in this note, a generalization of higher rank numerical range isintroduced and some of its properties are investigated

the rank-k numerical range has a close connection to the construction of quantum error correction code for a noisy quantum channel. for noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the associated joint rank-k numerical range is non-empty. in this paper the notion of joint rank-k numerical range is generalized and some statements of [2011, generaliz...

A complex square matrix A is said to be idempotent, or a projector, whenever A2 = A; when A is idempotent, and Hermitian (or real symmetric), it is often called an orthogonal projector, otherwise an oblique projector. Projectors are closely linked to generalized inverses of matrices. For example, for a given matrix A the product PA = AA + is well known as the orthogonal projector on the range (...

We study the spectrum and the rank of a linear combination of two orthogonal projectors. We characterize when this linear combination is EP, diagonalizable, idempotent, tripotent, involutive, nilpotent, generalized projector, and hypergeneralized projector. Also we derive the Moore-Penrose inverse of a linear combination of two orthogonal projectors in a particular case. The main tool used here...

The causal perturbation expansion defines the Dirac sea in the presence of a time-dependent external field. It yields an operator whose image generalizes the vacuum solutions of negative energy and thus gives a canonical splitting of the solution space into two subspaces. After giving a self-contained introduction to the ideas and techniques, we show that this operator is in general not idempot...

We show that modified Wigner projector technique and generalized Bloch theorem approach yield maximally efficient diagonalization of the Hamiltonian of the large symmetrical systems. For the sake of illustration, we perform a case study of the simplified DNA molecule model and solve the energy eigenproblem analytically by using the unit symmetry cell (symcell) and the corresponding low-dimensio...

The present paper is devoted to an improvement of the theory of the recently proposed generalized augmented matrix preconditioning method [5]. Namely, we compute a sharp lower bound on eigenvalues of the preconditioned matrix based on the properties of the projector involved in its definition.

By defining "a virtual gap" for the continuous spectrum through the notion of eigendifferential (Weyl's packet) and using the differential projector operator, we present a rigorous demonstration and discussion of the quantum adiabatic theorem for systems having a nondegenerate continuous spectrum. An explicit formula for a generalized geometrical phase is derived in terms of the eigenstates of ...