Some Sobolev Spaces as Pontryagin Spaces
|
|
- Dwight Booth
- 5 years ago
- Views:
Transcription
1 Some Sobolev Spaces as Pontryagin Spaces Vladimir A. Strauss Department of Pure Applied Mathematics, Simón Bolívar University, Sartenejas-Baruta, Apartado 89., Caracas, Venezuela Carsten Trun Technische Universität Ilmenau, Institut für Mathemati, Postfach 565, D Ilmenau, Germany Abstract We show that well nown Sobolev spaces can quite naturally be treated as Pontryagin spaces. This point of view gives a possibility to obtain new properties for some traditional objects such as simplest differential operators. Key words: Function spaces, Pontryagin spaces, selfadjoint operators, spectrum MSC: 47B5, 47E5, 47B25 Introduction Let H be a separable Hilbert space with a scalar product (,. H is said to be an indefinite metric space if it is equipped by a sesquilinear continuous Hermitian form (indefinite inner product [, ] such that the corresponding quadratic form has indefinite sign (i.e. [x, x] taes positive, negative zero values. The indefinite inner product can be represented in the form [, ] = (G,, where G is a so-called Gram operator. The operator G is bounded self-adjoint. If the Gram operator for an indefinite metric space is boundedly invertible its invariant subspace corresponding to the negative spectrum of G is finite-dimensional, lets say κ-dimensional, the space is called a Pontryagin space with κ negative squares. There are a lot of problems in different areas of mathematics, mechanics or physics that can be naturally considered as addresses: str@usb.ve (Vladimir A. Strauss, carsten.trun@tu-ilmenau.de (Carsten Trun.
2 problems in terms of Operator Theory in Pontryagin spaces. We have no aim to give here an overview on this theory its application. We refer only to the stard text boos [,2,] to [4] for a brief introduction. Our scope is a modest illustration of some singular situations that shows an essential difference between Operator Theory in Hilbert spaces in Pontryagin spaces. For this goal we use Sobolev spaces that represents a new approach. 2 Preliminaries A Krein space (K, [, ] is a linear space K which is equipped with an (indefinite inner product (i.e., a hermitian sesquilinear form [.,.] such that K can be written as K = G + [. ]G ( where (G ±, ±[.,.] are Hilbert spaces [. ] means that the sum of G + G is direct [G +, G ] =. The norm topology on a Krein space K is the norm topology of the orthogonal sum of the Hilbert spaces G ± in (. It can be shown that this norm topology is independent of the particular decomposition (; all topological notions in K refer to this norm topology denotes any of the equivalent norms. Krein spaces often arise as follows: In a given Hilbert space (G, (.,., every bounded self-adjoint operator G in G with ρ(g induces an inner product [x, y] := (Gx, y, x, y G, (2 such that (G, [.,.] becomes a Krein space; here, in the decomposition (, we can choose G + as the spectral subspace of G corresponding to the positive spectrum of G G as the spectral subspace of G corresponding to the negative spectrum of G. A subspace L of a linear space K with inner product [.,.] is called non- degenerated if there exists no x L, x, such that [x, L] =, otherwise L is called degenerated; note that a Krein space K is always nondegenerated, but it may have degenerated subspaces. An element x K is called positive (non-negative, negative, non-positive, neutral, respectively if [x, x] > (, <,, =, respectively; a subspace of K is called positive (non-negative, etc., respectively, if all its nonzero elements are positive (nonnegative, etc., respectively. For the definition simple properties of Krein spaces linear operators therein we refer to [2], [3] []. If in some decomposition ( one of the components G ± is of finite dimension, it is of the same dimension in all such decompositions, the Krein space (K, [.,.] is called a Pontryagin space. For the Pontryagin spaces K occurring in this paper, the negative component G is of finite dimension, say κ; in 2
3 this case, K is called a Pontryagin space with κ negative squares. If K arises from a Hilbert space G by means of a self-adjoint operator G with inner product (2, then K is a Pontryagin space with κ negative squares if only if the negative spectrum of the invertible operator G consists of exactly κ eigenvalues, counted according to their multiplicities. In a Pontryagin space K with κ negative squares each non-positive subspace is of dimension κ, a non-positive subspace is maximal non-positive (that is, it is not properly contained in another non-positive subspace if only if it is of dimension κ. If L is a non-degenerated linear space with inner product [.,.] such that for a κ-dimensional subspace L we have [x, x] <, x L, x, but there is no (κ + -dimensional subspace with this property, then there exists a Pontryagin space K with κ negative squares such that L is a dense subset of K. This means that L can be completed to a Pontryagin space in a similar way as a pre-hilbert space can be completed to a Hilbert space. The spectrum of a selfadjoint operator A in a Pontryagin space with κ negative squares is real with the possible exception of at most κ non-real pairs of eigenvalues λ, λ of finite type. We denote by L λ (A the algebraic eigenspace of A at λ. Then dim L λ (A = dim L λ (A the Jordan structure of A in L λ (A in L λ (A is the same. Further the relation κ = λ σ R κ λ (A + λ σ(a C + dim L λ (A holds, where σ denotes the set of all eigenvalues of A with a nonpositive eigenvector κ λ (A denotes the maximal dimension of a nonpositive subspace of L λ (A. Moreover, according to a theorem of Pontryagin, A has a κ-dimensional invariant