1 a Note on Spectral Theory
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1 Bounded and Unbounded Operators
1 1 Bounded and unbounded operators 1. Let X, Y be Banach spaces and D 2 X a linear space, not necessarily closed. 2. A linear operator is any linear map T : D ! Y . 3. D is the domain of T , sometimes written Dom (T ), or D (T ). 4. T (D) is the Range of T , Ran(T ). 5. The graph of T is Γ(T ) = f(x; T x)jx 2 D (T )g 6. The kernel of T is Ker(T ) = fx 2 D (T ): T x = 0g 1.1 Operations 1. aT1 + bT2 is defined on D (T1) \D (T2). 2. if T1 : D (T1) ⊂ X ! Y and T2 : D (T2) ⊂ Y ! Z then T2T1 : fx 2 D (T1): T1(x) 2 D (T2). In particular, inductively, D (T n) = fx 2 D (T n−1): T (x) 2 D (T )g. The domain may become trivial. 3. Inverse. The inverse is defined if Ker(T ) = f0g. This implies T is bijective. Then T −1 : Ran(T ) !D (T ) is defined as the usual function inverse, and is clearly linear. @ is not invertible on C1[0; 1]: Ker@ = C. 4. Closable operators. It is natural to extend functions by continuity, when possible. If xn ! x and T xn ! y we want to see whether we can define T x = y. Clearly, we must have xn ! 0 and T xn ! y ) y = 0; (1) since T (0) = 0 = y. Conversely, (1) implies the extension T x := y when- ever xn ! x and T xn ! y is consistent and defines a linear operator. -
On the Origin and Early History of Functional Analysis
U.U.D.M. Project Report 2008:1 On the origin and early history of functional analysis Jens Lindström Examensarbete i matematik, 30 hp Handledare och examinator: Sten Kaijser Januari 2008 Department of Mathematics Uppsala University Abstract In this report we will study the origins and history of functional analysis up until 1918. We begin by studying ordinary and partial differential equations in the 18th and 19th century to see why there was a need to develop the concepts of functions and limits. We will see how a general theory of infinite systems of equations and determinants by Helge von Koch were used in Ivar Fredholm’s 1900 paper on the integral equation b Z ϕ(s) = f(s) + λ K(s, t)f(t)dt (1) a which resulted in a vast study of integral equations. One of the most enthusiastic followers of Fredholm and integral equation theory was David Hilbert, and we will see how he further developed the theory of integral equations and spectral theory. The concept introduced by Fredholm to study sets of transformations, or operators, made Maurice Fr´echet realize that the focus should be shifted from particular objects to sets of objects and the algebraic properties of these sets. This led him to introduce abstract spaces and we will see how he introduced the axioms that defines them. Finally, we will investigate how the Lebesgue theory of integration were used by Frigyes Riesz who was able to connect all theory of Fredholm, Fr´echet and Lebesgue to form a general theory, and a new discipline of mathematics, now known as functional analysis. -
The Mathematical Sciences at Clemson
BIOMATHEMATICS IS The Geometry of Biological Time m Arthur Winfree, Purdue University The Geometry of Biological Time explains periodic processes in living systems >< and their nonliving analogues in the abstract terms of systems theory. Emphasis is on phase singularities, waves, and mutual synchronization in -n tissues composed of many clocklike units. Also provided are detailed de- )5-._U scriptions of the most commonly used experimental systems, such as electrical oscillations and waves, circadian clocks, the cell division cycle, and the crystal-like regularities observed in the regeneration of severed limbs. z No theoretical background is assumed: required notions are introduced through an extensive collection of illustrations and easily understood o examples. 1979/approx. 