Related papers: Biorthogonal vectors, sesquilinear forms and some …
We analyse special classes of biorthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with $\mathcal{G}$- quasi bases. We also discuss their relevance in some concrete quantum mechanical system driven by manifestly…
In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not…
The purpose of this article is twofold. First of all, the notion of $(D, E)$-quasi basis is introduced for a pair $(D, E)$ of dense subspaces of Hilbert spaces. This consists of two biorthogonal sequences $\{ \varphi_n \}$ and $\{ \psi_n…
In this paper we continue our analysis of intertwining relations for both self-adjoint and not self-adjoint operators. In particular, in this last situation, we discuss the connection with pseudo-hermitian quantum mechanics and the role of…
In \cite{b-i-t}, F. Bagarello, A. Inoue and C. Trapani investigated some operators defined by Riesz bases. These operators connect with ${\it quasi}$-${\it hermitian \; quantum \; mechanics}$, and its relatives. In this paper, we change the…
We analyze special classes of bi-orthogonal sets of vectors in Hilbert and in Krein spaces, and their relations with generalized Riesz systems. In this way, the notion of the first/second type sequences is introduced and studied. We also…
Led by the key example of the Korteweg-de Vries equation, we study pairs of Hamiltonian operators which are non-homogeneous and are given by the sum of a first-order operator and an ultralocal structure. We present a complete classification…
We discuss some features of non-self-adjoint Hamiltonians with real discrete simple spectrum under the assumption that the eigenvectors form a Riesz basis of Hilbert space. Among other things, {we give conditions under which these…
The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation…
We construct a two-parameters example of {\em pseudo-bosons}, and we show that they are not regular, in the sense previously introduced by the author. In particular, we show that two biorthogonal bases of $\Lc^2(\Bbb R)$ can be constructed,…
We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a…
It is shown that a pair of Hilbert space operators V and W such that V*W=I (called a biisometric pair) shares some common properties with unilateral shifts when orthonormal basis are replaced with biorthogonal sequences, and it is also…
Using as a prototype example the harmonic oscillator we show how losing self-adjointness of the hamiltonian $H$ changes drastically the related functional structure. In particular, we show that even a small deviation from strict…
We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schr\"odinger equation, of relevance…
An elementary 'quantum-mechanical' derivation of the conditions for a system of functions to form a Reisz basis of a Hilbert space on a finite interval is presented.
We describe how self-adjoint ordered operator spaces, also called non-unital operator systems in the literature, can be understood as $*$-vector spaces equipped with a matrix gauge structure. We explain how this perspective has several…
Let $D$ and $U$ be linear operators in a vector space (or more generally, elements of an associative algebra with a unit). We establish binomial-type identities for $D$ and $U$ assuming that either their commutator $[D,U]$ or the second…
We study maximal representations of nonnegative sesquilinear forms in real or complex Hilbert spaces, that are not necessarily closed or even closable. We associate positive self-adjoint operators with such forms, in a sense similar to…
Second order equations of the form $z'' + A_0 z + D z'=0$ in an abstract Hilbert space are considered. Such equations are often used as a model for transverse motions of thin beams in the presence of damping. We derive various properties of…
We discuss two physical examples of the so-called {\em pseudo-bosons}, recently introduced in connection with pseudo-hermitian quantum mechanics. In particular, we show that the so-called {\em extended harmonic oscillator} and the {\em…