Spectral Theorem for Definitizable Normal Linear Operators on Krein Spaces

In the present note a spectral theorem for normal definitizable linear operators on Krein spaces is derived by developing a functional calculus ϕ↦ϕ(N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi \mapsto \phi(N)}$$\end{document} which is the proper analogue of ϕ↦∫ϕdE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\phi \mapsto \int \phi \, dE}$$\end{document} in the Hilbert space situation. This paper is the first systematical study of definitizable normal operators on Krein spaces.


Introduction
A bounded linear operator N on a Krein space (K, [., .]) is called normal, if N commutes with its Krein space adjoint N + , i.e. NN + = N + N . This is equivalent to the fact that its real part A := N +N + 2 and its imaginary part B := N −N + 2i commute. We call N definitizable whenever the selfadjoint operators A and B are both definitizable in the classical sense. This means the existence of so-called definitizing polynomials p(z), q(z) ∈ R[z] \ {0} such that [p(A)x, x] ≥ 0 and [q(B)x, x] ≥ 0 for all x ∈ K; see [4].
In the Hilbert space setting the spectral theorem for bounded linear, normal operators is a well-known functional analysis result. It is almost as folklore as the spectral theorem for bounded linear, selfadjoint operators.
In the Krein space world there exists no similar result for general selfadjoint operators. Assuming in addition definitizability, a spectral theorem could be derived by Heinz Langer; cf. [4]. This theorem became an important starting point for various spectral results. The main difference to selfadjoint operators on Hilbert spaces is the appearance of a finite number of critical points, where the spectral projections no longer behave like a measure.
Only a rather small number of publications dealt with the situation of a normal (definitizable) operators on a Krein space. The Pontryagin space case 222 M. Kaltenbäck IEOT was studied up to a certain extent for example in [7] and [5]. Special normal operators on Krein spaces were considered for example in [1] and [6]. But until now no adequate version of a spectral theorem on normal definitizable operators in Krein spaces has been found.
In the present paper we derive a spectral theorem for bounded linear, normal, definitizable operators formulated in terms of a functional calculus generalizing the functional calculus φ → φ dE in the Hilbert space case. Thus, we provide the first systematical study of definitizable normal operators and their spectral properties on Krein spaces. In order to achieve this goal, we use the methods developed in [2] for definitizable selfadjoint operators and extend them for two commuting definitizable selfadjoint operators.
Let us anticipate a little more explicitly what happens in this note. Denoting by p(z) and q(z) the definitizing real polynomials for A and B, respectively, we build a Hilbert space V which is continuously and densely embedded in the given Krein space K. Denoting by T : V → K the adjoint of the embedding, we have T T + = p(A) + q(B). Then we use the * -homomorphism Θ : (T T + ) (⊆ B(K)) → (T + T ) (⊆ B(V)), C → (T × T ) −1 (C), studied in [2], in order to drag our normal operator N ∈ (T T + ) ⊆ B(K) into (T + T ) ⊆ B(V). The resulting normal operator Θ(N ) acts on a Hilbert space. Therefore, it has a spectral measure E(Δ) on C.
The proper family F N of functions suitable for the aimed functional calculus are bounded and measurable functions on σ(Θ(N )) ∪ (Z R p + iZ R q ) ∪ Z i (⊆ C∪C 2 ). Here Z R p = p −1 {0} ∩ R and Z R q = q −1 {0} ∩ R denote the real zeros of p(z) and q(z), respectively, and , values in C dp(Re z)·dq(Im z)+2 at z ∈ Z R p + iZ R q and values in C dp(ξ)·dq(η) at z = (ξ, η) ∈ Z i . Here d p (w) := min{j : p (j) (w) = 0} (d q (w) := min{j : q (j) (w) = 0}) denotes p's (q's) degree of zero at w. Finally, φ ∈ F N satisfies a growth regularity condition at all points from can be seen as a function s N ∈ F N . The nice thing about these, somewhat tediously defined functions φ ∈ F N is the fact that for a suitable polynomial s ∈ C[z, w] in two variables and a function g : σ(Θ(N )) g dE T + , show that this operator does not depend on the actual decomposition (1.1) and that φ → φ(N ) is indeed a * -homomorphism. Here R1,R2 σ(Θ(N )) g dE is the integral of g with respect to the spectral measure E taking into account the fact that g has values in Vol. 85 (2016) Spectral Theorem for Definitizable Normal 223 If φ stems from a characteristic function corresponding to a Borel subset Δ of C such that no point of Z R p + iZ R q belongs to the boundary of Δ, then φ(N ) is a selfadjoint projection on K.

