![]() ∑ i = 1 m ∥ T x i ∥ p ≤ ∥ k ∥ p ∑ i = 1 m | 〈 x i, h 〉 K | pįor each positive integer m and any choice of vectors x 1, x 2, …, x m in K.Įquipped with this example, we can construct many others by simply taking a linear combination of operators of the form (2.5) since Π p ( K ) is a linear subspace of B ( K ).Ĭonsider a Krein space K with the indefinite inner product 〈 ⋅, ⋅ 〉 K. There exists a circle Γ : = which separates the spectrum of T for which the scalar multiple ξ R ( ξ ) of the resolvent operator 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 K and having the following properties: More recently, this problem has been considered in a series of papers by Shkalikov and also by Azizov and Gridneva, Azizov and Khatskevich, and Pyatkov. Further details on the development of this problem till the 1990s can be found in. Krein and Langer, and independently Azizov, showed that Pontryagin’s theorem remains true for maximal dissipative operators. Langer proved the existence of maximal definite invariant subspaces for a wider class of operators, the definitizable operators. As this subject developed, theorems on the existence of T-invariant subspaces were obtained for other classes of operators. An important generalization of Pontryagin’s result was obtained by Krein and Langer. Krein obtained an analogue of Pontryagin’s theorem for unitary operators on π κ spaces and developed a new approach to the invariant subspace problem in spaces with an indefinite metric. In this regard, we note the articles by Krein, Langer, Azizov, and some others. 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. One year earlier, Sobolev had solved a similar problem for the case κ = 1. Then there exists a maximal non- negative T- invariant subspace ℳ ( of dimension κ) such that the spectrum of the restriction T | M lies in the closed upper half plane. Theorem 1.1 Let T be a self adjoint operator in a π κ space. In particular, he proved the following theorem. One of the first results in this direction was obtained by Pontryagin in 1944 for self-adjoint operators acting on π κ-spaces. 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. 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 K. More details on Krein space theory can be found in, and. Maximal non-positive (positive, negative, etc.) subspaces in K are similarly defined.īefore winding up this review on Krein spaces, we note that the Cauchy-Schwarz inequality, If a non-negative subspace ℒ admits no nontrivial non-negative extensions, then it is called a maximal non-negative subspace. We say that the subspace ℒ is definite if 〈 x, x 〉 = 0 if and only if x = 0. A non-positive ( negative, uniformly negative) subspace is defined in a similar way. An element k ∈ K is called positive, negative, or neutral if 〈 k, k 〉 K > 0, 〈 k, k 〉 K 0, 〈 x, x 〉 ≥ δ ∥ x ∥, ( δ > 0)) for all x in ℒ. ![]() ![]() The indefinite inner product 〈 ⋅, ⋅ 〉 K on a Krein space K gives rise to a classification of elements of K. ![]() Because of this construction, Krein spaces are sometimes called J-spaces.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |