No abstract

A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence \L ) of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) -£) fc (L k x, x)ui with respect to a Hilbert basis (wk)t 6 j, {I Q N). It is shown that with the normand inner-product {((Q , P))) = £ t {{L k , M l » , together with a multiplication denned componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators. INTRODUCTIONThe purpose of this paper is to show how a space of quadratic operators (that is homogeneous polynomial operators of degree 2, (see for example [3,4,7,8]) can be given a multiplication under which it becomes a linear algebra. The construction is new in that it does not make use of the composition of quadratic operators, nor of the composition of the linear maps associated with the tensor product of the quadratic operator's domain. The former attempt fails because the composition of two quadratic operators is generally of degree 4. The latter fails because even though a quadratic operator from one vector space E to another F is associated with a unique symmetric bilinear operator on the cartesian product E x E^*F and hence with a unique linear operator on the tensor product E®E-*F, those linear operators may only be composed in the extremely unusual case where F = E® E. More elaborate nonlinear algebras of homogeneous polynomial operators do exist of course [2], but these form a very different topic from the one studied here.

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Due to increase in the satelite launch activities from many countries around the world the orbital debris issue has become a major concern for the space agencies to plan a collision-free orbit design. The risk of collisions is calculated using the in situ measurements and available models. Spatial density models are useful in understanding the long-term likelihood of a collision in a particular region of space and also helpful in pre-launch orbit planning. In this paper, we present a method of estimating model parameters such as number of peaks and peak locations of spatial density model using continuous wavelets. The proposed methodology was experimented with two line element data and the results are presented.

This paper studies a few interesting properties of the rising sun function of a bounded real function defined on a closed and bounded interval [ , ] a b on the real line. An operator which we call the Rising Sun operator, defined on the space of all bounded real functions defined on [ , ] a b and its related operators are introduced and their properties are investigated.