576 pp./290 lllus./Cioth $32.00 _ (Biomathematics. Volume 8) ISBN 0-387-09373-7 z Mathematical Population Genetics G) Warren J. Ewens, University of Pennsylvania, Philadelphia Presents the mathematical theory of population genetics with emphasis on those aspects relevant to evolutionary studies. The opening chapter pro- vides an excellent general historical and biological background. Subsequent chapters treat deterministic and stochastic models, discrete and continuous time processes, theory concerning classical and molecular aspects, and one, two, and many loci in a concise and comprehensive manner, with ample references to additional literature. An essential working guide for population geneticists interested in the mathematical foundations of their field and mathematicians involved in genetic evolutionary processes. 1979/ approx. 330 pp./ 4111us/17 Tables/ Cloth $32.00 (Biomathematics. Volume 9) ISBN 0-387-09577-2 Diffusion and Ecological Problems: M~thematical Models Akira Okubo, State University of New York, Stony Brook The first comprehensive book on mathematical models of diffusion in an ecological context. -
Global Existence for Quasilinear Diffusion Equations in Nondivergence Form
GLOBAL EXISTENCE FOR QUASILINEAR DIFFUSION EQUATIONS IN NONDIVERGENCE FORM WOLFGANG ARENDT AND RALPH CHILL A. We consider the quasilinear parabolic equation u β(t, x, u, u)∆u = f (t, x, u, u) t − ∇ ∇ in a cylindrical domain, together with initial-boundary conditions, where the quasilinearity operates on the diffusion coefficient of the Laplacian. Un- der suitable conditions we prove global existence of a solution in the energy space. Our proof depends on maximal regularity of a nonautonomous lin- ear parabolic equation which we use to provide us with compactness in order to apply Schaefer’s fixed point theorem. 1. I We prove global existence of a solution of the quasilinear diffusion prob- lem u β(t, x, u, u)∆u = f (t, x, u, u) in (0, ) Ω, t − ∇ ∇ ∞ × (1) u = 0 in (0, ) ∂Ω, ∞ × u(0, ) = u0( ) in Ω, · · where Ω Rd is an open set, u H1(Ω) and ⊂ 0 ∈ 0 1 β : (0, ) Ω R1+d [ε, ](ε (0, 1) is fixed) and ∞ × × → ε ∈ f : (0, ) Ω R1+d R ∞ × × → are measurable functions which are continuous with respect to the last vari- able, for every (t, x) (0, ) Ω. The function f satisfies in addition a linear growth condition with∈ respect∞ × to the last variable. We prove in fact existence of a solution in the space H1 ([0, ); L2(Ω)) L2 ([0, ); D(∆ )) C([0, ); H1(Ω)), loc ∞ ∩ loc ∞ D ∩ ∞ 0 2 where D(∆D) is the domain of the Dirichlet-Laplacian in L (Ω). Date: October 21, 2008. 2000 Mathematics Subject Classification. 35P05, 35J70, 35K65. -
Dissipative Operators in a Banach Space
Pacific Journal of Mathematics DISSIPATIVE OPERATORS IN A BANACH SPACE GUNTER LUMER AND R. S. PHILLIPS Vol. 11, No. 2 December 1961 DISSIPATIVE OPERATORS IN A BANACH SPACE G. LUMER AND R. S. PHILLIPS 1. Introduction* The Hilbert space theory of dissipative operators1 was motivated by the Cauchy problem for systems of hyperbolic partial differential equations (see [5]), where a consideration of the energy of, say, an electromagnetic field leads to an L2 measure as the natural norm for the wave equation. However there are many interesting initial value problems in the theory of partial differential equations whose natural setting is not a Hilbert space, but rather a Banach space. Thus for the heat equation the natural measure is the supremum of the temperature whereas in the case of the diffusion equation the natural measure is the total mass given by an Lx norm. In the present paper a suitable extension of the theory of dissipative operators to arbitrary Banach spaces is initiated. An operator A with domain ®(A) contained in a Hilbert space H is called dissipative if (1.1) re(Ax, x) ^ 0 , x e ®(A) , and maximal dissipative if it is not the proper restriction of any other dissipative operator. As shown in [5] the maximal dissipative operators with dense domains precisely define the class of generators of strongly continuous semi-groups of contraction operators (i.e. bounded operators of norm =§ 1). In the case of the wave equation this furnishes us with a description of all solutions to the Cauchy problem for which the energy is nonincreasing in time. -