Multiple Embeddings
In the present section we fix a Krein space (K, [., .]) and Hilbert spaces (V, [., .]), (V 1 , [., .]) and (V 2 , [., .]). Moreover, let T 1 : V 1 → K, T 2 : V 2 → K and T : V → K be bounded linear, injective mappings such that one easily concludes that T + x → T + j x constitutes a well-defined, contractive linear mapping from ran T + onto ran T + j for j = 1, 2. As (ran T + ) ⊥ = ker T = {0} and (ran T + j ) ⊥ = ker T j = {0} these ranges are dense in the Hilbert spaces V and V j . Hence, there exists a unique bounded linear continuation of T + x → T + j x to V, which has dense range in V j . Denoting by R j the adjoint mapping of this continuation, we clearly have T j = T R j and ker R j = (ran R * ker T = {0} and the density of ran T + yields R 1 R * 1 + R 2 R * 2 = I. If T 1 T + 1 and T 2 T + 2 commute, then by T T + = T 1 T + 1 + T 2 T + 2 also T j T + j and T T + commute. Moreover, in this case Employing again T 's injectivity and the density of ran T + , we see that R j R * j and T + T commute for j = 1, 2. We also obtain Thus, we showed Lemma 2.1. With the above notations and assumptions there exist injective contractions R 1 : V 1 → V and R 2 : V 2 → V such that T 1 = T R 1 , T 2 = T R 2 and R 1 R * 1 + R 2 R * 2 = I. If T 1 T + 1 and T 2 T + 2 commute, then the operators R j R * j and T + T on V commute as well as the operators R * , j = 1, 2, and by Θ : (T T + ) (⊆ B(K)) → (T + T ) (⊆ B(V)) we shall denote the * -algebra homomorphisms mapping the identity operator to the identity operator as in Theorem 5.7 from [2] corresponding to the mappings T j , j = 1, 2, and T : (2.1) Here S denotes the commutant of the bounded linear operator S : X → X in B(X ). We can apply Theorem 5.7 in [2] also to the bounded linear, injective R j : V j → V, j = 1, 2, and denote the corresponding * -algebra homomorphisms by Γ j :

Proposition 2.2. With the above notations and assumptions we have
According to Theorem 5.7 in [2] for j = 1, 2 we have Θ j (C)T + j = T + j C and Θ(C) Applying this equation to C + and taking adjoints yields In particular, Θ(C) ∈ (R j R * j ) . Therefore, we can apply Γ j to Θ(C) and get . For the following assertion note that, by (2.3) and by the fact that Γ j is a * -algebra homomorphism mapping the identity operator to the identity operator, we have (j = 1, 2) for any bounded and measurable h : Since E(C \ K) = 0 and E j (C \ K) = 0 for a certain compact K ⊆ C and since C[z,z] is densely contained in C(K), we obtain from the uniqueness assertion of the Riesz Representation Theorem that for any bounded and measurable h.
Again the density of ran Recall from Theorem 5.10 in [2] the mappings (j = 1, 2) and Ξ : ( we shall denote the corresponding mappings outgoing from the mappings . According to Lemma 5.10 in [2], Λ j • Γ j (D) = DR j R * j . Hence, using the notation from Corollary 2.3 In case that T 1 T + 1 and T 2 T + 2 commute we have T 1 T + 1 , T 2 T + 2 ∈ (T T + ) and the later equality can be expressed as (j = 1, 2)  [4]. ♦ By Corollary 7.15 in [2] the definitizability of A and B is equivalent to the concept of definitizability in [2]. Also note that in Pontryagin spaces any bounded linear and normal operator is definitizable in the above sense; see Example 6.2 in [2].

]) and bounded linear and injective operators
for all x, y ∈ K we conclude that The fact that a normal operator is definitizable implies certain spectral properties of Θ(N ).  Proof. We are going to show the first inclusion. The second one is shown in the same manner. For this let n ∈ N and set By (3.1) this inequality can only hold for x = 0. Since Δ n is open, by the Spectral Theorem for normal operators on Hilbert spaces we have Δ n ⊆ ρ(N ). The asserted inclusion finally follows from and Proof. First note that the integrals on the right hand sides exist as bounded operators, because by Lemma 3.3 we have |p(Re z)| ≤ R 1 R * 1 · |p(Re z) + q(Im z)| and |q(Im z)| ≤ R 2 R * 2 · |p(Re z) + q(Im z)| on σ(Θ(N )). Clearly, both sides vanish on the range of E{z ∈ C : p(Re z) = 0 = q(Im z)}. Its orthogonal complement p(Re z) By a density argument the first asserted equality of the present corollary holds true on H and in turn on V. The second equality is shown in the same manner.