non-positive subspace L max. If q denotes the minimal polynomial of the resriction A L max, then the polynomial q q, where q (z = q(z, is independent of the particular choice of L max one can show that [q (Aq(Ax, x] for x D(A κ. As a consequence, a selfadjoint operator in a Pontryagin space possesses a spectral function with possible critical points. For details we refer to [,3]. The linear space of bounded linear operators defined on a Pontryagin or Krein space K with values in a Pontryagin or Krein space K 2 is denoted by L(K, K 2. If K := K = K 2 we write L(K. We study linear relations in K, that is, linear subspaces of K 2. The set of all closed linear relations in K is denoted by C(K. Linear operators are viewed as linear relations via their graphs. For the usual definitions of the linear operations with relations the inverse we refer to [7], [8], [9]. We recall only that the multivalued part mul S of a linear relation S is defined by mul S = { ( } y y S. 3
4 Let S be a closed linear relation in K. The resolvent set ρ(s of S is defined as the set of all λ C such that (S λ L(K. The spectrum σ(s of S is the complement of ρ(s in C. The extended spectrum σ(s of S is defined by σ(s = σ(s if S L(K σ(s = σ(s { } otherwise. We set ρ(s := C\ σ(s. The adjoint S + of S is defined as S + := {( h h [f, h] = [f, h ] for all ( f f S }. S is said to be symmetric (selfadjoint if S S + (resp. S = S +. For the description of the selfadjoint extensions of closed symmetric relations we use the so-called boundary value spaces (for the first time the corresponding approach was applied in fact by A.V. Strauss [5], [6] without employing the term boundary value space. Definition Let A be a closed symmetric relation in the Krein space (K, [, ]. We say that {G, Γ, Γ } is a boundary value space for A + if (G, (, is a Hilbert space there exist linear mappings Γ, Γ : A + G such that Γ := ( Γ Γ : A + G G is surjective, the relation [f, g] [f, g ] = (Γ ˆf, Γ ĝ (Γ ˆf, Γ ĝ (3 holds for all ˆf = ( f f, ĝ = ( g g A +. If a closed symmetric relation A has a selfadjoint extension  in K with ρ(â, then there exists a boundary value space {G, Γ, Γ } for A + such that  coincides with er Γ (see [4]. For basic facts on boundary value spaces further references see e.g. [3], [4], [5] [6]. We recall only a few important consequences. For the rest of this section let A be a closed symmetric relation assume that there exists a boundary value space {G, Γ, Γ } for A +. Then A := er Γ A := er Γ (4 are selfadjoint extensions of A. The mapping Γ = ( Γ Γ induces, via A Θ := Γ Θ = { ˆf A + Γ ˆf Θ }, Θ C(G, (5 a bijective correspondence Θ A Θ between C(G the set of closed extensions A Θ A + of A. In particular (5 gives a one-to-one correspondence between the closed symmetric (selfadjoint extensions of A the closed symmetric (resp. selfadjoint relations in G. Moreover, A Θ is an operator if only if Θ Γ {( h h mul A + } = {}. (6 4
5 If Θ is a closed operator in G, then the corresponding extension A Θ of A is determined by A Θ = er ( Γ ΘΓ. (7 Let N λ := er(a + λ = ran (A λ [ ] be the defect subspace of A set ˆN λ := {( f λf f Nλ }. Now we assume that the selfadjoint relation A in (4 has a nonempty resolvent set. For each λ ρ(a the relation A + can be written as a direct sum of (the subspaces A ˆN λ (see [4]. Denote by π the orthogonal projection onto the first component of K 2. The functions λ γ(λ := π (Γ ˆN λ L(G, K, λ ρ(a, λ M(λ := Γ (Γ ˆN λ L(G, λ ρ(a (8 are defined holomorphic on ρ(a are called the γ-field the Weyl function corresponding to A {G, Γ, Γ }. For λ, ζ ρ(a the relation (3 implies M(λ = M(λ γ(ζ = ( + (ζ λ(a ζ γ(λ (9 M(λ M(ζ = (λ ζγ(ζ + γ(λ ( hold (see [4]. Moreover, by [4], we have the following connection between the spectra of extensions of A the Weyl function. Lemma 2 If Θ C(G A Θ is the corresponding extension of A then a point λ ρ(a belongs to ρ(a Θ if only if belongs to ρ(θ M(λ. A point λ ρ(a belongs to σ i (A Θ if only if belongs to σ i (Θ M(λ, i = p, c, r. For λ ρ(a Θ ρ(a the well-nown resolvent formula (A Θ λ = (A λ + γ(λ ( Θ M(λ γ(λ + ( holds (for a proof see e.g. [4]. 3 The Underlying Space Let H,2 (, be the Sobolev space of all absolutely continuous functions f with f L 2 (,. Let be a positive real number, >. We define for 5
6 f, g H,2 (, [f, g] := (f, g L 2 (, (f, g L 2 (,. (2 If L is an arbitrary subset of H,2 (, we set L [ ] := {x H,2 (, : [x, y] = for all y L}. Then we have the following. Proposition 3 For the space (H,2 (,, [.,.] we have the following properties. ( If equals for some n N, then the function g H,2 (,, defined by n 2 π 2 g(x = cos(nπx belongs to the isotropic part of (H,2 (,, [.,.], that is [f, g] = for all f H,2 (,. (2 If >, then (H,2 (,, [.,.] π 2 is a Pontryagin space with one negative square. (3 If for all n N, then (H,2 (,, [.,.] π 2 n 2 π 2 is a Pontryagin space with a finite number of negative squares. Set H := span {f j j 2 π 2, j N}, where f j H,2 (, is defined by f j (x = sin(jπx. Then the number κ of negative squares of (H,2 (,, [.,.] satisfies κ = dim H +. Proof: Assertion ( is an easy calculation. We assume n 2 π 2 for all n N. Define the operator C by D(C := {g H,2 (, g H,2 (, g( = g( = }, C g := g for g D(C. Let us note that the functions f j (x = sin(jπx, j =, 2..., are eigen functions of C. Moreover, each function g in L 2 (, can be written as g = α j f j, j= Let us note that the expression (y (t 2 y(t 2 with t as the time is (up to a constant the Lagrangian for free small oscillations in one dimension (see [2], p. 58 for details. From this point of view the corresponding integral represents the action. 6
