Theory Of Pontryagin Spaces - TU Wien

A GUIDE TO THE READERix– R is the set of real numbers, and C is the set of complex numbers.– D is the open unit disk {x C : x 1}, and T is the unit circle {x C : x 1}.– C is the open upper half plane {x C : Im x 0}, and C is the open lower halfplane {x C : Im x 0}.Functions, and Sets of Functions:– The characteristic function of a set M is denoted as 1 M .– For x R we denote by ⌊x⌋ the largest integer not exceeding x.– For an open subset Ω we denote by H(Ω) the set of all complex-valued functiondefined and analytic on Ω. If M is an arbitrary subset of C, we denote by H(M) theset of all complex-valued function defined and analytic on some open set containingM.– For a subset M of C, we denote by M the set {x C : x M}. For acomplex-valued function defined on some subset of C, we denote by f the functiondefined on M and acting asf (x) : f (x),x M .Linear Algebra:– For n N we denote by Cn the set of all n-vectors with complex entries. We writean n-vector as the column of its entries. For n, m N we denote by Cn m the set ofall n m-matrices with complex entries (n rows and m columns).– For a matrix A (αi j ) i 1,.,n Cn m , we denote by At Cm n the transpose of A,j 1,.,mand by A Cm n the conjugate transpose of A. In formulas, that isAt (βi j )i 1,.,m ,βi j : α ji , i 1, . . . , m, j 1, . . . , n .j 1,.,nA (γi j )i 1,.,m ,βi j : γ ji , i 1, . . . , m, j 1, . . . , n .j 1,.,n– Unless the contrary is explicitly stated all linear spaces are understood over thescalar field C.– For a linear space V we denote by dim V its dimension. Unless the contrary isexplicitly stated all dimensions are understood as a nonnegative integer or , anddifferent cardinalities of infinity are not distinguished.– For a subset M of a linear space V, we denote by span M the linear span of M.This is the smallest linear subspace of V which contains M.– If L and M are linear subspaces of a linear space V, then L ̇M is the sum of Land M. If L and M satisfy L M {0}, we write L ̇M and speak of the directsum of L and M.– For a linear map ϕ : V W we denote ran ϕ : y W : x A : ϕx y ,and speak of the range and kernel of ϕ. ker ϕ : x A : ϕx 0 ,

xA GUIDE TO THE READERMeasures:– If µ is a complex measure, we denote by µ its total variation.– If µ is a positive measure, we denote by supp µ its support (which is the smallestclosed subset of R whose complement is a µ -zero set).

Chapter 1Linear Algebrachapter LINALGPreliminary version Fri 1 May 2015 13:22We lay out the algebraic basics of indefinite scalar product spaces. Topics include:orthocomplemented subspaces, angular operators, semidefinite subspaces, indexof positivity and negativity, skewly linked neutral subspaces. As an example weconsider the scalar product space generated by the Nevanlinna kernel of a rationalfunction and Cauchy-type transforms in some detail.Much of the material is standard linear algebra; the experienced reader may skipthis chapter and return when ar product spacesOrthogonalityFinite Dimensional SubspacesDefiniteness properties and maximalityFundamental DecompositionsAngular OperatorsIndices of positivity and negativitySkewly Linked SubspacesThe Nevanlinna Kernel of a Rational Function1 9.4.20157 13.4.201515 13.4.201519 13.4.201525 13.4.201529 19.4.201533 23.4.201539 23.4.201542 ?1.1 Scalar Product SpacescalarProductSpacesUnless explicitly stated, all linear spaces are assumed over the scalar field C ofcomplex numbers.1

