The invariant subspace problem for absolutely p-summing operators in Krein spaces
© Wanjala; licensee Springer 2012
Received: 3 July 2012
Accepted: 19 October 2012
Published: 31 October 2012
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 on it.
MSC:47B50, 46C20, 47B10.
for any and ,
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 λ and μ, 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 κ. In most of the literature, it is always assumed that for a -space.
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.
for any vectors f and g in , and
(ii) , , where denotes the Krein space adjoint of J.
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 ℒ in is called indefinite if it contains both positive and negative elements. We say that ℒ is semi-definite if it is not indefinite. A semi-definite subspace ℒ is called non-negative (positive, uniformly positive) if (, , ()) for all x in ℒ. A non-positive (negative, uniformly negative) subspace is defined in a similar way. We say that the subspace ℒ is definite if if and only if .
If a non-negative subspace ℒ 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.
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  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 ℳ (of dimension κ) such that the spectrum of the restriction lies in the closed upper half plane.
One year earlier, Sobolev  had solved a similar problem for the case . After Pontryagin’s 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 [10–12], Azizov [13, 14], and some others. Krein  obtained an analogue of Pontryagin’s 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 Pontryagin’s result was obtained by Krein  and Langer . 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 , and independently Azizov , showed that Pontryagin’s 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 [17–21] and also by Azizov and Gridneva , Azizov and Khatskevich , and Pyatkov [24, 25].
- (i)there exists a circle which separates the spectrum of T for which the scalar multiple of the resolvent operatoris 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.
We begin this section by defining the class of absolutely p-summing operators whose theory was developed in the late sixties mainly by Pietsch . The literature on these operators is very extensive and varied. References [27–29], and  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 Grothendieck’s paper , where one of the classical theorems of the time (Grothendieck’s theorem) was obtained. It says that every bounded linear operator from to is absolutely summing (see the definition below).
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 .
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 .
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 .
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 .
where we have set .
The reverse implication can be proved in a similar way and is omitted. □
for any continuous linear functional F on . Note that the integrand of the second integral in (2.7) is just a scalar valued function.
which is understood to be the strong limit of the integral sums, is a bounded linear operator on ℋ and is called a Riesz projector. The following theorem, which can be found in  enumerates some of the various properties of this integral.
The operator P does not depend on the choice of the contour Γ isolating the set and it is a projection;
The subspaces and are invariant under T and , .
If T is a bounded linear operator and , then .
In this theorem, it is understood that T is an everywhere defined operator on ℋ. Otherwise, the Riesz projector P has the additional property that , the domain of T in ℋ.
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 .
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 Γ.
- (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.
where . Hence, the third condition holds.
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. □
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.
and so the operator is expansive.
where we have used the Cauchy-Schwarz inequality to obtain the first inequality and the fact that and to obtain the second inequality.
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