Resolvent and Polynomial Numerical Hull
Helsinki University of Technology Institute of Mathematics Research Reports Espoo 2008 A546 RESOLVENT AND POLYNOMIAL NUMERICAL HULL Olavi Nevanlinna AB TEKNILLINEN KORKEAKOULU TEKNISKA HÖGSKOLAN HELSINKI UNIVERSITY OF TECHNOLOGY TECHNISCHE UNIVERSITÄT HELSINKI UNIVERSITE DE TECHNOLOGIE D’HELSINKI Helsinki University of Technology Institute of Mathematics Research Reports Espoo 2008 A546 RESOLVENT AND POLYNOMIAL NUMERICAL HULL Olavi Nevanlinna Helsinki University of Technology Faculty of Information and Natural Sciences Department of Mathematics and Systems Analysis Olavi Nevanlinna: Resolvent and polynomial numerical hull; Helsinki University of Technology Institute of Mathematics Research Reports A546 (2008). Abstract: Given any bounded operator T in a Banach space X we dis- cuss simple approximations for the resolvent (λ − T )−1, rational in λ and polynomial in T . We link the convergence speed of the approximation to the Green’s function for the outside of the spectrum of T and give an application to computing Riesz projections. AMS subject classifications: 47A10, 47A12, 47A66 Keywords: resolvent operator, spectrum, polynomial numerical hull, numerical range, quasialgebraic operator, Riesz projection Correspondence Olavi Nevanlinna Helsinki University of Technology Institute of Mathematics P.O. Box 1100 FI-02015 TKK Finland olavi.nevanlinna@tkk.fi ISBN 978-951-22-9406-0 (print) ISBN 978-951-22-9407-7 (PDF) ISSN 0784-3143 (print) ISSN 1797-5867 (PDF) Helsinki University of Technology Faculty of Information and Natural Sciences Department of Mathematics and Systems Analysis P.O. Box 1100, FI-02015 TKK, Finland email: math@tkk.fi http://math.tkk.fi/ 1 Introduction Let T be a bounded operator in a complex Banach space X and denote by σ(T ) its spectrum. -
Limiting Absorption Principle for Some Long Range Perturbations of Dirac Systems at Threshold Energies
LIMITING ABSORPTION PRINCIPLE FOR SOME LONG RANGE PERTURBATIONS OF DIRAC SYSTEMS AT THRESHOLD ENERGIES NABILE BOUSSAID AND SYLVAIN GOLENIA´ Abstract. We establish a limiting absorption principle for some long range perturbations of the Dirac systems at threshold energies. We cover multi-center interactions with small coupling constants. The analysis is reduced to studying a family of non-self-adjoint operators. The technique is based on a positive commutator theory for non self-adjoint operators, which we develop in appendix. We also discuss some applications to the dispersive Helmholtz model in the quantum regime. Contents 1. Introduction 1 2. Reduction of the problem 5 2.1. The non self-adjoint operator 5 2.2. From one limiting absorption principle to another 7 3. Positive commutator estimates. 9 4. Main result 12 Appendix A. Commutator expansions. 14 Appendix B. A non-selfadjoint weak Mourre theory 16 Appendix C. Application to non-relativistic dispersive Hamiltonians 19 References 20 1. Introduction We study properties of relativistic massive charged particles with spin-1=2 (e.g., electron, positron, (anti-)muon, (anti-)tauon,:::). We follow the Dirac formalism, see [17]. Because of the spin, the configu- ration space of the particle is vector valued. To simplify, we consider finite dimensional and trivial fiber. Let ≥ 2 be an integer. The movement of the free particle is given by the Dirac equation, @' i = D '; in L2( 3; 2 ); ~ @t m R C where m > 0 is the mass, c the speed of light, ~ the reduced Planck constant, and 3 2 X 2 (1.1) Dm := c~ ⋅ P + mc = −ic~ k@k + mc : k=1 Here we set := ( 1; 2; 3) and := 4. -
Spectrum (Functional Analysis) - Wikipedia, the Free Encyclopedia