2CHAPTER 1. LINEAR ALGEBRAI.141.1.1.1 Definition. Let V be a linear space, and let [., .] : V V C. If [., .] is linearin the first argument, i.e.,[αx βy, z] α[x, z] β[y, z],x, y, z V, α, β C ,and conjugate linear in the second argument, i.e.,[z, αx βy] α[z, x] β[z, y],x, y, z V, α, β C ,we call [., .] a sesquilinear form on V. If [., .] is a sesquilinear form and is hermitian,i.e.,[x, y] [y, x], x, y V ,we call [., .] a scalar product on V. If V is a linear space and [., .] is a scalar producton V, we call a tuple (V, [., .]) a scalar product space. When no confusion is possible, we will often drop explicit notation of the scalarproduct [., .] in (V, [., .]) and shortly speak of a scalar product space V.Let us point out that in contrast to most of Functional Analysis’ literature we do notassume that [x, x] 0, x V \ {0}.Moreover observe that if [., .] is linear in the first argument or conjugate linear in thesecond and is hermitian, then [., .] is a scalar product.Simple manipulations with the axioms show the following properties; the proof is leftto the reader.I.77.1.1.2 Lemma. Let V be a linear space and let [., .] : V V C be a sesquilinearform. Then the following statements hold.(1) We have [x, 0] [0, x] 0, x V.(2) It holds that4[x, y] [x y, x y] [x y, x y] i[x iy, x iy] i[x iy, x iy],x, y V .(1.1.1)This relation is called the polar identity.(3) The map [., .] is hermitian if and only if [x, x] R, x V.(4) For each y V the mapϕy :(Vx C7 [x, y]is a linear functional on V.proof omittedI.5. 1.1.3 Definition. Let (V, [., .]) and (W, ., . ) be two scalar product spaces, and let ϕbe a map from V to W. If ϕx, ϕy [x, y],x, y V ,we call ϕ isometric. If ϕ : V W is isometric, we also say that ϕ is an isometry ofV into W. I.9

1.1. SCALAR PRODUCT SPACES3Note that we do not include the requirement that ϕ is linear into the definition of anisometry. Of course, linear and isometric maps play the crucial role of being the mapspreserving the structure of a scalar product space, and are the most important ones inthe present context.A map between two scalar product spaces which is linear, isometric and bijective iscalled an isometric isomorphism. We say that two scalar product spaces areisometrically isomorphic, if there exists an isometric isomorphism between them.The next statement provides two easier-to-check conditions for linear maps to beisometric.I.86.1.1.4 Lemma. Let (V, [., .]) and (W, ., . ) be two scalar product spaces, and let ϕbe a linear map of V into W. Then the following statements are equivalent.(1) ϕ is isometric.(2) There exists a subset M of V with span M V, such that ϕx, ϕy [x, y] for allx, y M.(3) It holds that ϕx, ϕx [x, x] for all x V.Proof. If ϕ is isometric, then (2) and (3) obviously hold. In fact, (2) holds for everysubset M of V. Sufficiency of (2) in order that ϕ is isometric follows using linearityand conjugate linearity of the scalar products, and sufficiency of (3) follows using thepolar identity. As a first example we consider finite-dimensional spaces.I.2.1.1.5 Example. Let m N, and consider the space Cm . Moreover, let a matrixm mG (γi j )mwith G G be given. Seti, j 1 Cα1 .αm!β1., .βm! : β1 ! .α1G.αmβm! mXi, j 1α1βi · γi j · α j ,.αm!β1., .βm! Cm ,(1.1.2)I.19then [., .] is a scalar product on Cm . Thereby, linearity in the first argument is obviousand the assumption G G yields [x, y] [y, x], x, y Cm .Let us show that, conversely, every scalar product [., .] on Cm can be obtained in thisway. Denote by e1 , . . . , em the canonical basis vectors, i.e., ek : (δk j )mj 1 . Here δk jstands for the Kronecker-Delta, i.e., δk j 1 if k j and 0 otherwise. Setγi j : [e j , ei ],i, j 1, . . . , m .(1.1.3)I.78(1.1.4)I.79 The matrix G : (γi j )mi, j 1 satisfies G G , and α1.αm!β1., .βm! mhXj 1α je j,mXi 1mi Xβi · [e j , ei ] · α j βi ei i, j 1β 1 ! .βmThe matrix G defined by (1.1.3) is called the Gram matrix of [., .].Denote by (., .) the euclidean scalar product on Cm , i.e.,m α1 ! β1 ! X.: αi βi . , .αmβmi 1α1G.αm!.