The Proper Function Class
In order to introduce a functional calculus we have to introduce an algebra structure on A m,n := (C m ⊗ C n ) × C 2 C m·n+2 and on B m,n := C m ⊗ C n C m·n for m, n ∈ N. On B m,n we define addition, scalar multiplication, multiplication and conjugation in the same way only neglecting the entries with indices (m, 0) and (0, n).
Finally, for m, n ∈ N we introduce the projection π : A m,n → B m,n , (a k,l ) (k,l)∈Im,n → (a k,l ) 0≤k≤m−1 0≤l≤n−1 . On B m,n we assume π to be the identity.  Moreover, we shall denote the set of their real zeros by Z R p and Z R q , i.e.
Now we introduce the following classes of functions: (i) By M N we denote the set of functions φ defined on We provide M N pointwise with scalar multiplication, addition and multiplication, where the operations on A dp(Re z),dq(Im z) or B dp(ξ),dq(η) are as in Definition 4.1. We also define a conjugate linear involution . # on M N by , such that f •τ is sufficiently smooth-more exactly, at least max x,y∈R d p (x)+ d q (y) − 1 times continuously differentiable-on an open neighbourhood of Z R p + iZ R q , and such that f is holomorphic on an open neighbourhood of Z i . Then f can be considered as an element Since Re z is a zero of p of degree exactly d p (Re z) the entries with index (d p (Re z), 0) do not vanish. Moreover, p N (ξ, η) = 0 for all (ξ, η) ∈ Z i . Hence, p N ∈ R. Similarly, if q(w) is considered as an element of C[z, w], then q N (z) k,l = 0, (k, l) ∈ I dp(Re z),dq(Im z) \ {(0, d q (Im z))}, and We need an easy algebraic lemma based on the Euclidean algorithm.
with u(z, w), v(z, w), r(z, w) ∈ C[z, w] such that r's z-degree is less than m and its w-degree is less than n. Here u(z, w), v(z, w), r(z, w) can be found in restricted to the space of all polynomials from C[z, w] with z-degree less than m and w-degree less than n is bijective.
Proof. Applying the Euclidean algorithm to s(z, w) ∈ C[z, w] and a(z) we obtain s(z, w) = a(z)u(z, w) + t(z, w), where u(z, w), t(z, w) ∈ C[z, w] are such that t's z-degree is less than m. Applying the Euclidean algorithm to t(z, w) and b(w) we obtain where v(z, w), r(z, w) ∈ C[z, w] are such that r's z-degree is less than m and its w-degree is less than n. The resulting polynomials u(z, w), t(z, w), v(z, w), r(z, w) belong to In any case it is easy to check that then (s) = (r). Hence, r(z, w) = 0 yields s(z, w) ∈ ker . On the other hand, if 0 = (s) = (r), then for each fixed ζ ∈ a −1 {0} and k ∈ {0, . . . , d a (ζ) − 1} the function w → ∂ k ∂z k r(ζ, w) has zeros at all w ∈ b −1 {0} with multiplicity at least d b (w). Since w → ∂ k ∂z k r(ζ, w) is of w-degree less than n, it must be identically equal to zero.
This implies that for any η ∈ C the polynomial z → r(z, η) has zeros at all ζ ∈ a −1 {0} with multiplicity at least d a (ζ). Since the degree of this polynomial in z is less than m, we obtain r(z, η) = 0 for any z ∈ C. Thus, r ≡ 0.
Our description of ker shows in particular that restricted to the space of all polynomials from C[z, w] with z-degree less than m and w-degree less than n is one-to-one. Comparing dimensions shows that this restriction of is also onto.

Corollary 4.9. With the notation from Definition 4.3 for any
Proof. By Lemma 4.8 there exists an s ∈ C[z, w] such that (s) Re z,Im z = π(φ(z)) for all z ∈ Z R p + iZ R q , and such that (s) ξ,η = φ(ξ, η) for all (ξ, η) ∈ Z i . Hence, by the definition of R we obtain φ − s N ∈ R.
Vol. 85 (2016) Spectral Theorem for Definitizable Normal 233 Definition 4.11. With the notation from Definition 4.3 we denote by F N the set of all elements φ ∈ M N such that z → φ(z) is Borel measurable and bounded on σ(Θ(N )) \ (Z R p + iZ R q ), and such that for each w ∈ σ(Θ(N )) ∩ Example 4.12. For ζ ∈ (Z R p + iZ R q )∪Z i and a ∈ C(ζ) consider the functions aδ ζ ∈ M N which assumes the value a at ζ and the value zero on the rest of  Proof. For a fixed w ∈ σ(Θ(N )) ∩ (Z R p + iZ R q ) and z ∈ σ(Θ(N )) \ (Z R p + iZ R q ) by Remark 4.13 the expression In order to be able to prove spectral results for our functional calculus, we need that with φ also z → φ(z) −1 belongs to F N if φ is bounded away from zero.
Proof. By the first assumption φ −1 is a well-defined object belonging to M N .
is bounded on this set. It remains to verify the boundedness of (4.1) on a certain neighbourhood The expression in (4.3) can be written as is bounded by assumption. The assumed invertibility of φ(w) means φ(w) 0,0 = 0. Hence, 1 dp(Re w)−1 k=0 for z → w. Thus, φ −1 ∈ F N since we verified that the expression (4.2) is a O(max(| Re(z − w)| dp(Re w) , | Im(z − w)| dq(Im w) )).

Functional Calculus
In this section we employ the same assumptions and notation as in the previous one.  N )) with values in C on σ(Θ(N )) \ (Z R p + iZ R q ) and values in C 2 on σ(Θ(N )) ∩ (Z R p + iZ R q ) such that φ − s N ∈ R, such that g is bounded and measurable on σ(Θ(N )) \ (Z R p + iZ R q ), and such that φ(z) = s N (z) + (p N + q N )(z) · g(z), z ∈ σ(Θ(N )), (5.1) where the multiplication here has to be understood in the sense of Remark 4.10.