### A simple proof of polar decomposition in pseudo-Euclidean geometry

We give a simple direct proof of the polar decomposition for separated linear maps in pseudo-Euclidean geometry.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

We give a simple direct proof of the polar decomposition for separated linear maps in pseudo-Euclidean geometry.

Consider the Hilbert space (H,〈• , •〉) equipped with the indefinite inner product[u,v]=v*J u,u,v∈ H, where J is an indefinite self-adjoint involution acting on H. The Krein space numerical range WJ(T) of an operator T acting on H is the set of all the values attained by the quadratic form [Tu,u], with u ∈H satisfying [u,u]=± 1. We develop, implement and test an alternative algorithm to compute WJ(T) in the finite dimensional case, constructing 2 by 2 matrix compressions of T and their easily determined...

The convexity and compactness in the weak operator topology of the image and pre-image of a generalized fractional linear transformation is established. As an application the exponential dichotomy of solutions to evolution problems of the parabolic type is proved.

Let H and K be complex complete indefinite inner product spaces, and ℬ(H,K) (ℬ(H) if K = H) the set of all bounded linear operators from H into K. For every T ∈ ℬ(H,K), denote by ${T}^{\u2020}$ the indefinite conjugate of T. Suppose that Φ: ℬ(H) → ℬ(K) is a bijective linear map. We prove that Φ satisfies $\Phi \left(A\right)\Phi \left(B\right)=\Phi \left(B\right)\Phi {\left(A\right)}^{\u2020}$ for all A, B ∈ ℬ(H) with $AB=B{A}^{\u2020}$ if and only if there exist a nonzero real number c and a generalized indefinite unitary operator U ∈ ℬ(H,K) such that $\Phi \left(A\right)=cUA{U}^{\u2020}$ for all A ∈ ℬ(H).

In this paper we study $J$-EP matrices, as a generalization of EP-matrices in indefinite inner product spaces, with respect to indefinite matrix product. We give some properties concerning EP and $J$-EP matrices and find connection between them. Also, we present some results for reverse order law for Moore-Penrose inverse in indefinite setting. Finally, we deal with the star partial ordering and improve some results given in the “EP matrices in indefinite inner product spaces” (2012), by relaxing some...

The present paper consists of two parts. In Section 1 we consider fractional-linear transformations (f.-l.t. for brevity) F in the space $\mathcal{L}({X}_{1},{X}_{2})$ of all linear bounded operators acting from ${X}_{1}$ into ${X}_{2}$, where ${X}_{1},{X}_{2}$ are Banach spaces. We show that in the case of Hilbert spaces ${X}_{1},{X}_{2}$ the image F(ℬ) of any (open or closed) ball ℬ ⊂ D(F) is convex, and if ℬ is closed, then F(ℬ) is compact in the weak operator topology (w.o.t.) (Theorem 1.2). These results extend the corresponding results on compactness obtained in [3],...