4CHAPTER 1. LINEAR ALGEBRAThen the relation (1.1.4) can be written in the form α1 ! β1 ! α1 ! β1 ! . G . , . , .αmβmαmβmThus, we may say that the Gram matrix of [., .] realises the switch from (., .) to [., .]. As another example for a scalar product space, we consider a construction whichappears in the context of distributions.I.6.1.1.6 Example. For an open set D Rn (n N) we denote by C (D) the linear space (D) the linear spaceof all complex-valued infinitely differentiable functions, and by C00of all complex-valued, infinitely differentiable and compactly supported functions.Let E be a linear subspace of C (D) which is closed under complex conjugation andmultiplication, let φ : E C be a linear functional on E such that φ( f ) R for allreal-valued functions f E, and set[ f, g]φ : φ( f · ḡ),f, g E .Then [., .]φ is a scalar product on E. To illustrate Rthis construction in a concrete instance, consider D : R, E : C00(R) 2and φ( f ) : R f (x) dx, f C00 (R). Then [., .]φ is nothing but the usual L (R)-scalarproduct on E. A universal model for scalar product spaces can be obtained with the help ofhermitian kernels.I.80.1.1.7 Definition. Let M be a nonempty set and let K : M M C be a function. IfK(ζ, η) K(η, ζ),ζ, η M ,(1.1.5)we call K a hermitian kernel on M.I.4 Let M be a nonempty set. Then we denote by F (M) the set of all finitely supportedcomplex valued functions on M, i.e.,noF (M) : f C M : f 1 (C \ {0}) is finite ,and consider F (M) as a linear space endowed with the pointwise defined operations.For each ξ M let a function δξ be defined as 1 , ζ ξδξ (ζ) : , ζ M.(1.1.6) 0 , ζ , ξI.50Then {δξ : ξ M} is a basis of F (M).Given a hermitian kernel a scalar product space can be constructed.I.81.1.1.8 Lemma. Let M be a nonempty set, let K be a hermitian kernel on M, and setXg(η) · K(η, ζ) · f (ζ), f, g F (M) .(1.1.7)[ f, g]K : η,ζ MThen the tuple (F (M), [., .]K ) is a scalar product space.I.20

1.1. SCALAR PRODUCT SPACES5Proof. The sum on the right side of (1.1.7) contains only finitely many nonzerosummands and hence [., .]K is well-defined. Linearity in the first argument is obviousand (1.1.5) ensures that [., .] is hermitian. I.82.1.1.9 Proposition. Let (V, [., .]) be a scalar product space, let M be a nonempty set,and let ι : M V be a map. ThenK(η, ζ) : [ι(ζ), ι(η)],is a hermitian kernel on M. The map(F (M)ϕ:fζ, η M , VP7 ζ M f (ζ)ι(ζ)(1.1.8)I.84(1.1.9)I.85is linear and isometric. We have ran ϕ span(ran ι). The map ϕ is injective if and onlyif ι is injective and ran ι is linearly independent.Proof. We have K(ζ, η) [ι(η), ι(ζ)] [ι(ζ), ι(η)] K(η, ζ), i.e., K is a hermitiankernel. The map ϕ is well-defined since f (ζ) is equal to 0 for all but finitely manypoints ζ M. Linearity of ϕ follows since the algebraic operations of F (M) aredefined pointwise.To show that ϕ is isometric, it is enough to check the isometry condition for elementsof the basis {δξ : ξ M} of F (M). This, however, is built in the definition:[ϕ(δζ ), ϕ(δη )] [ι(ζ), ι(η)] K(η, ζ) [δζ , δη ]K ,ζ, η M .The equality ran ϕ span(ran ι) is obvious.It remains to show the stated characeterisation of injectivity. First, if ι is not injective,then certainly ϕ cannot be injective. Assume that ι is injective. Then ran ι is linearlydependent if and only if there exist n N, α1 , . . . , αn C not all zero, andPζ1 , . . . , ζn M pairwise different, such that ni 1 αi ι(ζi ) 0. This is equivalent toexistence of a nonzero function f F (M) with ϕ( f ) 0, the connection being αi , ζ ζi , i 1, . . . , nf (ζ) 0 , otherwise As a corollary we obtain that indeed spaces of the form (F (M), [., .]K ) form auniversal model for scalar product spaces (up to isometric isomorphism).I.83.1.1.10 Corollary. Let (V, [., .]) be a scalar product space and let M V be a basisof V. Let ι : M V be the set-theoretic inclusion map, let K be the hermitian kernel(1.1.8) and ϕ the map (1.1.9). Then ϕ is an isometric isomorphism of F (M) onto V. Example 1.1.5 may be considered as a particular case of the above construction. Assume that a matrix G (γi j )mi, j 1 with G G is given, and setM : {1, . . . , m},K(ζ, η) : γζη , ζ, η M .Then the space F (M) is nothing else but Cm , and the scalar product (1.1.7) coincideswith the scalar product defined by (1.1.2). The basis {δ1 , . . . , δm } of F (M) is nothingbut the canonical basis of Cm .proof omitted (COR)