7 where α j, j =, 2..., are some constants. For g D(C H, where H denotes the orthogonal complement with respect to the usual scalar product (, L 2 (, but within the Hilbert space H,2 (,, we have also that (f, g L 2 (, = for all f H. Thus, g has the representation g = j> α π j f j, where the sum converges in the norm of L 2 (,. This implies that there exists an ɛ > with (C g, g L 2 (, > ( + ɛ(g, g L 2 (, for all g D(C H. Therefore there exists constants c, c > with [g, g] > c(c g, g L 2 (, + ɛ(g, g L 2 (, c(g, g H,2 (, for g D(C H, so D(C H is a uniformly positive linear manifold. It is easy to see that for f H we have [f, f] <. Let us note, that [f j, f l ] = (j 2 π 2 (f j, f l L 2 (, = (j 2 π 2 δ jl, where δ jl is the symbol of Kronecer. This means that (D(C H [ ] H. Thus, D(C is a pre- Pontryagin space with respect to the product [, ], where the number of negative squares equals dim H. Let us note also that the subspace D(C [ ] is two-dimensional. Indeed, we define h, h 2 H,2 (, by h (x = sin( x + cos( x, h 2 (x = sin( x cos( x z, z 2 H,2 (, by z (x = cos( x, z sin 2 (x = cot( cos( x sin( x. For every g H,2 (, we have [g, z ] = g( [g, z 2 ] = g(. Thus, (D(C [ ] = sp {z, z 2 } = sp {h, h 2 }. Moreover, we have [h, h ] = 2 (sin 2 [h 2, h 2 ] = 2 (sin 2. This proves (3. If > π 2, then H = {} the above considerations imply (2. 7
8 4 A Symmetric Operator Associated to the Second Derivative of Defect Four For the rest of this paper, we assume that is such, that sin. Then, according to Proposition 3, the space (H,2 (,, [.,.] is a Pontryagin space. We consider the following operator A, defined by D(A := {g H,2 (, g, g H,2 (, with g( = g( = g ( = g ( = g ( = g ( = } Ag := g, g D(A. Let us calculate A +. By the definition g D(A + if only if there is g + H,2 (, such that [Af, g] = [f, g + ] for every f D(A. Let f H,2 (, be smooth f(x = for x [, δ (ɛ, ] for some δ, ɛ >. Then the expression f (tg (tdt f (tg(tdt can be considered as an action of the generalized function g IV g on the test function f. From the other h the expression [f, g + ] can be consider as an action of the generalized function (g + g + on the same probe function f. Thus, g IV (t + g (t = (g + (t + g + (t. The latter yields where f, f 2 are defined by g + (t = g (t + αf (t + βf 2 (t, (3 f (x = sin x, f 2 (x = cos x, g H,2 (,. (4 Now the direct calculation shows that the equality [Af, g] = [f, g + ] is fulfilled for every f D(A, if g is under Condition (4 g + is defined by (3. So, A + = {( g g + ( αf +βf 2 g, g H,2 (,, α, β C }. mul A + = sp {f, f 2 }, (5 Lemma 4 Then A is a closed symmetric operator in (H,2 (,, [.,.]. 8
9 Proof: Obviously, A is symmetric. The best way to show the closeness is via the calculation of A ++. We leave it to the reader. Let ( ( f g f +α f +β f 2 g +α 2 f +β 2 f 2 be elements from A + with f, g D(A α, α 2, β, β 2 C. Then we have [ f +α f + β f 2, g] [f, g + α 2 f + β 2 f 2 ] = = fg +f g f g +f g + + (α f 2 β f g (α 2 f 2 β 2 f. We define mappings Γ, Γ : A + C 4 by ( f(+f ( f Γ f +α f +β f 2 = f(+f ( f( f( f ( ( f f Γ f +α f +β f 2 = ( α (α cos β sin for ( f f +α f +β f 2 A +. Theorem 5 The triplet {Γ, Γ } is a boundary value space for A +. In particular A := er Γ is an operator a selfadjoint extension of A, i.e. D(A := {g H,2 (, g, g H,2 (, with g ( = g ( = } A g := g, g D(A. Moreover, for λ ρ(a, the Weyl function is given by M(λ = ( λ λ tan λ tan ( λ λ sin + λ sin tan ( λ λ λ sin + λ sin ( λ tan λ tan sin tan tan sin λ tan sin sin tan λ sin λ sin λ tan Proof: The above calculations imply that {Γ, Γ } is a boundary value space for A +. Let λ C \ R. Define g, g 2 H,2 (, by g (x = cos( λx g 2 (x = sin( λx. (6 Then we have er (A + λ = sp {g, g 2, f, f 2 }. 9
10 Let f = αg + βg 2 + γf + δf 2 for some α, β, γ, δ C. Then Γ ( f λf = Γ ( f f +γ(λ f +δ(λ f 2 = α( λ α( λ cos λ+β( λ sin λ α+δ α cos λ+β sin λ+γ sin +δ cos ( Γ f λf = β λ γ α λ sin λ+β λ cos λ+γ cos δ sin γ (λ γ (λ cos δ (λ sin Now, by (8, it follows that M is of the above form. Now, via (5 we can parameterize all selfadjoint extensions of A via all selfadjoint relations Θ in C 4. Theorem 6 Let Θ be a selfadjoint relation in C 4. Then A Θ is a selfadjoint extension of A. If for all α, β C ( α β sin α cos / mul Θ \ {} (7 holds, then A Θ is an operator. If, in particular, Θ is a selfadjoint matrix, then A Θ is a selfadjoint operator an extension of A with domain D(A Θ := g H,2 (, g, g H,2 (,, ( g ( g ( g(+g ( = Θ g(+g ( g(. g( Proof: Relation (7 follows from (6, (5 the definitions of Γ Γ. If Θ is a matrix, (7 is satisfied the description of D(A Θ follows from (7. 5 A Symmetric Operator Associated to the Second Derivative of Defect Two We start this Section opposite to Section 4. For this we put D(à := {g H,2 (, g, g H,2 (, } (8 Ãg := g, g D(Ã. Thus, the operator à corresponds to the same formal differential expression as the operator considered in the previous section, but with a different domain
11 which is in some sense maximal. Let us calculate Ã+. For f, g D(Ã we have [Ãf, g] = f (tg (tdt + f (tg(tdt = = ( f (tg (t f (tg (t + ( f (tg(t f(tg (t f (tg (tdt + f(tg (tdt = = ( f ( + f( g ( + f ( ( g ( + g( f ( ( g ( + g( + ( f ( + f( g ( + [f, Ãg] Note that the maps f(t ( f ( + f(, f(t f (, f(t ( f ( + f( f(t f ( represent unbounded linear functionals on H,2 (,. Thus, the expression [Ãf, g] gives a continuous linear functional (with respect to f on H,2 (, if only if g ( = ( g ( + g( = ( g ( + g( = g ( = by the definition of the adjoint operator the latter conditions restrict the domain of Ã+. For brevity below we set A: = Ã+. Thus, we have the following operator A, defined by D(A := {g H,2 (, g, g H,2 (, with g ( = g ( =, Ag := g, g( + g ( = g( + g ( = } g D(A. Then A is a closed symmetric operator in (H,2 (,, [.,.], which is, in contrast to Section 4, densely defined. In particular A + = Ã = {( g g g, g H,2 (, } is an operator therefore all selfadjoint extensions of A are operators. We define mappings Γ, Γ : A + C 2 by ( ( Γ f f = f(+f ( f(+f ( ( ( Γ f f = f ( f ( for ( f f A +. Theorem 7 The triplet {Γ, Γ } is a boundary value space for A +. The Weyl function is given by M(λ = λ ( λ tan λ λ ( λ sin λ λ ( λ sin λ λ ( λ tan λ, λ ρ(a. Proof: The above calculations imply that {Γ, Γ } is a boundary value space
