valued Function (BVF) of the VSVF. In many cases, we can first find at every point t of M a neighborhood of t where we can find many BVFs of regularity R of the VSVF defined on this neighborhood. A difficult problem is to choose one BVF on every neighborhood in such a way that these BVFs agree on the intersections of neighborhoods and therefore form a BVF of regularity R defined on the whole manifold. Example 1. Many problems lead to a system of linear equations. Such a system can be written in matrix form as AX = B, where A is a given m by n matrix, X is a column matrix n by 1 containing the n unknowns of the system, and B is a given column matrix m by 1. In problems of the real world, it is exceptional that the entries of the given matrices are constant. Most of the time they depend on a variable t moving on a manifold M, and our system becomes A(t)X(t) = B(t) at every point t of M. When the Matrix-Valued Functions (MVF) A and B are of regularity R, an important problem is to find a VVF X solution of regularity R so that the VVF X gives a solution X(t) of the system A(t)X(t) = B(t) at every point t of M. In a first step, we have to find a BVF of the kernel of A, and this kernel is a VSVF. Example 2. In problems involving a square MVF A whose entries are complex-valued functions of t on M, it is very helpful to reduce A to block-diagonal form. For this, we need to find for every eigenvalue of A depending on t a BVFs of the corresponding generalized eigensubspace of A depending on t.
Example 3. I consider a smooth manifold M. I consider a finite-dimensional real vector spaces V. For every point t on the manifold M, I consider a set G(t) = {v1(t), v2(t), …, vn(t)} of n vectors of V. I assume that the vector-valued functions v1, v2, …, vn are smooth functions from the smooth manifold M into the finite-dimensional vector space V. For every point t on the manifold M, I denote by S(t) the vector subspace of V generated by the set G(t) of vectors of V. I assume that the dimension D of S(t) does not depend on t. I consider a positive integer m that does not exceed D. I consider two points a and b on the manifold M. I consider a vector subspace W(a) of dimension m of the vector subspace S(a) of the vector space V. I consider a vector subspace W(b) of dimension m of the vector subspace S(b) of the vector space V. The problem is to find conditions on the type of manifold M such that for every point t on M, we can find a set B(t) = {b1(t), b2(t), …, bm(t)} of m vectors of V satisfying the following conditions: 1. For every point t on the manifold M, the set B(t) is a subset of the vector subspace S(t). 2. For every point t on the manifold M, the vectors of the set of vectors B(t) are linearly independent. 3. The vector-valued functions b1, b2, …, bm from the manifold M into the finite-dimensional vector space V are smooth. 4. The set of vectors B(a) is a basis of the vector subspace W(a). 5. The set of vectors B(b) is a basis of the vector subspace W(b). Many very difficult problems on generalizations of such problems have been solved and published, notably, generalizations from finite-dimensional cases to infinite-dimensional cases. My project is not to find more such results. My project is to make some of the published results easy to learn for everyone who needs them. For this, my idea is to construct a book starting from the easiest such problems, and then consider more and more difficult problems, giving the priority to the results that are needed by the largest number of scientists.
