This paper investigates isometries on orthogonal direct sums of Kreĭn spaces and subspaces. Internal sums (i-sums) of pairwise orthogonal regular subspaces 𝔎_n , n≥ 0 of a Kreĭn space 𝔎 associated with given fundamental symmetries J_n , n≥ 0 are introduced as subspaces isomorphic with external sums. A characterization of multiplication operators between external and internal sums is provided in terms of the uniform norm boundedness of the restrictions to the components. We introduce internal shifts (i-shifts) as operators unitarily equivalent to standard shifts on Hardy-type Kreĭn spaces and establish several equivalent characterizations. A Wold-type decomposition theorem for isometries on Kreĭn spaces is proved, decomposing an isometry into a unitary part and an i-shift under suitable regularity conditions. We study (minimal) unitary extensions U on a Kreĭn space 𝔎 of bounded isometries acting on a Kreĭn subspace ℌ . The regularity of the minimal subspace ⋁ _n≤ 0U^nℌ is characterized in terms of the convergence of (U^*nPU^n)_n≥ 0 , where P is the orthogonal projection onto ℌ . Conditions for U^*|_𝔎⊖ℌ to be a shift or an i-shift are established. We study the possibility that shifts on Kreĭn spaces have minimal unitary extensions of bilateral shift type. We prove, for example, that V and U^*|_𝔎⊖ℌ are i-shifts if and only if each k∈𝔎 can be represented as k=∑ _n=-∞^∞ U^nr_n , where the sequence r_n=(U^*PU-P)U^*nk , n≥ 0 of the Fourier coefficients is norm square summable. Finally, necessary and sufficient conditions for the uniqueness of unitary extensions are given in terms of the nature of V^*.
We extend the study of normal projections in Krein spaces to the unbounded case. We characterize both weakly normal and normal projections, and prove that every normal projection admits a decomposition as the sum of a selfadjoint projection and a closed projection with neutral range. We show that every closed subspace 𝒮 is the range of a (possibly unbounded) normal projection and parametrize the set of normal projections onto 𝒮 using the notion of normal companions.
Maz’ya’s Problem 43 asks whether the sharp planar condition for L^p -dissipativity of the Lamé operator remains sufficient in dimension three. We give a negative answer by an exact computer-assisted construction. At Poisson ratio ν =0 and p=18+12√(2) , the proposed condition holds with equality. We exhibit an explicit nonvanishing periodic vector field and certify, through a finite Fourier dual certificate evaluated in exact rational arithmetic, that the associated transformed form has strictly negative mean. A quantitative localization argument converts this periodic obstruction into a compactly supported smooth test field in every nonempty open subset of ℝ^3 . The certificate, two verification scripts, a readable coefficient export, and cryptographic checksums are provided in the accompanying supplementary archive.
In this paper S is a closed symmetric linear relation in a Krein space ℌ with adjoint S^* , finite and equal defect numbers d, and a boundary mapping : S^* →ℂ^2d with Gram matrix 𝖰 . We introduce a class 𝔸_S, of self-adjoint extensions of S in a Krein space ℌ containing ℌ as a Krein subspace of finite codimension, together with a class ℙ_𝖰 of d × 2d matrix polynomials. Both classes are equipped with natural equivalence relations. Given 𝒫(z) ∈ℙ_𝖰 , we consider a boundary eigenvalue problem defined by the condition 𝒫(z)({f,g})=0 , {f,g}∈ S^* , and look for its linearizations. By a linearization we mean a linear relation A∈𝔸_S, 𝖻 such that the Shtraus extension T_A(z) of S determined by A coincides, for all z ∈ℂ , with the linear relation {{f,g}∈ S^*: 𝒫(z)𝖻({f,g})=0} . We prove that this correspondence defines a bijection between the equivalence classes in 𝔸_S, 𝖻 and those in ℙ_𝖰 . Moreover, we provide a condition under which the resulting linearizations are spectrally equivalent to the boundary eigenvalue problem: their regular and spectral points coincide, and for each eigenvalue in ℂ there is a bijection between their Jordan chains.