Spectrum (functional analysis) - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Spectrum_(functional_analysis) Spectrum (functional analysis) From Wikipedia, the free encyclopedia In functional analysis, the concept of the spectrum of a bounded operator is a generalisation of the concept of eigenvalues for matrices. Specifically, a complex number λ is said to be in the spectrum of a bounded linear operator T if λI − T is not invertible, where I is the identity operator. The study of spectra and related properties is known as spectral theory, which has numerous applications, most notably the mathematical formulation of quantum mechanics. The spectrum of an operator on a finite-dimensional vector space is precisely the set of eigenvalues. However an operator on an infinite-dimensional space may have additional elements in its spectrum, and may have no eigenvalues. For example, consider the right shift operator R on the Hilbert space ℓ2, This has no eigenvalues, since if Rx=λx then by expanding this expression we see that x1=0, x2=0, etc. On the other hand 0 is in the spectrum because the operator R − 0 (i.e. R itself) is not invertible: it is not surjective since any vector with non-zero first component is not in its range. In fact every bounded linear operator on a complex Banach space must have a non-empty spectrum. The notion of spectrum extends to densely-defined unbounded operators. In this case a complex number λ is said to be in the spectrum of such an operator T:D→X (where D is dense in X) if there is no bounded inverse (λI − T)−1:X→D. -
Functional Analysis TMA401/MMA400
Functional analysis TMA401/MMA400 Peter Kumlin 2018-09-21 1 Course diary — What has happened? Week 1 Discussion of introductory example, see section 1. Definition of real/complex vector space, remark on existence of unique zero vector and inverse vectors, example of real vector spaces (sequence spaces and function spaces). Hölder and Minkowski inequal- ities. Introducing (the to all students very well-known concepts) linear combination, linear independence, span of a set, (vector space-) basis (= Hamel basis) with examples. All vector spaces have basis (using Axiom of choice/Zorn’s lemma; it was not proven but stated). Introducing norms on vector spaces with examples, equivalent norms, con- vergence of sequences in normed spaces, showed that C([0; 1]) can be equipped with norms that are not equivalent. Stated and proved that all norms on finite-dimensional vector spaces are equivalent. A proof of this is supplied below. Also mentioned that all infinte-dimensional vector spaces can be equipped with norms that are not equivalent (easy to prove once we have a Hamel basis). Theorem 0.1. Suppose E is a vector space with dim(E) < 1. Then all norms on E are equivalent. Proof: We observe that the relation that two norms are equivalent is transitive, so it is enough to show that an arbitrary norm k · k on E is equivalent to a fixed norm k · k∗ on E. Let x1; x2; : : : ; xn, where n = dim(E), be a basis for E. This means that for every x 2 E there are uniquely defined scalars αk(x), k = 1; 2; : : : ; n, such that x = α1(x)x1 + α2(x)x2 + ::: + αn(x)xn: Set kxk∗ = jα1(x)j + jα2(x)j + ::: jαn(x)j for x 2 E. -
Hermitian Geometry on Resolvent Set
Hermitian geometry on resolvent set (I) Ronald G. Douglas and Rongwei Yang Abstract. For a tuple A = (A1, A2, ..., An) of elements in a unital Banach n algebra B, its projective joint spectrum P (A) is the collection of z ∈ C such that A(z)= z1A1 + z2A2 + · · · + znAn is not invertible. It is known that the −1 B-valued 1-form ωA(z) = A (z)dA(z) contains much topological informa- c tion about the joint resolvent set P (A). This paper studies geometric properties c of P (A) with respect to Hermitian metrics defined through the B-valued fun- ∗ damental form ΩA = −ωA ∧ ωA and its coupling with faithful states φ on B, i.e. φ(ΩA). The connection between the tuple A and the metric is the main subject of this paper. In particular, it shows that the K¨ahlerness of the metric is tied with the commutativity of the tuple, and its completeness is related to the Fuglede-Kadison determinant. Mathematics Subject Classification (2010). Primary 47A13; Secondary 53A35. Keywords. Maurer-Cartan form, projective joint spectrum, Hermitian metric, arXiv:1608.05990v4 [math.FA] 27 Feb 2020 Ricci tensor, Fuglede-Kadison determinant. 