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The invariant subspace problem for absolutely p-summing operators in Krein spaces
Journal of Inequalities and Applications volumeĀ 2012, ArticleĀ number:Ā 254 (2012)
Abstract
Let , and let T be a bounded linear operator defined on a Krein space . We prove the existence of a non-positive subspace of invariant under T with the assumption that T is absolutely p-summing with some further conditions imposed onĀ it.
MSC:47B50, 46C20, 47B10.
1 Introduction
Let . In this article we consider the question of the existence of a non-negative subspace invariant under an absolutely p-summing operator T defined on a Krein space . Recall that a complex linear space (or more precisely, ) on which a Hermitian form is defined, that is, a complex valued function defined on such that
-
(i)
,
-
(ii)
for any and ,
is called a Krein space if in there are two linear manifolds and such that
and are Hilbert spaces and . By a linear manifold here we mean a set with the property that for any two vectors and any complex numbers Ī» and Ī¼, we have . It is always assumed that . Otherwise, or is a Hilbert space. The Hermitian form is called the indefinite inner product of the Krein space . If, in particular, , then is called a -space or a Pontryagin space of index Īŗ. In most of the literature, it is always assumed that for a -space.
The representation (1.1) is called a fundamental decomposition of the Krein space and is not unique in general. Using this decomposition, a Hilbert space inner product can be defined on as follows:
where , , with .
Although decomposition (1.1) is not unique in general, its components are uniquely determined and the Hilbert norms generated by different decompositions (1.1) according to (1.2) are equivalent and therefore define the same norm topology. All topological notions such as continuity and convergence in a Krein space are understood to be with respect to this norm topology.
If in the Hilbert space , the orthogonal projections onto and are denoted by and respectively, then the operator
is called the fundamental symmetry associated with decomposition (1.1) and has the following properties: (i)
for any vectors f and g in , and
(ii) , , where denotes the Krein space adjoint of J.
On the other hand, given a Hilbert space and an operator J with the above two properties defined on it (or, more generally, an operator G with and , the resolvent set of G), then an indefinite inner product is defined on by (1.4) (or, respectively, by the relation
and is a Krein space. Because of this construction, Krein spaces are sometimes called J-spaces.
The indefinite inner product on a Krein space gives rise to a classification of elements of . An element is called positive, negative, or neutral if , , or respectively. A linear manifold or a subspace ā in is called indefinite if it contains both positive and negative elements. We say that ā is semi-definite if it is not indefinite. A semi-definite subspace ā is called non-negative (positive, uniformly positive) if (, , ()) for all x in ā. A non-positive (negative, uniformly negative) subspace is defined in a similar way. We say that the subspace ā is definite if if and only if .
If a non-negative subspace ā admits no nontrivial non-negative extensions, then it is called a maximal non-negative subspace. Maximal non-positive (positive, negative, etc.) subspaces in are similarly defined.
Before winding up this review on Krein spaces, we note that the Cauchy-Schwarz inequality,
holds in a Krein space setting. This can be deduced from the following:
More details on Krein space theory can be found in [1ā4], and [5].
We now turn to the main problem under consideration here, which is the question of the existence of semi-definite invariant subspaces for absolutely p-summing operators on a Krein space . For various classes of operators, this problem has been a subject of research since the early days of the theory of operators in spaces with an indefinite metric. One of the first results in this direction was obtained by Pontryagin [6] in 1944 for self-adjoint operators acting on -spaces. In particular, he proved the following theorem.
Theorem 1.1 Let T be a self adjoint operator in a space. Then there exists a maximal non-negative T-invariant subspace ā³ (of dimension Īŗ) such that the spectrum of the restriction lies in the closed upper half plane.
One year earlier, Sobolev [7] had solved a similar problem for the case . After Pontryaginās result, the problem on the existence of invariant maximal semi-definite subspaces turned out to be the focus of attention in the theory of operators in Pontryagin and Krein spaces. In this regard, we note the articles by Krein [8, 9], Langer [10ā12], Azizov [13, 14], and some others. Krein [8] obtained an analogue of Pontryaginās theorem for unitary operators on spaces and developed a new approach to the invariant subspace problem in spaces with an indefinite metric. An important generalization of Pontryaginās result was obtained by Krein [9] and Langer [10]. As this subject developed, theorems on the existence of T-invariant subspaces were obtained for other classes of operators. Langer [11, 12] proved the existence of maximal definite invariant subspaces for a wider class of operators, the definitizable operators. Krein and Langer [15], and independently Azizov [14], showed that Pontryaginās theorem remains true for maximal dissipative operators. Further details on the development of this problem till the 1990s can be found in [2, 16]. More recently, this problem has been considered in a series of papers by Shkalikov [17ā21] and also by Azizov and Gridneva [22], Azizov and Khatskevich [23], and Pyatkov [24, 25].