12 for A +. Let λ C \ R g, g 2 H,2 (, as in (6. Then we have er (A + λ = sp {g, g 2 }. Let f = αg + βg 2 for some α, β C. Then ( ( ( Γ f λf = f α( λ Γ f = α( λ cos λ+β( λ sin λ ( ( ( Γ f λf = f Γ f = β λ α λ sin λ+β λ cos λ Now, by (8, it is follows that M is of the above form. Lemma 8 The operator A = er Γ is a selfadjoint extension of A with a compact resolvent σ(a = σ p (A = {, π 2, 4π 2, 9π 2, 6π 2,...}. (9 Proof: The operator A = er Γ is selfadjoint in the Hilbert space H,2 (,. We have for f D(A ((A + If, f H,2 (, = f 2 L 2 (, + f 2 H 2,2 (,, where H 2,2 (, is the Sobolev space of all functions f H,2 (, with f H,2 (,. This gives f 2 H 2,2 (, (A + If H,2 (, f H 2,2 (,. Therefore, as the embedding of H 2,2 (, into H,2 (, is compact, the selfadjoint operator A has a compact resolvent. By ( the difference between the resolvents of A A is of finite ran, hence A has a compact resolvent. We have σ(a = σ p (A. Now (9 follows from a simple calculation. Proposition 9 Let α R, α Then the operator A α defined by D(A α := {g H,2 (, g, g H,2 (, with α < 2. (2 αg ( = g( + g ( αg ( = g( + g (} A α g := g, g D(A α. is a selfadjoint extension of A with non-real eigenvalues. 2
13 In the case α = 2 we have that the selfadjoint extension A 2 of A has a Jordan chain of length two corresponding to the eigenvalue. Proof: Set Θ = α.. α Then A Θ = A α, hence, by Lemma 2 the fact that σ(a R (see Lemma 8, we have for all non-real λ that λ σ p (A α if only if = det (M(λ Θ = 2 ( λ 2 + λα2 α 2 ( λ 2 2λ 2 +. (2 2 Hence, λ,2 = α2 2 ± α α are the solutions of Equation (2. Assertion (2 implies now the existence of two non-real eigenvalues of A α. In the case α = 2 we have that the functions h, h D(A 2 given by h (x = e x h (x = x 2 ex satisfy ( A 2 + h = h ( A 2 + h =, i.e. {h, h } is a Jordan chain of A 2 corresponding to the eigenvalue. References [] T.Ya. Azizov, I.S. Iohvidov: Linear Operators in Spaces with an Indefinite Metric, John Wiley & Sons, Ltd., Chichester, 989. [2] J. Bognár: Indefinite Inner Product Spaces, Springer Verlag, New Yor- Heidelberg, 974. [3] V.A. Derach: On Weyl Function Generalized Resolvents of a Hermitian Operator in a Krein Space, Integral Equations Operator Theory 23 (995, [4] V.A. Derach: On Generalized Resolvents of Hermitian Relations in Krein Spaces, J. Math. Sci. (New Yor 97 (999, [5] V.A. Derach, M.M. Malamud: Generalized Resolvents the Boundary Value Problems for Hermitian Operators with Gaps, J. Funct. Anal. 95 (99,
14 [6] V.A. Derach, M.M. Malamud: The Extension Theory of Hermitian Operators the Moment Problem, J. Math. Sci. (New Yor 73 (995, [7] A. Dijsma, H.S.V. de Snoo: Symmetric Selfadjoint Relations in Krein Spaces I, Operator Theory: Advances Applications 24 (987, Birhäuser Verlag Basel, [8] A. Dijsma, H.S.V. de Snoo: Symmetric Selfadjoint Relations in Krein Spaces II, Ann. Acad. Sci. Fenn. Math. 2, (987, [9] M. Haase: The Functional Calculus for Sectorial Operators, Birhäuser Verlag, Basel, 26. [] I.S. Iohvidov, M.G. Krein, H. Langer: Introduction to the Spectral Theory of Operators in Spaces with an Indefinite Metric, Mathematical Research, Vol. 9, Aademie-Verlag, Berlin, 982. [] M.G. Krein, H. Langer: On the Spectral Functions of a Self-Adjoint Operator in a Space with Indefinite Metric, Dol. Aad. Nau SSSR, 52 (963, [2] L.D. Lau, E.M. Lifshitz: Mechanics, Pergamon Press, 96. [3] H. Langer: Spectral Functions of Definitizable Operators in Krein Spaces, in: Functional Analysis Proceedings of a Conference held at Dubrovni, Yugoslavia, November 2-4, 98, Lecture Notes in Mathematics 948, Springer Verlag Berlin-Heidelberg-New Yor (982, pp [4] H. Langer, B. Najman, C. Tretter: Spectral theory of the Klein-Gordon equation in Pontryagin spaces, Comm. Math. Phys. 267 (26, No., [5] A.V. Strauss: On Selfadjoint Extensions in an Orthogonal Sum of Hilbert Spaces, Dol. Aad. Nau SSSR 44 (962, No. 5, [6] A.V. Strauss: Characteristic Functions of Linear Operators, Izv. Aad. Nau SSR, Ser.Mat. 24 (96, No.,
Perturbation Theory for Self-Adjoint Operators in Krein spaces
Perturbation Theory for Self-Adjoint Operators in Krein spaces Carsten Trunk Institut für Mathematik, Technische Universität Ilmenau, Postfach 10 05 65, 98684 Ilmenau, Germany E-mail: carsten.trunk@tu-ilmenau.de
More informationFinite Rank Perturbations of Locally Definitizable Self-adjoint Operators in Krein Spaces. Jussi Behrndt
Finite Rank Perturbations of Locally Definitizable Self-adjoint Operators in Krein Spaces Jussi Behrndt Abstract. In this paper we generalize the well-known result that a definitizable operator in a Krein
More informationTechnische Universität Ilmenau Institut für Mathematik
Technische Universität Ilmenau Institut für Mathematik Preprint No. M 07/22 Accumulation of complex eigenvalues of indefinite Sturm- Liouville operators Behrndt, Jussi; Katatbeh, Qutaibeh; Trunk, Carsten
More informationQUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE. Denitizable operators in Krein spaces have spectral properties similar to those
QUASI-UNIFORMLY POSITIVE OPERATORS IN KREIN SPACE BRANKO CURGUS and BRANKO NAJMAN Denitizable operators in Krein spaces have spectral properties similar to those of selfadjoint operators in Hilbert spaces.