6CHAPTER 1. LINEAR ALGEBRANext, we discuss some constructions which can be carried out with scalar productspaces. Verification of these facts is straightforward; we leave the details to the reader.I.8.1.1.11 Proposition. The following constructions can be carried out within the classof scalar product spaces.(1) Let V be a linear space and let (W, ., . ) be a scalar product space. Moreover,let ϕ : V W be a linear map, and set[x, y] : ϕx, ϕy ,x, y V .Then [., .] is a scalar product on V, and ϕ is a linear and isometric map of(V, [., .]) into (W, ., . ).The scalar product [., .] is the unique scalar product on V such that ϕ becomesisometric. We speak of [., .] as the scalar product on V defined by requiringisometry of ϕ.(2) Let (V, [., .]) be a scalar product space and let W be a linear space. Letϕ : V W be a linear map, and assume that[x, y] 0,x ker ϕ, y V .Then there exists a scalar product ., . on W, such that ϕ is a linear andisometric map of (V, [., .]) into (W, ., . ).Assume that ϕ is surjective. Then the scalar product ., . such that ϕ becomesisometric is unique. We speak of ., . as the scalar product on W defined byrequiring isometry of ϕ.(3) Let V be a linear space, let [., .]i , i 1, . . . , n, be scalar products on V, and set[x, y] : nX[x, y]i ,i 1x, y V .Then [., .] is a scalar product on V. We refer to [., .] as the sum scalar product onV of [., .]1 , . . . , [., .]n .proof omitted The constructions from Proposition 1.1.11 apply in particular with the usual universalconstructions of linear spaces.I.87.1.1.12 Corollary. The following constructions can be carried out within the class ofscalar product spaces.(1) Let (V, [., .]) be a scalar product space and let L be a linear subspace of V.Denote by [., .] L L the restricton of [., .] to elements of L, and consider L as alinear space endowed with the operations inherited from V. Then (L, [., .] L L ) isa scalar product space.The set-theoretic inclusion map of L into V is linear and isometric.

ion--OrthogonalityI.3.1.2. ORTHOGONALITY7(2) Let (V, [., .]) be a scalar product space and let N be a linear subspace of V with[x, y] 0,x N, y V .Then a scalar product ., . on V/N is well-defined by x N, y N : [x, y],x, y V .The canonical projection π : V V/N, x 7 x N, is linear and isometric.Here V/N is endowed with the natural linear operations defined viarepresentants. We refer to ., . as the factor scalar product on V/N.(3) For each j {1, . . . , n} let (V j , [., .] j ) be a scalar product space, and denoten X(x1 ; . . . ; xn ), (y1 ; . . . ; yn ) : [x j , y j ] j ,j 1(x1 ; . . . ; xn ), (y1 ; . . . ; yn ) nYj 1Vj .QnQn Thenj 1 V j , [., .] is a scalar product space, wherej 1 V j is endowed withthe natural linear operations defined in a componentwise manner.QFor each j {1, . . . , n} the embedding ι j : V j nj 1 V j defined asι j x : (0, . . . , 0, x , 0, . . . , 0), j-th placex Vj ,is linear and isometric.We speak of [., .] as the sum scalar product onQnj 1V j.Proof. For (1) use the set-theoretic inclusion map and Proposition 1.1.11, (1). For (2)use the canonical projection and Proposition 1.1.11, (2). For (3) use the canonicalprojections πi onto the i-th component and Proposition 1.1.11, (2), to define scalarproducts on V and then apply Proposition 1.1.11, (3). 1.2 Orthogonality1.2.1 Definition. Let (V, [., .]) be a scalar product space.(1) If x, y V and[x, y] 0 ,we call x and y orthogonal w.r.t. [., .], or shortly [., .]-orthogonal. We write x[ ]yto express this fact.(2) If M, N V andx[ ]y,x M, y N ,we call M and N orthogonal w.r.t. [., .] or shortly [., .]-orthogonal. We writeM[ ]N to express this fact.

