A finite-dimensional complex space with indefinite scalar product [·, ·] having v− = 2 negative squares and v+ ≥ 2 positive ones is considered. The paper presents a classification of operators that are normal with respect to this product. It relates to the paper [1], where the similar classification was obtained by Gohberg and Reichstein for the case v = min{v−, v+} = 1.
IntroductionConsider a complex linear space C n with an indefinite scalar product [· , ·]. By definition, the latter is a nondegenerate sesquilinear Hermitian form. If the ordinary scalar product (· , ·) is fixed, then there exists a nondegenerate Hermitian operator H such that [x, y] an operator U is called H-unitary if U U [ * ] = I, where I is the identity transformation. Let V be a nontrivial subspace of C n . V is called neutral if [x, y] = 0 for all x, y ∈ V . In this case we may write [V, V ] = 0. V is called nondegenerate if from x ∈ V and ∀y ∈ V [x, y] = 0 it follows that x = 0. The subspace V [⊥] is defined as the set of all vectors x ∈ C n : [x, y] = 0 ∀y ∈ V . If V is nondegenerate, then V [⊥] is also nondegenerate and V+V [⊥] = C n .A linear operator A acting in C n is called decomposable if there exists a nondegenerate subspace V ⊂ C n such that both V and V [⊥] are invariant for A. Then A is the orthogonal sum of A 1 = A| V and A 2 = A| V [⊥] . Since the conditions AV [⊥] ⊆ V [⊥] and A [ * ] V ⊆ V are equivalent, an operator A is decomposable if there exists a nondegenerate subspace V which is invariant both for A and A [ * ]
Key words Krein space, normal operator, spectral function MSC (2010) 47B50, 47B15Sign 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.
A real finite dimensional space with indefinite scalar product having v− negative squares and v+ positive ones is considered. The paper presents a classification of operators that are normal with respect to this product for the cases min{v−, v+} = 1, 2. The approach to be used here was developed in the papers [1] and [2], where the similar classification was obtained for complex spaces with v = min{v−, v+} = 1, 2, respectively.
In this paper we investigate the one-dimensional harmonic oscillator with a leftright boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point.The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.