More informationAN OPERATOR THEORETIC APPROACH TO DEGENERATED NEVANLINNA-PICK INTERPOLATION
1 AN OPERATOR THEORETIC APPROACH TO DEGENERATED NEVANLINNA-PICK 1 Introduction INTERPOLATION Harald Woracek Institut für Technische Mathematik Technische Universität Wien A-1040 Wien, Austria In this paper
More informationInfinite-dimensional perturbations, maximally nondensely defined symmetric operators, and some matrix representations
Available online at www.sciencedirect.com ScienceDirect Indagationes Mathematicae 23 (2012) 1087 1117 www.elsevier.com/locate/indag Infinite-dimensional perturbations, maximally nondensely defined symmetric
More informationOn Unitary Relations between Kre n Spaces
RUDI WIETSMA On Unitary Relations between Kre n Spaces PROCEEDINGS OF THE UNIVERSITY OF VAASA WORKING PAPERS 2 MATHEMATICS 1 VAASA 2011 III Publisher Date of publication Vaasan yliopisto August 2011 Author(s)
More informationOn the number of negative eigenvalues of the Laplacian on a metric graph
On the number of negative eigenvalues of the Laplacian on a metric graph Jussi Behrndt Institut für Mathematik, MA 6-4, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany E-mail:
More informationHilbert space methods for quantum mechanics. S. Richard
Hilbert space methods for quantum mechanics S. Richard Spring Semester 2016 2 Contents 1 Hilbert space and bounded linear operators 5 1.1 Hilbert space................................ 5 1.2 Vector-valued
More informationUNBOUNDED OPERATORS ON HILBERT SPACES. Let X and Y be normed linear spaces, and suppose A : X Y is a linear map.
UNBOUNDED OPERATORS ON HILBERT SPACES EFTON PARK Let X and Y be normed linear spaces, and suppose A : X Y is a linear map. Define { } Ax A op = sup x : x 0 = { Ax : x 1} = { Ax : x = 1} If A
More informationCOMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE
COMMON COMPLEMENTS OF TWO SUBSPACES OF A HILBERT SPACE MICHAEL LAUZON AND SERGEI TREIL Abstract. In this paper we find a necessary and sufficient condition for two closed subspaces, X and Y, of a Hilbert
More informationSPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT
SPECTRAL THEOREM FOR SYMMETRIC OPERATORS WITH COMPACT RESOLVENT Abstract. These are the letcure notes prepared for the workshop on Functional Analysis and Operator Algebras to be held at NIT-Karnataka,
More information1. General Vector Spaces
1.1. Vector space axioms. 1. General Vector Spaces Definition 1.1. Let V be a nonempty set of objects on which the operations of addition and scalar multiplication are defined. By addition we mean a rule
More information1 Math 241A-B Homework Problem List for F2015 and W2016
1 Math 241A-B Homework Problem List for F2015 W2016 1.1 Homework 1. Due Wednesday, October 7, 2015 Notation 1.1 Let U be any set, g be a positive function on U, Y be a normed space. For any f : U Y let
More informationLocal Definitizability of T [ ] T and T T [ ]
Local Definitizability of T [ ] T and T T [ ] Friedrich Philipp, André C.M. Ran and Micha l Wojtylak Abstract. The spectral properties of two products AB and BA of possibly unbounded operators A and B
More informationLinear relations and the Kronecker canonical form
Linear relations and the Kronecker canonical form Thomas Berger Carsten Trunk Henrik Winkler August 5, 215 Abstract We show that the Kronecker canonical form (which is a canonical decomposition for pairs
More informationarxiv: v1 [math.sp] 12 Nov 2009
A remark on Schatten von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains arxiv:0911.244v1 [math.sp] 12 Nov 2009 Jussi Behrndt Institut für Mathematik, Technische
More informationBOUNDARY VALUE PROBLEMS IN KREĬN SPACES. Branko Ćurgus Western Washington University, USA
GLASNIK MATEMATIČKI Vol. 35(55(2000, 45 58 BOUNDARY VALUE PROBLEMS IN KREĬN SPACES Branko Ćurgus Western Washington University, USA Dedicated to the memory of Branko Najman. Abstract. Three abstract boundary
More informationKrein s formula and perturbation theory
ISSN: 40-567 Krein s formula and perturbation theory P.Kurasov S.T.Kuroda Research Reports in athematics Number 6, 2000 Department of athematics Stockholm University Electronic versions of this document
More informationAnalysis Preliminary Exam Workshop: Hilbert Spaces
Analysis Preliminary Exam Workshop: Hilbert Spaces 1. Hilbert spaces A Hilbert space H is a complete real or complex inner product space. Consider complex Hilbert spaces for definiteness. If (, ) : H H
More informationLINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM
LINEAR ALGEBRA BOOT CAMP WEEK 4: THE SPECTRAL THEOREM Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F is R or C. Definition 1. A linear operator
More informationScattering Theory for Open Quantum Systems
Scattering Theory for Open Quantum Systems Jussi Behrndt Technische Universität Berlin Institut für Mathematik Straße des 17. Juni 136 D 10623 Berlin, Germany Hagen Neidhardt WIAS Berlin Mohrenstr. 39
More informationCharacterization of half-radial matrices
Characterization of half-radial matrices Iveta Hnětynková, Petr Tichý Faculty of Mathematics and Physics, Charles University, Sokolovská 83, Prague 8, Czech Republic Abstract Numerical radius r(a) is the
More informationSelfadjoint operators in S-spaces
Journal of Functional Analysis 60 011) 1045 1059 www.elsevier.com/locate/jfa Selfadjoint operators in S-spaces Friedrich Philipp a, Franciszek Hugon Szafraniec b, Carsten Trunk a, a Technische Universität
More informationSPECTRAL THEORY EVAN JENKINS
SPECTRAL THEORY EVAN JENKINS Abstract. These are notes from two lectures given in MATH 27200, Basic Functional Analysis, at the University of Chicago in March 2010. The proof of the spectral theorem for
More informationOn Eisenbud s and Wigner s R-matrix: A general approach
On Eisenbud s and Wigner s R-matrix: A general approach J. Behrndt a, H. Neidhardt b, E. R. Racec c, P. N. Racec d, U. Wulf e January 8, 2007 a) Technische Universität Berlin, Institut für Mathematik,
More informationTitle: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on
Title: Localized self-adjointness of Schrödinger-type operators on Riemannian manifolds. Proposed running head: Schrödinger-type operators on manifolds. Author: Ognjen Milatovic Department Address: Department
More informationThe spectral zeta function
The spectral zeta function Bernd Ammann June 4, 215 Abstract In this talk we introduce spectral zeta functions. The spectral zeta function of the Laplace-Beltrami operator was already introduced by Minakshisundaram
More informationHyponormal matrices and semidefinite invariant subspaces in indefinite inner products
Electronic Journal of Linear Algebra Volume 11 Article 16 2004 Hyponormal matrices and semidefinite invariant subspaces in indefinite inner products Christian Mehl Andre C M Ran Leiba Rodman lxrodm@mathwmedu
More informationReview problems for MA 54, Fall 2004.