Let . It is the aim of this paper to prove the existence of a non-positive invariant subspace for an absolutely p-summing operator T (defined below) acting on a Krein space and having the following properties:
-
(i)
there exists a circle which separates the spectrum of T for which the scalar multiple of the resolvent operator
is expansive, that is,
for all ;
-
(ii)
there exists some real number such that
The subset of p-absolutely summing operators having both properties stated above is nonempty as we shall show at the end of this article.
2 Preliminaries
We begin this section by defining the class of absolutely p-summing operators whose theory was developed in the late sixties mainly by Pietsch [26]. The literature on these operators is very extensive and varied. References [27ā29], and [30] are probably among the most extensive ones in this regard.
The first results marking the beginning of the theory of these operators are contained in Grothendieckās paper [31], where one of the classical theorems of the time (Grothendieckās theorem) was obtained. It says that every bounded linear operator from to is absolutely summing (see the definition below).
Let and be Hilbert spaces with inner products and respectively, and let . A bounded linear operator is called absolutely p-summing if there exists a constant such that for each positive integer m and any vectors in , we have
or equivalently,
where is the topological dual of . The smallest constant c for which (2.1) holds is denoted by , while stands for the set of all absolutely p-summing operators from into . If , then this set is denoted by . For the case , these operators are simply referred to as absolutely summing operators. Using the Minkowski inequality for sums, one can easily prove that is a linear subspace of , the set of all bounded linear operators from into .
Closely related to the class of absolutely p-summing operators is the class of absolutely -summing operators for , . If inequality (2.1) is replaced by
then the operator T is called absolutely -summing. In this case, the smallest constant c for which (2.2) holds is denoted by , while denotes the class of absolutely -summing operators from into . The class can also be characterized as a collection of operators which take weakly q-summable sequences in to strongly p-summable sequences in . We note that if , then . As before, we write for .
We now look at the above definitions in the context of Krein spaces. Let and be Krein spaces with inner products and respectively, and let be a bounded linear operator. If we replace (2.1) and (2.2) with
and
respectively, we get the definitions of absolutely p-summing and absolutely -summing operators between Krein spaces and . The rest of the notation remains as in the Hilbert space case.
Below is an example of an absolutely p-summing operator on an arbitrary Krein space for .
Example 2.1 Let be a Krein space with the indefinite inner product , and let h and k be fixed elements of with . Let and define an operator by
Then T is clearly well defined and linear. The fact that this operator is bounded and absolutely p-summing with follows from
since
and so
for each positive integer m and any choice of vectors in .
Equipped with this example, we can construct many others by simply taking a linear combination of operators of the form (2.5) since is a linear subspace of .
Consider a Krein space with the indefinite inner product . By we denote the Hilbert space associated with the Krein space , that is, the space together with the positive definite inner product , where J is the operator defined in (1.3).
Remark 2.2 Let and be Krein spaces, and let be a bounded linear operator. Then T is absolutely p-summing if and only if it is absolutely p-summing as an operator from into .
Proof Let and suppose that is absolutely p-summing. Since and J maps the open unit ball onto itself in a one-to-one way, then for some constant and any positive integer m, we have
where we have set .
The reverse implication can be proved in a similar way and is omitted.āā”
We conclude this section by introducing the concept of a Riesz projector (see [32] for more properties of this operator). Let Ī be a Cauchy contour, and let be a continuous function on Ī with values in a complete normed linear space . Then (as in complex function theory) the integral
is defined as a Stieltjes integral, where the corresponding Stieltjes sum converges in the norm of . Thus, the value of (2.6) is a vector in which appears as a limit (in the norm of ) of the corresponding Stieltjes sum. From this definition, it is clear that
for any continuous linear functional F on . Note that the integrand of the second integral in (2.7) is just a scalar valued function.
