REPRESENTATION THEOREMS FOR INDEFINITE QUADRATIC FORMS REVISITED (original) (raw)

The Tan 2 ‚ Theorem for indefinite quadratic forms

2010

A version of the Davis–Kahan Tan 2‚ theorem [3] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a result by Motovilov and Selin [13]. Mathematics Subject Classification (2010). Primary 47A55, 47A07; Secondary 34L05.

Representation of non-semibounded quadratic forms and orthogonal additivity

arXiv: Functional Analysis, 2018

In this article we give a representation theorem for non-semibounded Hermitean quadratic forms in terms of a (non-semibounded) self-adjoint operator. The main assumptions are closability of the Hermitean quadratic form, the direct integral structure of the underlying Hilbert space and orthogonal additivity. We apply this result to several examples, including the position operator in quantum mechanics and quadratic forms invariant under a unitary representation of a separable locally compact group. The case of invariance under a compact group is also discussed in detail.

Quadratic forms with Applications

2013

In this dissertation, our aim is to review the Theory of Quadratic Forms on Euclidean and Hermitian spaces, to give an idea of its generalization to Hilbert spaces and to mention some common applications including the Linear Regression, the Mean Square Approximation, the Rayleigh-Ristz method and the Lax-Milgram Theorem (the bounded as well as the unbounded cases).

The Tan 2$\Theta$ Theorem for indefinite quadratic forms

Journal of Spectral Theory, 2013

A version of the Davis-Kahan Tan 2Θ theorem [SIAM J. Numer. Anal. 7 (1970), 1 -46] for not necessarily semibounded linear operators defined by quadratic forms is proven. This theorem generalizes a recent result by Motovilov and Selin [Integr. Equat. Oper. Theory 56 (2006), 511 -542].

An alternative theorem for quadratic forms and extensions

Linear Algebra and its Applications, 1995

This note concerns an alternative theorem for quadratic forms established recently by Y. Yuan. We explore some directions in which Yuan's result can be extended, and we indicate also some important situations where an alternative theorem a la Yuan does not hold.

Indefinite Quadratic Forms and their Neighbours

2010

The aim of this work is to derive the connection among cycle, proper cycle, right and left neighbour of indefinite reduced binary quadratic form F(x, y) = ax 2 + bxy + cy 2 of discriminant Δ = b 2 − 4ac.

Hermitian Forms and Systems of Quadratic Forms

Documenta Mathematica

We associate to every symmetric (antisymmetric) hermitian form a system of quadratic forms over the base field which determines its isotropy and metabolicity behaviour. It is shown that two even hermitian forms are isometric if and only if their associated systems are equivalent. As an application, it is also shown that an anisotropic symmetric hermitian form over a quaternion division algebra in characteristic two remains anisotropic over all odd degree extensions of the ground field.

Nonnegativity of degenerated quadratic forms of the calculus of variations

Journal of Mathematical Sciences, 1994

In the classical .calculus of variations the application of second-order conditions leads to the necessity of investigating the definiteness of the quadratic form 1 + (B(0x(t),x(t)) + 2(C(t>(t),x(t)))d,+ U(x) 0 x(1)), (0.1) This form is considered on the set of vector-functions x which belong to the space W~, 1[0, 1] (this space consist of absolutely continuous n-dimensional vector-functions x for which ~ E L~ [0, 1]) which, in addition, satisfy the boundary constraints Nox(O) 4-Nix(l) = 0. (0.2) Here, for every t, A(t), B(t), and C(t) are square n x n matrices, and the specified functions A, B, C are sufficiently smooth, fl is a given 2n x 2n symmetric matrix, Ni : R n-+ R a are given linear operators, and n and h are specified natural numbers. Recall that the index of a quadratic form is the maximum dimension of the subspaces of dimensions on which it is negative definite. It is well known [1, 2] that if the index U is finite, then the quadratic form satisfies the Legendre condition A(t) >_ 0 for almost all t. Now if it satisfies the strong Legendre condition, i.e.,there exists a 5 > 0 such that A(t) >__ 5I, for almost all t E [0, I] (0.a) (I is the identity matrix), then the index U is finite and the positiveness of the quadratic form is equivalent to the absence of conjugate points on [0, 1]. As to the nonnegativity of U, it is equivalent to the absence of conjugate points on the half-open interval [0, 1). Thus, in the case when the strong Legendre condition is satisfied, the problem of investigating the definiteness of the form U is completely solved, at least from the point of view of the theory. A different situation arises when the Legendre condition is satisfied but degenerates, i.e., the strong Legendre condition is violated. For instance, if A(t) = t~Ao, where a > 1, and A0 is a positive-definite matrix, then the strong Legendre condition is violated at a single point. However, in this case the classical results and methods are no longer applicable. It is the investigation of these degenerated quadratic forms that comprises the content of this paper. The following examples will serve as the necessary explanations. E x a m p l e 0.1. n = I.

Quadratic D-forms with applications to hermitian forms

Journal of Pure and Applied Algebra, 2019

We study some properties of quadratic forms with values in a field whose underlying vector spaces are endowed with the structure of right vector spaces over a division ring extension of that field. Some generalized notions of isotropy, metabolicity and isometry are introduced and used to find a Witt decomposition for these forms. We then associate to every (skew) hermitian form over a division algebra with involution of the first kind a quadratic form defined on its underlying vector space. It is shown that this quadratic form, with its generalized notions of isotropy and isometry, can be used to determine the isotropy behaviour and the isometry class of (skew) hermitian forms.

Representation of bilinear forms in non-Archimedean Hilbert space by linear operators

2006

The paper considers representing symmetric, non-degenerate, bilinear forms on some non-Archimedean Hilbert spaces by linear operators. Namely, upon making some assumptions it will be shown that if φ is a symmetric, non-degenerate bilinear form on a non-Archimedean Hilbert space, then φ is representable by a unique self-adjoint (possibly unbounded) operator A.

Positive forms on Banach spaces

Acta Mathematica Hungarica, 2003

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case.

From Weyl-Heisenberg frames to infinite quadratic forms

Let a, b be two fixed positive constants. A function g ∈ L 2 (R) is called a mother Weyl-Heisenberg frame wavelet for (a, b) if g generates a frame for L 2 (R) under modulates by b and translates by a, i.e., {e imbt g(t− na} m,n∈Z is a frame for L 2 (R). In this paper, we establish a connection between mother Weyl-Heisenberg frame wavelets of certain special forms and certain strongly positive definite quadratic forms of infinite dimension. Some examples of application in matrix algebra are provided.