Let $$\mathcal {H}$$ be a Hilbert space, A positive definite operator in and $$\langle f,g\rangle _A=\langle Af,g\rangle $$ , $$f,g\in \mathcal the A-inner product. This paper studies geometry of set $$\begin{aligned} {I}_A^a:=\{\text { adjointable isometries for } \langle \ \rangle _A\}. \end{aligned}$$ It is proved that {I}_A^a$$ submanifold Banach algebra operators, homogeneous space group i...

In this paper, we constructed relationships with the differents 2D elasticity tensor invariants. Indeed, let ${\bf A}$ be a tensor. Rotation group action leads to pair of Lax in linear elasticity. This five independent invariants chosen among six. The definite positive criteria are established determined We believe that approach finds interesting applications, as one elastic material classifica...

In this note, unless we say otherwise every vector space or algebra we speak about is over C. If A is a Banach algebra and e ∈ A satisfies xe = x and ex = x for all x ∈ A, and also ‖e‖ = 1, we say that e is unity and that A is unital. If A is a unital Banach algebra and x ∈ A, the spectrum of x is the set σ(x) of those λ ∈ C for which λe−x is not invertible. It is a fact that if A is a unital B...

In this paper, by using Mond-Pečarić method we provide some inequalities for connections of positive definite matrices. Next, we discuss specifications of the obtained results for some special cases. In doing so, we use α-arithmetic, α-geometric and α-harmonic operator means.

Inspired by the great success of sparse coding for vector valued data, our goal is to represent symmetric positive definite (SPD) data matrices as sparse linear combinations of atoms from a dictionary, where each atom itself is an SPD matrix. Since SPD matrices follow a non-Euclidean (in fact a Riemannian) geometry, existing sparse coding techniques for Euclidean data cannot be directly extende...

Conditionally positive definite kernels provide a powerful tool for scattered data approximation. Many nice properties of such methods follow from an underlying reproducing kernel structure. While the connection between positive definite kernels and reproducing kernel Hilbert spaces is well understood, the analog relation between conditionally positive definite kernels and reproducing kernel Po...

In this paper, the stabilization of quadratic-input nonlinear systems with bounded controls is considered. According to the type of quadratic-input forms, two cases, namely, positive definite and positive semi-definite, are considered. For the case of positive definiteness, a universal formula for bounded stabilizers is given via a known Lyapunov control function. For the case of positive semid...