Review problems for MA 54, Fall 2004. Below are the review problems for the final. They are mostly homework problems, or very similar. If you are comfortable doing these problems, you should be fine on
More informationMath 350 Fall 2011 Notes about inner product spaces. In this notes we state and prove some important properties of inner product spaces.
Math 350 Fall 2011 Notes about inner product spaces In this notes we state and prove some important properties of inner product spaces. First, recall the dot product on R n : if x, y R n, say x = (x 1,...,
More information13. Families of extensions
13.1 13. Families of extensions 13.1 A general parametrization of extensions. We have seen in Chapter 12 that a closed, densely defined symmetric operator S in general has several selfadjoint extensions:
More informationA Brief Introduction to Functional Analysis
A Brief Introduction to Functional Analysis Sungwook Lee Department of Mathematics University of Southern Mississippi sunglee@usm.edu July 5, 2007 Definition 1. An algebra A is a vector space over C with
More informationA note on the σ-algebra of cylinder sets and all that
A note on the σ-algebra of cylinder sets and all that José Luis Silva CCM, Univ. da Madeira, P-9000 Funchal Madeira BiBoS, Univ. of Bielefeld, Germany (luis@dragoeiro.uma.pt) September 1999 Abstract In
More informationISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION
ISOLATED SEMIDEFINITE SOLUTIONS OF THE CONTINUOUS-TIME ALGEBRAIC RICCATI EQUATION Harald K. Wimmer 1 The set of all negative-semidefinite solutions of the CARE A X + XA + XBB X C C = 0 is homeomorphic
More information5 Compact linear operators
5 Compact linear operators One of the most important results of Linear Algebra is that for every selfadjoint linear map A on a finite-dimensional space, there exists a basis consisting of eigenvectors.
More information08a. Operators on Hilbert spaces. 1. Boundedness, continuity, operator norms
(February 24, 2017) 08a. Operators on Hilbert spaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ [This document is http://www.math.umn.edu/ garrett/m/real/notes 2016-17/08a-ops
More informationFive Mini-Courses on Analysis
Christopher Heil Five Mini-Courses on Analysis Metrics, Norms, Inner Products, and Topology Lebesgue Measure and Integral Operator Theory and Functional Analysis Borel and Radon Measures Topological Vector
More informationPerturbation of eigenvalues of Klein-Gordon operators. Ivica Nakić
Perturbation of eigenvalues of Klein-Gordon operators Ivica Nakić Department of Mathematics, University of Zagreb (based on the joint work with K. Veselić) OTIND, Vienna, 17-20 December 2016 Funded by
More informationSJÄLVSTÄNDIGA ARBETEN I MATEMATIK
SJÄLVSTÄNDIGA ARBETEN I MATEMATIK MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET Matching Conditions for the Laplace and Squared Laplace Operator on a Y-graph av Jean-Claude Kiik 2014 - No 1 MATEMATISKA
More informationSectorial Forms and m-sectorial Operators
Technische Universität Berlin SEMINARARBEIT ZUM FACH FUNKTIONALANALYSIS Sectorial Forms and m-sectorial Operators Misagheh Khanalizadeh, Berlin, den 21.10.2013 Contents 1 Bounded Coercive Sesquilinear
More informationContents. Preface for the Instructor. Preface for the Student. xvii. Acknowledgments. 1 Vector Spaces 1 1.A R n and C n 2
Contents Preface for the Instructor xi Preface for the Student xv Acknowledgments xvii 1 Vector Spaces 1 1.A R n and C n 2 Complex Numbers 2 Lists 5 F n 6 Digression on Fields 10 Exercises 1.A 11 1.B Definition
More informationAlgebra II. Paulius Drungilas and Jonas Jankauskas
Algebra II Paulius Drungilas and Jonas Jankauskas Contents 1. Quadratic forms 3 What is quadratic form? 3 Change of variables. 3 Equivalence of quadratic forms. 4 Canonical form. 4 Normal form. 7 Positive
More informationMathematics Department Stanford University Math 61CM/DM Inner products
Mathematics Department Stanford University Math 61CM/DM Inner products Recall the definition of an inner product space; see Appendix A.8 of the textbook. Definition 1 An inner product space V is a vector
More informationLinear Passive Stationary Scattering Systems with Pontryagin State Spaces
mn header will be provided by the publisher Linear Passive Stationary Scattering Systems with Pontryagin State Spaces D. Z. Arov 1, J. Rovnyak 2, and S. M. Saprikin 1 1 Department of Physics and Mathematics,
More informationKREIN S RESOLVENT FORMULA AND PERTURBATION THEORY
J. OPERATOR THEORY 52004), 32 334 c Copyright by Theta, 2004 KREIN S RESOLVENT FORMULA AND PERTURBATION THEORY P. KURASOV and S.T. KURODA Communicated by Florian-Horia Vasilescu Abstract. The difference
More informationReview and problem list for Applied Math I
Review and problem list for Applied Math I (This is a first version of a serious review sheet; it may contain errors and it certainly omits a number of topic which were covered in the course. Let me know
More informationClasses of Linear Operators Vol. I
Classes of Linear Operators Vol. I Israel Gohberg Seymour Goldberg Marinus A. Kaashoek Birkhäuser Verlag Basel Boston Berlin TABLE OF CONTENTS VOLUME I Preface Table of Contents of Volume I Table of Contents
More informationIr O D = D = ( ) Section 2.6 Example 1. (Bottom of page 119) dim(v ) = dim(l(v, W )) = dim(v ) dim(f ) = dim(v )
Section 3.2 Theorem 3.6. Let A be an m n matrix of rank r. Then r m, r n, and, by means of a finite number of elementary row and column operations, A can be transformed into the matrix ( ) Ir O D = 1 O
More informationFunctional Analysis Exercise Class
Functional Analysis Exercise Class Week: January 18 Deadline to hand in the homework: your exercise class on week January 5 9. Exercises with solutions (1) a) Show that for every unitary operators U, V,
More informationSYLLABUS. 1 Linear maps and matrices
Dr. K. Bellová Mathematics 2 (10-PHY-BIPMA2) SYLLABUS 1 Linear maps and matrices Operations with linear maps. Prop 1.1.1: 1) sum, scalar multiple, composition of linear maps are linear maps; 2) L(U, V
More informationTHE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS
THE FORM SUM AND THE FRIEDRICHS EXTENSION OF SCHRÖDINGER-TYPE OPERATORS ON RIEMANNIAN MANIFOLDS OGNJEN MILATOVIC Abstract. We consider H V = M +V, where (M, g) is a Riemannian manifold (not necessarily
More informationWhere is matrix multiplication locally open?
