Salvatore Triolo

Gibbs states defined by biorthogonal sequences

Motivated by the growing interest on PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and on their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended version of the Heisenberg algebraic dynamics, deducing some of their properties, useful for our purposes.

Invariant analytic orthonormalization procedure with an application to coherent states

We discuss a general strategy which produces an orthonormal set of vectors, stable under the action of a given set of unitary operators Aj, j=1,2,…,n, starting from a fixed normalized vector in H and from a set of unitary operators. We discuss several examples of this procedure and, in particular, we show how a set of coherentlike vectors can be produced and in which condition over the lattice spacing this can be done.

Representable states on quasilocal quasi *-algebras

Continuing a previous analysis originally motivated by physics, we consider representable states on quasilocal quasi *-algebras, starting with examining the possibility for a compatible family of local states to give rise to a global state. Some properties of local modifications of representable states and some aspects of their asymptotic behavior are also considered.

An invariant analytic orthonormalization procedure with applications

We apply the orthonormalization procedure previously introduced by Bagarello and Triolo [J. Math. Phys. 48, 043505 (2007)] and adopted in connection with coherent states to Gabor frames and other examples. For instance, for Gabor frames, we show how to construct g(x)∊L2(R) in such a way the functions gn(x)=eian1xg(x+an2), n∊Z2 and a some positive real number, are mutually orthogonal. We discuss in some detail the role of the lattice naturally associated with the procedure in this analysis.

Coordinate representation for non-Hermitian position and momentum operators

In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q^ and p^ similar, in a suitable sense, to the self-adjoint position and momentum operators q^0 and p^0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions, and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q^ and p^, based on the so-called quasi *-algebras.

A note on faithful traces on a von Neumann algebra

In this short note we give some techniques for constructing, starting from asufficient family ℱ of semifinite or finite traces on a von Neumann algebraM, a new trace which is faithful.

Property (gab) through localized SVEP

In this article we study the property (gab) for a bounded linear operator T ϵ L(X) on a Banach space X which is a stronger variant of Browder’s theorem. We shall give several characterizations of property (gab). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gab) holds for large classes of operators and prove the stability of property (gab) under some commuting perturbations.