8CHAPTER 1. LINEAR ALGEBRA(3) For a subset M V we call M [ ] : x V : x[ ]y for all y Mthe orthogonal complement of M in (V, [., .]). Concerning orthogonal complements, two words of caution are in order. One, theorthogonal complement M [ ] must always be understood w.r.t. a given scalar productspace, and not only w.r.t. the scalar product [., .]. In fact, as M could be a subset ofdifferent scalar product spaces, one should be specific about the base space. However,to avoid cumbersome notation like “M (V,[.,.]) ”, we do not indicate the base spaceexplicitly. Two, the terminology “orthogonal complement” may be slightlymisleading. The space M [ ] is in general neither complementary to M in the sensethat M M [ ] {0} nor in the sense that M M [ ] V.Observe that for each subset M V the orthogonal complement can be represented as\M [ ] ker[., x] .x MFrom this is follow immediately that M [ ] is a linear subspace of V.We start with collecting some properties of orthogonal complements; proofs arestraightforward and are left to the reader.I.90.1.2.2 Lemma. Let (V, [., .]) be a scalar product space. Then the following statementshold.(1) Let M V, then M [ ] (span M)[ ] .(2) Let M, N V with M N. Then N [ ] M [ ] .(3) Let I be a set and let Mi V, i I. Then [ [ ] \ [ ]Mi Mi .i Ii IIf I is finite, then also Xspan Mii I [ ] \Mi[ ] .i I(4) Let L be a linear subspace of V. Then L L[ ] if and only if [x, x] 0, x L.proof omittedI.91. 1.2.3 Lemma. Let n N and let (V j , [., .] j ), j 1, . . . , n, be scalar product spaces.QConsider the space V : nj 1 V j endowed with the sum scalar product [., .]. Let, foreach j {1, . . . , n}, a subset M j of V j be given. ThennYj 1[ ] jMj n Yj 1Mj [ ].If each of the sets M j contains the zero element of the respective space V j , thenequality holds.

1.2. ORTHOGONALITY9[ ] jProof. Assume that x (x1 ; . . . ; xn ) V with x j M jQelements y (y1 ; . . . ; yn ) nj 1 M j we have[x, y] nX, j 1, . . . , n. Then for all[x j , y j ] j 0 .j 1Assume in addition that 0 M j , j 1, . . . , n, and let x (x1 ; . . . ; xn ) be given. Taking scalar products with the elements(0; . . . ; 0; y j ; 0; . . . ; 0), j-th place[ ] jyields that x j M jQnj 1Mj [ ]y j M j , j 1, . . . , n ,, j 1, . . . , n. Next, we show how orthogonal complements transfer via isometries.I.89.1.2.4 Lemma. Let (V, [., .]) and (W, ., . ) be two scalar product spaces, and letϕ : V W be linear and isometric. Then it holds that M [ ] ϕ 1 ϕ(M) , M V .If in addition ϕ is surjective, then N ϕ ϕ 1 (N)[ ] ,N W.Proof. To see the first equality, let M V be given and computex M [ ] [x, y] 0, y M ϕx, ϕy 0, y M ϕx (ϕ(M)) .The second equality is deduced from this one. Let N W be given, and apply whatwe already showed with M : ϕ 1 (N). This gives ϕ 1 (N)[ ] ϕ 1 ϕ ϕ 1 (N) , {z } Nand hence ϕ ϕ 1 (N)[ ]

Bibliography 129. Todo list . New York-Heidelberg-Berlin: Springer-Verlag, 1974, pp. xi 304. lang:1987 [Lan87] S. Lang. Linear algebra. Third edition. Undergraduate Texts in Mathematics. Springer-Verlag, New York, 1987, pp. x 285. . The purpose is to provide the reader with precise references to our selected standard literature.