Linear Algebra and its Applications 517 (2017) 167 176 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa Where is matrix multiplication locally open?
More informationChapter 8 Integral Operators
Chapter 8 Integral Operators In our development of metrics, norms, inner products, and operator theory in Chapters 1 7 we only tangentially considered topics that involved the use of Lebesgue measure,
More informationComputations of Critical Groups at a Degenerate Critical Point for Strongly Indefinite Functionals
Journal of Mathematical Analysis and Applications 256, 462 477 (2001) doi:10.1006/jmaa.2000.7292, available online at http://www.idealibrary.com on Computations of Critical Groups at a Degenerate Critical
More informationCHAPTER 6. Representations of compact groups
CHAPTER 6 Representations of compact groups Throughout this chapter, denotes a compact group. 6.1. Examples of compact groups A standard theorem in elementary analysis says that a subset of C m (m a positive
More informationChapter 7. Canonical Forms. 7.1 Eigenvalues and Eigenvectors
Chapter 7 Canonical Forms 7.1 Eigenvalues and Eigenvectors Definition 7.1.1. Let V be a vector space over the field F and let T be a linear operator on V. An eigenvalue of T is a scalar λ F such that there
More informationBoundary triples for Schrödinger operators with singular interactions on hypersurfaces
NANOSYSTEMS: PHYSICS, CHEMISTRY, MATHEMATICS, 2012, 0 0, P. 1 13 Boundary triples for Schrödinger operators with singular interactions on hypersurfaces Jussi Behrndt Institut für Numerische Mathematik,
More informationLECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori
LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.
More informationCOMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna
Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED
More informationNormal Matrices in Degenerate Indefinite Inner Product Spaces
Normal Matrices in Degenerate Indefinite Inner Product Spaces Christian Mehl Carsten Trunk October 25, 2006 Abstract Complex matrices that are structured with respect to a possibl degenerate indefinite
More informationChapter 2 Unbounded Operators and Abstract Functional Calculus
Chapter 2 Unbounded Operators and Abstract Functional Calculus 2.1 Unbounded Operators We start with an introduction to unbounded operators and their spectral theory. This will be brief and cursory so
More informationON A PROBLEM OF ELEMENTARY DIFFERENTIAL GEOMETRY AND THE NUMBER OF ITS SOLUTIONS
ON A PROBLEM OF ELEMENTARY DIFFERENTIAL GEOMETRY AND THE NUMBER OF ITS SOLUTIONS JOHANNES WALLNER Abstract. If M and N are submanifolds of R k, and a, b are points in R k, we may ask for points x M and
More informationInstability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs
Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs Zhiwu Lin and Chongchun Zeng School of Mathematics Georgia Institute of Technology Atlanta, GA 30332, USA Abstract Consider
More informationYour first day at work MATH 806 (Fall 2015)
Your first day at work MATH 806 (Fall 2015) 1. Let X be a set (with no particular algebraic structure). A function d : X X R is called a metric on X (and then X is called a metric space) when d satisfies
More informationOn the simplest expression of the perturbed Moore Penrose metric generalized inverse
Annals of the University of Bucharest (mathematical series) 4 (LXII) (2013), 433 446 On the simplest expression of the perturbed Moore Penrose metric generalized inverse Jianbing Cao and Yifeng Xue Communicated
More informationNormality of adjointable module maps
MATHEMATICAL COMMUNICATIONS 187 Math. Commun. 17(2012), 187 193 Normality of adjointable module maps Kamran Sharifi 1, 1 Department of Mathematics, Shahrood University of Technology, P. O. Box 3619995161-316,
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationUniversität Regensburg Mathematik
Universität Regensburg Mathematik Harmonic spinors and local deformations of the metric Bernd Ammann, Mattias Dahl, and Emmanuel Humbert Preprint Nr. 03/2010 HARMONIC SPINORS AND LOCAL DEFORMATIONS OF
More informationScattering matrices and Weyl functions
Scattering matrices and Weyl functions Jussi Behrndt a, Mark M. Malamud b, Hagen Neidhardt c April 6, 2006 a)technische Universität Berlin, Institut für Mathematik, Straße des 7. Juni 36, D 0623 Berlin,
More informationCHAPTER VIII HILBERT SPACES
CHAPTER VIII HILBERT SPACES DEFINITION Let X and Y be two complex vector spaces. A map T : X Y is called a conjugate-linear transformation if it is a reallinear transformation from X into Y, and if T (λx)
More informationFinding eigenvalues for matrices acting on subspaces
Finding eigenvalues for matrices acting on subspaces Jakeniah Christiansen Department of Mathematics and Statistics Calvin College Grand Rapids, MI 49546 Faculty advisor: Prof Todd Kapitula Department
More informationFoundations of Matrix Analysis
1 Foundations of Matrix Analysis In this chapter we recall the basic elements of linear algebra which will be employed in the remainder of the text For most of the proofs as well as for the details, the
More informationVectors in Function Spaces
Jim Lambers MAT 66 Spring Semester 15-16 Lecture 18 Notes These notes correspond to Section 6.3 in the text. Vectors in Function Spaces We begin with some necessary terminology. A vector space V, also
More informationMP463 QUANTUM MECHANICS
MP463 QUANTUM MECHANICS Introduction Quantum theory of angular momentum Quantum theory of a particle in a central potential - Hydrogen atom - Three-dimensional isotropic harmonic oscillator (a model of
More informationTHE DIRICHLET-TO-NEUMANN MAP FOR SCHRÖDINGER OPERATORS WITH COMPLEX POTENTIALS. Jussi Behrndt. A. F. M. ter Elst
DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2017033 DYNAMICAL SYSTEMS SERIES S Volume 10, Number 4, August 2017 pp. 661 671 THE DIRICHLET-TO-NEUMANN MAP FOR SCHRÖDINGER OPERATORS WITH COMPLEX POTENTIALS
More informationPh.D. Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2) EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified.
