In this paper we give an algorithm for inverting complex tridiagonal Hermitian matrices with optimal computational efforts. For of a special form and, in particular, Toeplitz matrices, the derived formulas lead to closed-form expressions elements inverse matrices.

Torsion objects of von Neumann categories describe the phenomenon ”spectrum near zero” discovered by S. Novikov and M. Shubin. In this paper we classify Hermitian forms on torsion objects of a finite von Neumann category. We prove that any such Hermitian form can be represented as the discriminant form of a degenerate Hermitian form on a projective module. We also find the relation between the ...

Let E ⊕ F be a direct sum decomposition of a complex Banach lattice X. Garth Dales asked recently whether the equation ‖x+ y‖ = ‖ |x| ∨ |y| ‖ for all x ∈ E and y ∈ F implies that E and F are bands. We show that this is the case by using the theory of hermitian operators. We then show that the same result holds if we replace Dales’s condition by ‖x+ y‖ = ‖(|x|p + |y|p)1/p‖ for any p = 2. To do t...

Classes of non-Hermitian operators that have only real eigenvalues are presented. Such operators appear in quantum mechanics and are expressed in terms of the generators of the Weyl-Heisenberg algebra. For each non-Hermitian operator A, a Hermitian involutive operator Ĵ such that A is Ĵ-Hermitian, that is, ĴA = AĴ , is found. Moreover, we construct a positive definite Hermitian Q such that A is...

In this paper, a discontinuity in beams whose intensity is adjusted by the spring stiffness factor is modeled using a torsional spring. Adapting two analyses in strong and weak forms for discontinuous beams, the improved governing differential equations and the modified stiffness matrix are derived respectively. In the strong form, two different solution methods have been presented to make an a...

This lecture takes a closer look at Hermitian matrices and at their eigenvalues. After a few generalities about Hermitian matrices, we prove a minimax and maximin characterization of their eigenvalues, known as Courant–Fischer theorem. We then derive some consequences of this characterization, such as Weyl theorem for the sum of two Hermitian matrices, an interlacing theorem for the sum of two ...

Some recent work of Gross and Prasad [14] suggests that the root numbers attached to certain symplectic representations of the Weil-Deligne group of a local field F control certain branching rules for representations of orthogonal groups over F . On a global level, this local phenomenon should have implications for the structure of the value at the center of symmetry for certain L-functions of ...

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly timedependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator p...

A Hermitian and an anti-Hermitian first-order intertwining operators are introduced and a class of η-weak-pseudo-Hermitian positiondependent mass (PDM) Hamiltonians are constructed. A corresponding reference-target η-weak-pseudo-Hermitian PDM – Hamiltonians’ map is suggested. Some η-weak-pseudo-Hermitian PT -symmetric Scarf II and periodic-type models are used as illustrative examples. Energy-l...

By further sharpening the Helgason-Johnson bound in 1969, this paper classifies all irreducible unitary representations with non-zero Dirac cohomology of Hermitian symmetric real form E7(−25).