Factorisation of orthogonal projectors

We study the problem of a special factorisation of an orthogonal projector~$P$ acting in the Hilbert space $L_2(\mathbb R)$ with $\dim\ker P<\infty$. In particular, we prove that the orthogonal projector~$P$ admits a special factorisation in the form$P=VV^*$, where $V$ is an isometric upper-triangular operator in the Banach algebra of all linear continuous operators in $L_2(\mathbb R)$. Moreover, wegive an explicit formula for the operator $V$.

On a New Family of Extremal Positive Maps of Three-Dimensional Matrix Algebra

We present a new one-parameter family of extremal positive maps on the three-dimensional matrix algebra. The new elements are characterized as mappings that preserve a one-dimensional orthogonal projector.

Weighted -Integral Representations of -Functions in

For -functions , given in the complex space , integral representations of the form are obtained. Here, is the orthogonal projector of the space onto its subspace of entire functions and the integral operator appears by means of explicitly constructed kernel Φ which is investigated in detail.

An iterative algorithm to compute the Bott-Duffin inverse and generalized Bott-Duffin inverse

Let L be a subspace of Cn and PL be the orthogonal projector of Cn onto L. For A?Cn?n, the generalized Bott-Duffin (B-D) inverse A(+)(L) is given by A(+)(L)= PL(APL + PL?)?. In this paper, by defined a non-standard inner product, a finite formulae is presented to compute Bott-Duffin inverse A(?)(L) = PL(APL+P?)? and generalized Bott-Duffin inverse A(?)(L)= PL (APL+PL?)? under the condition A is L?zero (i.e., AL?L?={0}). By this iterative method, when taken the initial matrix X0 = PLA?PL, the Bott-Duffin inverse A(?1)(L) and generalized Bott-duffin inverse A(?)(L) can be obtained within a finite number of iterations in absence of roundoff errors. Finally a given numerical example illustrates that the iterative algorithm dose converge.

On the state of pure shear

The algebraic proof of the fundamental theorem concerning pure shear, by making use only of the notion of orthogonal projector, is presented. It has been shown that the state of pure shear is the same for all singular symmetric traceless tensors in E3, up to the rotation.