Related papers: Completion by perturbations
In a finite dimensional Hilbert space, each normalized vector (state) can be chosen as a member of an orthonormal basis of the space. We give a proof of this statement in a manner that seems to be more comprehensible for physics students…
I prove that a Hilbert space has the property that each of its dense (not necessarily closed) subspaces contains an orthoormal basis if and only if it is separable.
Criteria and constructive methods for the completion of an incomplete basis of, or context in, a four-dimensional Hilbert space by (in)decomposable vectors are given.
In this paper, we firstly give a matrix approach to the bases of a separable Hilbert space and then correct a mistake appearing in both review and the English translation of the Olevskii's paper. After this, we show that even a diagonal…
Orthonormal systems in commutative $L_2$ spaces can be used to classify Banach spaces. When the system is complete and satisfies certain norm condition the unconditionality with respect to the system characterizes Hilbert spaces. As a…
In this note, we prove some results related to small perturbations of a frame for a Hilbert space $\mathcal{H}$ in order to have a woven pair for $\mathcal{H}$. Our results complete those known in the literature. In addition we study a…
In this short note we present a far generalization of the following very well-known assertion: assume that we have two orthonormal sequences in a Hilbert space and these sequences are quadratically close to each other. Then if one of these…
We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has…
Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\,…
In this paper, we study perturbation of Hilbert-Schmidt frames under structured modifications, where the perturbation takes the form of replacing finitely or infinitely many frame elements. We establish explicit criteria under which the…
We discuss some perturbation results concerning certain pairs of sequences of vectors in a Hilbert space $\Hil$ and producing new sequences which share, with the original ones, { reconstruction formulas on a dense subspace of $\Hil$ or on…
Let $\{v_n\}$ be a complete minimal system in a Hilbert space $\mathcal{H}$ and let $\{w_m\}$ be its biorthogonal system. It is well known that $\{w_m\}$ is not necessarily complete. However the situation may change if we consider systems…
For any $n$-tuple $(\alpha_1,...,\alpha_n)$ of linearly independent vectors in Hilbert space $H$, we construct a unique orthonormal basis $(\epsilon_1,...,\epsilon_n)$ of $span\{\alpha_1,...,\alpha_n\}$ satisfying:…
Quantum mechanics of unitary systems is considered in quasi-Hermitian representation. In this framework the concept of perturbation is found counterintuitive, for three reasons. The first one is that in this formalism we are allowed to…
We prove that any incomplete system of complex exponentials $\{e^{i\lambda_n t}\}$ in $L^2(-\pi,\pi)$ is a subset of some complete and minimal system of exponentials. In addition, we prove analogous statement for systems of reproducing…
We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. A result of N.J. Kalton is included which shows that this is best possible in that: A frame can be represented as a…
We present a new route to ergodicity breaking via Hilbert space fragmentation that displays an unprecedented level of robustness. Our construction relies on a single emergent (prethermal) conservation law. In the limit when the conservation…
We give a general framework for the treatment of perturbations of types and structures in continuous logic, allowing to specify which parts of the logic may be perturbed. We prove that separable, elementarily equivalent structures which are…
We study finite systems of subspaces of a complex Hilbert space such that each pair of subspaces satisfies a certain condition as described in the following. For each subspace excepting the first one an angle between this subspace and the…
Let H be a Hilbert space and let F be the family of all countable subsets of an orthonormal basis of H. We show that if F is infinite then F is equipollent with every linear basis of the vector space H. In doing so we also present a short…