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We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schrödinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy. [ABSTRACT FROM AUTHOR]

We investigate the techniques and ideas used in Shefi and Teboulle (SIAM J Optim 24(1), 269–297, 2014) in the convergence analysis of two proximal ADMM algorithms for solving convex optimization problems involving compositions with linear operators. Besides this, we formulate a variant of the ADMM algorithm that is able to handle convex optimization problems involving an additional smooth function in its objective, and which is evaluated through its gradient. Moreover, in each iteration, we allow the use of variable metrics, while the investigations are carried out in the setting of infinite-dimensional Hilbert spaces. This algorithmic scheme is investigated from the point of view of its convergence properties. [ABSTRACT FROM AUTHOR]

We determine the structure of all bijections on the cone of positive semidefinite operators which preserve the quantum f -divergence for an arbitrary strictly convex function f defined on the positive halfline. It turns out that any such transformation is implemented by either a unitary or an antiunitary operator. [ABSTRACT FROM AUTHOR]

From flat band ferromagnetism, we learned that the lowest energy half-filled flat band gives always ferromagnetism if the localized Wannier states on the flat band satisfy the connectivity condition. If the connectivity conditions are not satisfied, ferromagnetism does not appear. We show that this is not always the case namely, we show that ferromagnetism due to flat bands can appear even if the connectivity condition does not hold due to a peculiar behavior of the band situated just above the flat band. [ABSTRACT FROM AUTHOR]

In this article we consider the set of all positive semidefinite linear operators on a finite-dimensional complex Hilbert space equipped with the so-called logarithmic product. We describe the general form of all automorphisms of this structure which are continuous at 0. [ABSTRACT FROM AUTHOR]

A technique based on the transformation in positive semidefinite form of the Hamiltonian (Ĥ) is developed to allow the study of the enlargement possibilities of the emergence domain of a given state. For this reason the calculation of exact ground states for the polyphenylene type of non-integrable chains is performed. Using different exact transcriptions of Ĥ in positive semidefinite form and different solutions of the matching equations, the same type of ground state is deduced in different regions of the parameter space, hence a more global view of its emergence possibilities is obtained. With this procedure an enlarged view of the appearance in the phase diagram of different studied phases can be achieved. [ABSTRACT FROM PUBLISHER]

Using a positive semidefinite operator technique we deduce exact ground states for a zig-zag hexagon chain described by a non-integrable Hubbard model with on-site repulsion. Flat bands are not present in the bare band structure, and the operators [image omitted], introducing the electrons into the ground state, are all extended operators and confined in the quasi-1D chain structure of the system. Consequently, by increasing the number of carriers, the [image omitted] operators become connected, i.e. touch each other on several lattice sites. Hence the spin projection of the carriers becomes correlated in order to minimize the ground-state energy by reducing as much as possible the double occupancy leading to a ferromagnetic ground state. This result demonstrates in exact terms in a many-body frame that the conjecture made at the two-particle level by G. Brocks et al., [Phys. Rev. 93 (2004) p.146405.] that the Coulomb interaction is expected to stabilize correlated magnetic ground states in acenes, is clearly viable, and opens up new directions in the search for routes in obtaining organic ferromagnetism. Due to the itinerant nature of the obtained ferromagnetic ground state, the systems under discussion may also have direct application possibilities in spintronics. [ABSTRACT FROM AUTHOR]

Let A be a selfadjoint operator and P be an orthogonal projection both operating on a Hilbert space H. We say that A is P-complementable if A-μP⩾0 holds for some μ∈R. In this case we define IP(A)=max{μ∈R:A-μP⩾0}. As a tool for computing IP(A) we introduce a natural generalization of the Schur complement or shorted operator of A to S=R(P), denoted by Σ(A,P). We give expressions and a characterization for IP(A) that generalize some known results for particular choices of P. We also study some aspects of the shorted operator Σ(A,P) for P-complementable A, under the hypothesis of compatibility of the pair (A,S). We give some applications in the finite dimensional context. [Copyright &y& Elsevier]