Local spectral theory for normal operators in Krein spaces
Friedrich Philipp, Vladimir Strauss and Carsten Trunk
Abstract
Sign type spectra are an important tool in the investigation of spectral properties of selfadjoint operators in Krein spaces. It is our aim to show that also sign type spectra for normal operators in Krein spaces provide insight in the spectral nature of the operator: If the real part and the imaginary part of a normal operator in a Krein space have real spectra only and if the growth of the resolvent of the imaginary part (close to the real axis) is of finite order, then the normal operator possesses a local spectral function defined for Borel subsets of the spectrum which belong to positive (negative) type spectrum. Moreover, the restriction of the normal operator to the spectral subspace corresponding to such a Borel subset is a normal operator in some Hilbert space. In particular, if the spectrum consists entirely out of positive and negative type spectrum, then the operator is similar to a normal operator in some Hilbert space. We use this result to show the existence of operator roots of a class of quadratic operator polynomials with normal coefficients.
Introduction
Recall that a bounded operator in a Krein space is normal if , where denotes the adjoint operator of with respect to the Krein space (indefinite) inner product . In contrast to (definitizable) selfadjoint operators in Krein spaces, the knowledge about normal operators is very restricted.
Some results exist for normal operators in Pontryagin spaces. The starting point is a result of M.A. Naimark, see [27], which implies that for a normal operator N in a Pontryagin space there exists a dimensional nonpositive common invariant subspace for and its adjoint . In [23, 30] spectral properties of normal operators in Pontryagin spaces were considered and, in the case , a classification of the normal operators is given.
There is only a very limited number of results in the study of normal operators in spaces others than Pontryagin spaces. In [13] a definition of definitizable normal operators was given and it was proved that a bounded normal definitizable operator in a Banach space with a regular Hermitian form has a spectral function with finitely many critical points. Let us note that in this case the spectral function is a homomorphism from the Borel sets containing no critical points on their boundaries to a commutative algebra of normal projections, see also [3]. Some advances for Krein spaces without the assumption of definitizability can be found in [5]. We mention that [3] contains some perturbation results for fundamentally reducible normal operators. The case of fundamentally reducible and strongly stable normal operators is considered in [6, 7].
On the other hand, the spectral theory for definitizable (and locally definitizable) selfadjoint operators in Krein spaces is welldeveloped (see, e.g., [21, 15, 4] and references therein). One of the main features of definitizable selfadjoint operators in Krein spaces is their property to act locally (with the exception of at most finitely many points) similarly as a selfadjoint operator in some Hilbert space. More precisely, the spectrum of a definitizable operator consists of spectral points of positive and of negative type, and of finitely many exceptional (i.e. nonreal or critical) points, see [20]. For a real point of positive (negative) type of a selfadjoint operator in a Krein space there exists a local spectral function such that (resp. ) is a Hilbert space for (small) neighbourhoods of .
In [19, 22] a characterization for spectral points of positive (negative) type was given in terms of normed approximate eigensequences. If all accumulation points of the sequence for each normed approximate eigensequences corresponding to are positive (resp. negative) then is a spectral point of positive (resp. negative) type. Obviously, the above characterization can be used as a definition for spectral points of positive (negative) type for arbitrary (not necessarily selfadjoint) operators in Krein spaces (as it was done in [2]). It is the main result of this paper that also for a normal operator in a Krein space positive and negative type spectrum implies the existence of a local spectral function for . However, for this we have to impose some additional assumptions: The spectra of the real and imaginary part of are real and the growth of the resolvent of the imaginary part (close to the real axis) of is of finite order. Under these assumptions we are able to show that has a local spectral function on each closed rectangle which consists only of spectral points of positive type or of points from the resolvent set of . The local spectral function is then defined for all Borel subsets of this rectangle and is a selfadjoint projection in the Krein space . It has the property that is a Hilbert spaces for all such . This implies that the restriction of to the spectral subspace is a normal operator in the Hilbert space .
We emphasize that this result provides a simple sufficient condition for the normal operator to be similar to a normal operator in a Hilbert space: If each spectral point of is of positive or of negative type and if the spectra of the real and imaginary part of are real and the growth of the resolvent of the imaginary part is of finite order, then is similar to a normal operator in a Hilbert space. Actually, in the final section, we use this result to prove the existence of an operator root of a quadratic operator pencil with normal coefficients.
1 Some auxiliary statements
In this section we collect some statements on bounded operators in Banach spaces. As usual, by we denote the set of all bounded linear operators acting between Banach spaces and and set . In this paper a subspace is always a closed linear manifold. The approximate point spectrum of a bounded linear operator in a Banach space is the set of all for which there exists a sequence with for all and as . A point in is called an approximate eigenvalue of . We have
(1.1) 
see [10, Chapter VII, Proposition 6.7]. Therefore, if .