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Über Algebren stetiger Operatorfunktionen I.Z. Gochberg and J. Leiterer Studia Mathematica, Volume 57, issue 1, DOI: 10.4064/sm-57-1-1-26 January 1976 https://www.researchgate.net/publication/266985361_Uber_Algebren_stetiger_Operatorfunktionen https://www.impan.pl/en/publishing-house/journals-and-series/studiamathematica/all/57/1/101261/uber-algebren-stetiger-operatorfunktionen http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.bwnjournal-articlesmv57i1p1bwm?q=bwmeta1.element.bwnjournal-number-sm-1976-57-1;0&qt=CHILDREN-STATELESS Free pdf copy http://matwbn.icm.edu.pl/ksiazki/sm/sm57/sm5711.pdf Topics in Operator Theory Volume 1: Operators, Matrices and Analytic functions Editors Joseph A. Ball Vladimir Bolotnikov J. William Helton Leiba Rodman Series Title: Operator Theory: Advances and Applications 202 ISBN-13: 9783034601573 600 + xxxviii pages, Birkhäuser, 2010 https://www.springer.com/us/book/9783034601573 M.J. Dupr´e, J.F. Glazebrook and E. Previato Curvature of Universal Bundles of Banach Algebras . . . . . . . . . . . . . . . . . . 195--222 Conditions for a vector subspace E(t) and for a projector P(t) not to depend on t Jean-Claude Evard Linear Algebra and its Applications 91:121-131 DOI10.1016/0024-3795(87)90066-8 June 1987 Information on Research Gate: https://www.researchgate.net/publication/242984651_Conditions_for_a_vector_subspace_Et_and_for_a_ projector_Pt_not_to_depend_on_t Full text available on the following Web page of Science Direct: https://www.sciencedirect.com/science/article/pii/0024379587900668 pdf copy: https://ac.els-cdn.com/0024379587900668/1-s2.0-0024379587900668-main.pdf?_tid=3d0ee8e6-9d334843-b6b7-f67c65966729&acdnat=1524458354_523bc45b7fff2bc215b75090f2cecc11 On the existence of bases of class Cp of the kernel and the image of a matrix function Jean-Claude Evard Linear Algebra and its Applications 135(1):33-67 DOI: 10.1016/0024-3795(90)90115-S July 1990 Information on Research Gate: https://www.researchgate.net/publication/242985156_On_the_existence_of_bases_of_class_Cp_of_the_k ernel_and_the_image_of_a_matrix_function Information on the Web site of Science Direct: https://www.sciencedirect.com/science/article/pii/002437959090115S Free pdf copy on the following Web page of Science Direct: https://ac.els-cdn.com/002437959090115S/1-s2.0-002437959090115S-main.pdf?_tid=bd05c30d-cc824b3c-9a29-2870e5d20a50&acdnat=1524459812_c5a27527b5b659a5b9e860e9c55773b3
Differentiable families of subspaces J. Ferrer, M. Garćia, and F. Puerta Linear Algebra and its Applications Volume 199, Supplement 1, pages 229-252 March 1994 Information on the Web site of Science Direct: https://www.sciencedirect.com/science/article/pii/0024379594903514 Free pdf copy on the following Web page of Science Direct: https://ac.els-cdn.com/0024379594903514/1-s2.0-0024379594903514-main.pdf?_tid=29dc7a77-22694f42-b1f6-93a8ccce0f52&acdnat=1524416131_63dfc715256760cb8daa6bc90a06b850 Smooth parametrization of subspaces in a Banach space Maurice Dupre, Jean-Claude Evard, and James Glazebrook Revista de la Unión Matemática Argentina 41(2) January 1998 Information posted on Research Gate: https://www.researchgate.net/publication/268170314_Smooth_parametrization_of_subspaces_in_a_Bana ch_space https://www.semanticscholar.org/ https://pdfs.semanticscholar.org/b38a/e12c32baa9303f4befa3b82bbef03e780faf.pdf The Stiefel bundle of a Banach algebra Maurice Dupre and James Glazebrook Integral Equations and Operator Theory 41(3):264-287 DOI10.1007/BF01203172 September 2001 Full text available on the following Web page of Research Gate: https://www.researchgate.net/publication/226277456_The_Stiefel_bundle_of_a_Banach_algebra Holomorphic Framings for Projections in a Banach Algebra Maurice Dupre and James Glazebrook Georgian Mathematical Journal Volume 9, issue 3, pages 481--494 DOI: 10.1515/GMJ.2002.481 January 2002 Full text available on the following Web pages: https://pdfs.semanticscholar.org/3893/6d871387c08126bbfbfc18a8526b2d3c7f36.pdf https://www.researchgate.net/publication/251456901_Holomorphic_Framings_for_Projections_in_a_Bana ch_Algebra A Banach Algebra Version of the Sato Grassmannian and Commutative Rings of Differential Operators Maurice J Dupre, James F. Glazebrook, and Emma Previato Acta Applicandae Mathematicae 92(3):241-267 Volume 92, issue 3, pages 241--267 DOI: 10.1007/s10440-006-9052-3 October 2006 https://www.researchgate.net/publication/225733075_A_Banach_Algebra_Version_of_the_Sato_Grassma nnian_and_Commutative_Rings_of_Differential_Operators
Under construction Related documents Similarities and equivalences of matrix-valued functions: https://www.researchgate.net/project/Similarities-and-equivalences-of-matrix-valued-functions
Matrix differential equations: https://www.researchgate.net/project/Matrix-Differential-Equations Temporary end of this document