0. Introduction In [4], Cowen and the first-named author introduced geometric concepts such as holomorphic bundle and curvature into Operator Theory. This gave rise to complete and computable invariants for certain operators with rich spectral structure, i.e, the Cowen-Douglasoperators. The idea was extended to multivariable cases in the study of Hilbert modules in analytic function spaces, where curvature invariant is defined for some canonical tuples of commuting operators. -
Fractional Powers of Non-Densely Defined Operators
ANNALI DELLA SCUOLA NORMALE SUPERIORE DI PISA Classe di Scienze CELSO MARTINEZ MIGUEL SANZ Fractional powers of non-densely defined operators Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4e série, tome 18, no 3 (1991), p. 443-454 <http://www.numdam.org/item?id=ASNSP_1991_4_18_3_443_0> © Scuola Normale Superiore, Pisa, 1991, tous droits réservés. L’accès aux archives de la revue « Annali della Scuola Normale Superiore di Pisa, Classe di Scienze » (http://www.sns.it/it/edizioni/riviste/annaliscienze/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisa- tion commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Fractional Powers of Non-densely Defined Operators CELSO MARTINEZ - MIGUEL SANZ* 1. - Introduction and notation In [1] Balakrishnan extended the concept of fractional powers to closed linear operators A, defined on a Banach space X, such that ) - oo, 0[ is included in the resolvent set p(A) and the resolvent operator satisfies (Following Komatsu’s terminology in [10], we shall call these operators non- negative). Balakrishnan defined the power with base A and exponent a complex number a (Re a > 0) as the closure of a closable operator, JA, whose expression is: and and For ) and , For ) and , * Partially supported by D.G.C.Y.T., grant PS88-0115. Spain. Pervenuto alla Redazione il 25 Giugno 1990 e in forma definitiva il 13 Marzo 1991. -
Semigroups of Linear Operators
Semigroups of Linear Operators Sheree L. LeVarge December 4, 2003 Abstract This paper will serve as a basic introduction to semigroups of linear operators. It will define a semigroup in the context of a physical problem which will serve to motivate further (elementary) theoretical development of linear semigroups including the Hille- Yosida Theorem. Applications and examples will also be discussed. 1 Introduction Before defining what a semigroup is, one needs to recognize their global importance. Of course their importance cannot be fully realized until we have a clear definition and developed theory. However, in general, semigroups can be used to solve a large class of problems commonly known as evolution equations. These types of equations appear in many disciplines including physics, chemistry, biology, engineering, and economics. They are usually described by an initial value problem (IVP) for a differential equation which can be ordinary or partial. When we view the evolution of a system in the context of semigroups we break it down into transitional steps (i.e. the system evolves from state A to state B, and then from state B to state C). When we recognize that we have a semigroup, instead of studying the IVP directly, we can study it via the semigroup and its applicable theory. The theory of linear semigroups is very well developed [1]. For example, linear semigroup theory actually provides necessary and sufficient conditions to determine the well-posedness of a problem [3]. There is also theory for nonlinear semigroups which this paper will not address. This paper will focus on a special class of linear semigroups called C0 semigroups which are semigroups of strongly continuous bounded linear operators.