Of interest to us is the case when is the Banach space consisting of all bounded linear operators from the Banach space into the Banach space . So, let and be Banach spaces, and let be a continuous function. Then the value of the integral (2.6) is a bounded linear operator from into , and for each , we have
Let T be a bounded linear operator on an arbitrary Hilbert space ā. Suppose that T has a separated spectrum, and let be an isolated part of the spectrum of T. Let Ī be a Jordan closed rectifiable contour lying in , the resolvent set of T, and containing . Suppose further that lies outside this contour. Then the value of the integral
which is understood to be the strong limit of the integral sums, is a bounded linear operator on ā and is called a Riesz projector. The following theorem, which can be found in [2] enumerates some of the various properties of this integral.
Theorem 2.3
-
(a)
The operator P does not depend on the choice of the contour Ī isolating the set and it is a projection;
-
(b)
The subspaces and are invariant under T and , .
-
(c)
If T is a bounded linear operator and , then .
In this theorem, it is understood that T is an everywhere defined operator on ā. Otherwise, the Riesz projector P has the additional property that , the domain of T in ā.
3 Non-positive invariant subspace
Let p be such that . In this section we show the existence of a non-positive invariant subspace for an absolutely p-summing operators T acting on a Krein space satisfying the two conditions stated at the end of Section 1.
Let , and let be a Krein space. The following lemma establishes a contractive-like relation for any with the constant .
Lemma 3.1 Let be a Krein space with some fixed fundamental decomposition, and for , let be such that . Then with respect to this fixed fundamental decomposition, the inequality
holds.
Proof The Cauchy-Schwarz inequality together with the fact that yields the following series of inequalities:
āā”
Let be a Krein space. For , let . In the theorem below, which is our main result, we prove the existence of a non-positive invariant subspace for T under the assumption that the operator is contractive, where for some specified contour Ī.
Theorem 3.2 Let be a Krein space, and for , let . Assume that T has the following properties:
-
(i)
there exists a positive real number r for which the circle separates the spectrum of T and is such that the scalar multiple of the resolvent operator
is expansive, that is, ,
-
(ii)
there exists some real number such that
Then there exists a non-positive subspace which is invariant under T.
Proof For , let and assume that T satisfies the conditions of the theorem. For any real number a with , the operator aT belongs to and satisfies the conditions of the theorem with Ī¾ replaced by aĪ¾. It is clear that the first two conditions are satisfied. To see that the third condition is satisfied, we first recall that . Then
where . Hence, the third condition holds.
Since the operators T and aT have the same invariant subspaces, we will consider the case when the constant . So, without loss of generality, assume that the operator T is such that . Lemma 3.1 implies that for with and , we have
Let . Then and so . This implies that
From inequality (3.1) we see that
Since
we see that
Since by assumption
for some real number Ī±, inequality (3.2) can be written as
which is equivalent to
We now introduce the Riesz projector
First, we put in (3.3), divide either side of this equation by 2Ļ, and integrate from 0 to 2Ļ with respect to Īø to get
Hence,
which we write as
We now use the fact that and are bounded linear functionals on together with (2.7) to rewrite the above inequality as
where
Denote by the subspace . Inequality (3.4) implies that for , we have , and Theorem 2.3 ensures that this is the required invariant subspace. This concludes the proof.āā”
Note that the requirement in the theorem cannot be dropped. If we do, then (3.4) yields the invalid inequality
for all .
We conclude this article by showing that an operator with the properties stated in Theorem 3.2 above does exist, and therefore, our result does not hold on an empty set.
Example 3.3 Let be a Krein space and fix a negative element with the property that , and there exists a circle Ī centered at the origin and small enough such that and for all . Such an element h exists (for example, we can pick a suitable h in ). Define a bounded linear operator T on by
This operator satisfies all the conditions of Theorem 3.2 and clearly has a non-positive invariant subspace. To show that this is the case, we first note that from Example 2.1, one readily sees that T is absolutely p-summing with for . The only eigenvalues of this operator are and . Since T has finite rank, these eigenvalues are also the only spectral values of T. Hence, the circle Ī lies in the resolvent set of T and separates its spectrum. For any , the resolvent operator is defined. A little manipulation shows that is given by
Now
The last inequality holds because . Hence,
and so the operator is expansive.
We now show that there exists some real number such that
Now,
where we have used the Cauchy-Schwarz inequality to obtain the first inequality and the fact that and to obtain the second inequality.
We also have that
With , we see that
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Wanjala, G. The invariant subspace problem for absolutely p-summing operators in Krein spaces. J Inequal Appl 2012, 254 (2012). https://doi.org/10.1186/1029-242X-2012-254
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DOI: https://doi.org/10.1186/1029-242X-2012-254