The following Lemmas 1.1–1.3 are wellknown. For their proofs we refer to Lemma 0.11, Theorem 1.3, Theorem 0.8 and Corollary 0.13 in [28].
Lemma 1.1.
Let and be two commuting bounded operators in a Banach space and let be a polynomial in two variables. Then
If, in addition, the operators and have real spectra, i.e.
(1.2) 
then the following identity holds:
In particular, we have
Lemma 1.2.
Let be a bounded operator in a Banach space and let be a subspace of which is invariant with respect to . Then
where is the union of all bounded connected components of . In particular, if , we have
Lemma 1.3 (Rosenblum’s Corollary).
Let and be bounded operators in the Banach spaces and , respectively. If , then for every the operator equation
has a unique solution . In particular, implies .
Remark 1.4.
Let be a bounded operator in a Banach space and let be a compact set. We say that a subspace is the maximal spectral subspace of corresponding to if is invariant, and if holds for every invariant subspace with . Recall that such a subspace is hyperinvariant with respect to , i.e. it is invariant with respect to each bounded operator which commutes with (see [9, Chapter 1, Proposition 3.2]).
If the spectrum of the bounded operator is real, we say that the growth of the resolvent of is of finite order , , if for some there exists an , such that
(1.3) 
Since the function , , satisfies the Levinson condition (cf. [24, formula (2.1.2)]), it is a consequence of (1.3) and [24, Chapter II, §2, Theorem 5] that to each compact interval the maximal spectral subspace of corresponding to exists.
By we denote the spectral radius of a bounded operator in a Banach space.
Lemma 1.5.
Let be a bounded operator in a Banach space with real spectrum such that the growth of its resolvent is of order . Then for all we have
where .
Proof.
For we define the function
It is obvious that this function is nondecreasing and continuous. Therefore, the infimum exists. We have .
Let . Let be the circle with center and radius . For we have
(1.4) 
Observe that for , , the function is holomorphic outside of . Due to as , the Cauchy integral theorem and standard estimates of contour integrals we obtain
Therefore, the relation
yields
where denotes the smallest integer larger than . Since and for , together with (1.4) this gives
and hence
Letting we obtain
We have , which leads to the desired estimate with . ∎
For a finite interval we denote by the length of .
Corollary 1.6.
Let be as in Lemma 1.5. Then there exists such that for each , each and each compact interval with and we have
where denotes the maximal spectral subspace of corresponding to .
Proof.
We have , where . Clearly, the growth of the resolvent of is of order . Since and , Lemma 1.5 gives the estimate
with . As is real,
which is independent of , and . ∎
2 Spectral points of positive type of bounded operators in Gspaces
Recall that an inner product space is called a Krein space if there exist subspaces and such that and are Hilbert spaces and
(2.1) 
where denotes the direct sum of subspaces. We refer to (2.1) as a fundamental decomposition of the Krein space .
An inner product space is called a space if is a Hilbert space and the inner product is continuous with respect to the norm on , that is, there exists such that
Let be a Hilbert space inner product on inducing . Then the inner products and are connected via
where is a uniquely determined selfadjoint operator in . It is well known that is a Krein space if and only if is boundedly invertible, see, e.g. [8, 1]. A bounded operator in the space is said to be selfadjoint or selfadjoint if
(2.2) 
holds for all .
Remark 2.1.
Note that in a space it is in general not possible to define a bounded adjoint with respect to of a bounded operator. However, in a Krein space this is possible. In this case, the usual notion of selfadjointness in a Krein space coincides with selfadjointness in spaces.
Spectral points of definite type, defined below for bounded operators in a space, were defined for selfadjoint operators in spaces in [22] and in [2] for arbitrary operators (and relations) in Krein spaces.
Definition 2.2.
For a bounded operator in the space a point is called a spectral point of positive (negative) type of if for every sequence with and as , we have
We denote the set of all points of positive (negative) type of by (, respectively). A set is said to be of positive (negative) type with respect to if every approximate eigenvalue of in belongs to (, respectively).
Remark 2.3.
If the operator is selfadjoint, then the sets and are contained in (cf. [22]).
The following lemma is well known for selfadjoint operators in Krein spaces and selfadjoint operators in spaces (see e.g. [4, 22]). The proof for arbitrary bounded operators remains essentially the same. However, for the convenience of the reader we give a short proof here.
Lemma 2.4.
Let be a bounded operator in the space . Then a compact set is of positive type with respect to if and only if there exist a neighbourhood of in and numbers such that for all and each we have
In this case, the set is of positive type with respect to .
Proof.
Assume that is a compact set of positive type with respect to , i.e. . Let . Then it follows from Definition 2.2 and the properties of the points of regular type of that there exist such that for all we have
From this we easily conclude that for all and all with we have
Since was an arbitrary point in , the assertion follows from the compactness of . The converse statement is evident. ∎
One of the main results of [22] is that under a certain condition a selfadjoint operator in a space has a local spectral function of positive type on intervals which are of positive type with respect to the operator. Let us recall the definition of such a local spectral function and the exact statement for selfadjoint operators.
Definition 2.5.
Let be a space, and . A set function mapping from the system of Borelmeasurable subsets of whose closure is also contained in to is called a local spectral function of positive type of the operator on if for all the following conditions are satisfied:

is a Hilbert space and is selfadjoint.

.

If are mutually disjoint, then
where the sum converges in the strong operator topology.

. for every

.

.
Note that (ii) implies that is a projection for all and that from (iii) (or (v)) it follows that . By () we denote the open upper (lower, respectively) halfplane of the complex plane .
Theorem 2.6.
Let be a selfadjoint operator in the space . If the interval is of positive type with respect to and if each of the sets and accumulates to each point of , respectively, then has a local spectral function of positive type on . For each closed interval the subspace is the maximal spectral subspace of corresponding to .
3 Locally definite normal operators in Krein spaces
For the rest of this paper let be a Krein space. It is our aim to extend Theorem 2.6 to normal operators in Krein spaces. Recall that a bounded operator in a Krein space is called normal if it commutes with its adjoint , i.e.
By definition the real part of a bounded operator in a Krein space is the operator and the imaginary part is given by . It is clear that both real and imaginary part of an arbitrary bounded operator are selfadjoint. Moreover, it is easy to see that a bounded operator in is normal if and only if its real part and its imaginary part commute.
Lemma 3.1.
Let be a normal operator in the Krein space . If and have real spectra only, then .
Proof.
The following theorem is the main result of this section.
Theorem 3.2.
Let be a normal operator in the Krein space . If and have real spectra and the growth of the resolvent of is of finite order, then has a local spectral function of positive type on each closed rectangle which is of positive type with respect to .
Proof.
Let be of positive type with respect to . Together with Lemma 3.1 we have
By Lemma 2.4 there exist an open neighbourhood of in and numbers such that
(3.1) 
By Corollary 1.6, there exists a value such that for each compact interval with length and any we have
(3.2) 
where is the order of growth of the resolvent of and is the maximal spectral subspace of corresponding to the interval .
The proof will be divided into three steps. In the first step we define the spectral subspace corresponding to rectangles with . In the second step we prove some properties of the spectral subspaces defined in step 1. In the third step we define the spectral subspace corresponding to the rectangle and complete the proof.
1. Let and be compact intervals such that and . Note that the inner product space is a space which is not necessarily a Krein space. Since a maximal spectral subspace is hyperinvariant (see, e.g. [9]), the space is invariant with respect to , , and . By , , and denote the restrictions of , , and to , respectively. Then we have, see Lemma 1.2,
(3.3) 
Moreover, from , , (3.3) and Lemma 1.1 we conclude
hence
The operator is obviously selfadjoint. In the following we will show
(3.4) 
To this end set
We may assume that . Otherwise, the assertion of Theorem 3.2 follows directly from Theorem 2.6. We will show that for all and for all we have
which then implies (3.4), see Lemma 2.4. If , then it follows from (3.3) that , and nothing needs to be shown. Otherwise, there exists . Let and , , and suppose that . Let us prove that for all we have
(3.5) 
For this is a direct consequence of (3.2). Assume now that (3.5) holds for all where but does not hold for , i.e.
(3.6) 
Then we have
As , it follows from (3.1) that
Owing to , relation (3.5) holds for by assumption, and thus
follows. But this contradicts (3.6). Hence, (3.5) holds for , and, by induction, for . Hence,
Due to Theorem 2.6 the operator has a local spectral function of positive type on , and the subspace
is the maximal spectral subspace of corresponding to . Moreover, is a Hilbert space with respect to the inner product . Since is invariant with respect to both and , the orthogonal complement
is also  and invariant and is a Krein space, see e.g. [21]. Moreover, we have
2. Let be a rectangle as in step 1. By () we denote the complex (real, respectively) interior of the set (, respectively). In this step of the proof we shall show that the subspaces and , defined in the first step, have the following properties.