PhD Katarína Bellová Page 1 Mathematics 2 (10-PHY-BIPMA2 EXAM - Solutions, 20 July 2017, 10:00 12:00 All answers to be justified Problem 1 [ points]: For which parameters λ R does the following system
More informationKrein s resolvent formula and perturbation theory
Krein s resolvent formula and perturbation theory P. Kurasov and S. T. Kuroda Abstract. The difference between the resolvents of two selfadjoint extensions of a certain symmetric operator A is described
More information3. (a) What is a simple function? What is an integrable function? How is f dµ defined? Define it first
Math 632/6321: Theory of Functions of a Real Variable Sample Preinary Exam Questions 1. Let (, M, µ) be a measure space. (a) Prove that if µ() < and if 1 p < q
More informationMAT 445/ INTRODUCTION TO REPRESENTATION THEORY
MAT 445/1196 - INTRODUCTION TO REPRESENTATION THEORY CHAPTER 1 Representation Theory of Groups - Algebraic Foundations 1.1 Basic definitions, Schur s Lemma 1.2 Tensor products 1.3 Unitary representations
More informationIterative Solution of a Matrix Riccati Equation Arising in Stochastic Control
Iterative Solution of a Matrix Riccati Equation Arising in Stochastic Control Chun-Hua Guo Dedicated to Peter Lancaster on the occasion of his 70th birthday We consider iterative methods for finding the
More informationFunctional Analysis Review
Outline 9.520: Statistical Learning Theory and Applications February 8, 2010 Outline 1 2 3 4 Vector Space Outline A vector space is a set V with binary operations +: V V V and : R V V such that for all
More informationLinear Algebra Massoud Malek
CSUEB Linear Algebra Massoud Malek Inner Product and Normed Space In all that follows, the n n identity matrix is denoted by I n, the n n zero matrix by Z n, and the zero vector by θ n An inner product
More informationRepresentation Theory
Representation Theory Representations Let G be a group and V a vector space over a field k. A representation of G on V is a group homomorphism ρ : G Aut(V ). The degree (or dimension) of ρ is just dim
More informationPatryk Pagacz. Characterization of strong stability of power-bounded operators. Uniwersytet Jagielloński
Patryk Pagacz Uniwersytet Jagielloński Characterization of strong stability of power-bounded operators Praca semestralna nr 3 (semestr zimowy 2011/12) Opiekun pracy: Jaroslav Zemanek CHARACTERIZATION OF
More informationC.6 Adjoints for Operators on Hilbert Spaces
C.6 Adjoints for Operators on Hilbert Spaces 317 Additional Problems C.11. Let E R be measurable. Given 1 p and a measurable weight function w: E (0, ), the weighted L p space L p s (R) consists of all
More informationOrdinary Differential Equations II
Ordinary Differential Equations II February 23 2017 Separation of variables Wave eq. (PDE) 2 u t (t, x) = 2 u 2 c2 (t, x), x2 c > 0 constant. Describes small vibrations in a homogeneous string. u(t, x)
More informationMATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators.
MATH 423 Linear Algebra II Lecture 33: Diagonalization of normal operators. Adjoint operator and adjoint matrix Given a linear operator L on an inner product space V, the adjoint of L is a transformation
More informationSung-Wook Park*, Hyuk Han**, and Se Won Park***
JOURNAL OF THE CHUNGCHEONG MATHEMATICAL SOCIETY Volume 16, No. 1, June 2003 CONTINUITY OF LINEAR OPERATOR INTERTWINING WITH DECOMPOSABLE OPERATORS AND PURE HYPONORMAL OPERATORS Sung-Wook Park*, Hyuk Han**,
More informationThe i/s/o resolvent set and the i/s/o resolvent matrix of an i/s/o system in continuous time
21st International Symposium on Mathematical Theory of Networks and Systems July 7-11, 2014. The i/s/o resolvent set and the i/s/o resolvent matrix of an i/s/o system in continuous time Damir Z. Arov 1,
More informationSPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS
SPECTRAL PROPERTIES OF THE LAPLACIAN ON BOUNDED DOMAINS TSOGTGEREL GANTUMUR Abstract. After establishing discrete spectra for a large class of elliptic operators, we present some fundamental spectral properties
More informationUNBOUNDED OPERATORS ON HILBERT C -MODULES OVER C -ALGEBRAS OF COMPACT OPERATORS
J. OPERATOR THEORY 59:1(2008), 179 192 Copyright by THETA, 2008 UNBOUNDED OPERATORS ON HILBERT C -MODULES OVER C -ALGEBRAS OF COMPACT OPERATORS BORIS GULJAŠ Communicated by Şerban Strătilă ABSTRACT. The
More informationNEVANLINNA-PICK INTERPOLATION: THE DEGENERATED CASE
NEVANLINNA-PICK INTERPOLATION: THE DEGENERATED CASE Harald Woracek Institut für Technische Mathematik Technische Universität Wien A-040 Wien, Austria Introduction Let f be a complex valued function which
More informationChapter 3 Transformations
Chapter 3 Transformations An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Linear Transformations A function is called a linear transformation if 1. for every and 2. for every If we fix the bases
More informationLecture notes: Applied linear algebra Part 1. Version 2
Lecture notes: Applied linear algebra Part 1. Version 2 Michael Karow Berlin University of Technology karow@math.tu-berlin.de October 2, 2008 1 Notation, basic notions and facts 1.1 Subspaces, range and
More informationOn compact operators
On compact operators Alen Aleanderian * Abstract In this basic note, we consider some basic properties of compact operators. We also consider the spectrum of compact operators on Hilbert spaces. A basic
More informationMET Workshop: Exercises
MET Workshop: Exercises Alex Blumenthal and Anthony Quas May 7, 206 Notation. R d is endowed with the standard inner product (, ) and Euclidean norm. M d d (R) denotes the space of n n real matrices. When
More informationLinear Algebra and Dirac Notation, Pt. 2
Linear Algebra and Dirac Notation, Pt. 2 PHYS 500 - Southern Illinois University February 1, 2017 PHYS 500 - Southern Illinois University Linear Algebra and Dirac Notation, Pt. 2 February 1, 2017 1 / 14
More informationOperators with numerical range in a closed halfplane
Operators with numerical range in a closed halfplane Wai-Shun Cheung 1 Department of Mathematics, University of Hong Kong, Hong Kong, P. R. China. wshun@graduate.hku.hk Chi-Kwong Li 2 Department of Mathematics,
More informationMATH 20F: LINEAR ALGEBRA LECTURE B00 (T. KEMP)
MATH 20F: LINEAR ALGEBRA LECTURE B00 (T KEMP) Definition 01 If T (x) = Ax is a linear transformation from R n to R m then Nul (T ) = {x R n : T (x) = 0} = Nul (A) Ran (T ) = {Ax R m : x R n } = {b R m
More information