.

If is a subspace which is both  and invariant such that
then .

If then .

.

If the bounded operator commutes with then both and are invariant.

is the maximal spectral subspace of corresponding to .
By Lemma 1.2 and (3.3) we have
In addition,
From this and Lemma 1.1 we obtain
Since the spectrum of a normal operator in a Hilbert space coincides with its approximate point spectrum, (a) follows.
Let be a subspace as in (b). By Lemma 1.1 we have
As is the maximal spectral subspace of corresponding to , we conclude from the first relation that . From the second relation we obtain (b) since is the maximal spectral subspace of corresponding to the interval , cf. Theorem 2.6.
Let us prove (c). By definition of it follows from that . Hence, by Lemma 1.1 we have
(3.7) 
Let be a closed interval which contains and let and be the two (closed) components of . By and denote the maximal spectral subspaces of corresponding to the intervals and , respectively. Set
Obviously, we have
(3.8) 
And by [24, Chapter II, Theorem 4] and [24, Chapter I, §4.4] we have
(3.9) 
It is an immediate consequence of (b) that for . Hence, due to (3.7) and (3.9), it remains to show that . But this follows directly from (3.8) and Lemma 1.1.
Set . In order to show (d) we prove
(3.10) 
Since
and by Lemma 3.1, it suffices to show
(3.11) 
Let be a point contained in the set on the left hand side of this relation. Then there exists a compact rectangle with , and
Observe that the normal operator in the Krein space satisfies the conditions of Theorem 3.2. In particular, relation (3.1) holds with the same values and and with replaced by . Hence, there exists a subspace of which is  and invariant and has the properties

,

.
By virtue of (b) we conclude from () and Lemma 3.1 that . But since is also a subspace of , we have which by () implies . Hence, and therefore (3.11) holds.
In order to prove (e) let be closed rectangles such that , for all and
From (a) and (b) it follows that . Now, it is not difficult to see that , and (b) gives
(3.12) 
Let and be the orthogonal projections onto the Hilbert spaces and , respectively. As these spaces are invariant with respect to both and , the projections commute with . Let be a bounded operator which commutes with and let be the restriction of to . We obtain
The spectra of and are disjoint by (a) and (d), and Rosenblum’s Corollary (Theorem 1.3) implies , i.e. for every . By (3.12) this yields . Similarly, one shows that for all . From
and (3.12) we deduce
Hence, for there exists a sequence with each in some such that as . Since and as , we conclude .
After all which has been proved above, for (f) we only have to show that every invariant subspace with is a subspace of . Let be such a subspace. Then let be a sequence of rectangles as in the proof of (e). From
and Rosenblum’s Corollary we conclude . Therefore, for every and follows from (3.12).
3. In this step we complete the proof. Let and