Some results on frames by pre-frame operators in Q-Hilbert spaces

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

As earlier conjectured by several authors and much later established by Solèr, from the lattice-theory point of view, Quantum Mechanics may be formulated in real, complex or quaternionic Hilbert spaces only. On the other hand, no quantum systems seem to exist that are naturally described in a real or quaternionic Hilbert space. In a previous paper [23], we showed that any quantum system which is elementary from the viewpoint of the Poincaré symmetry group and it is initially described in a real Hilbert space, it can also be described within the standard complex Hilbert space framework. This complex description is unique and more precise than the real one as, for instance, in the complex description, all self-adjoint operators represent observables defined by the symmetry group. The complex picture fulfils the thesis of Solér’s theorem and permits the standard formulation of the quantum Noether’s theorem. The present work is devoted to investigate the remaining case, namely, the possibility of a description of a relativistic elementary quantum system in a quaternionic Hilbert space. Everything is done exploiting recent results of the quaternionic spectral theory that were independently developed. In the initial part of this work, we extend some results of group representation theory and von Neumann algebra theory from the real and complex cases to the quaternionic Hilbert space case. We prove the double commutant theorem also for quaternionic von Neumann algebras (whose proof requires a different procedure with respect to the real and complex cases) and we extend to the quaternionic case a result established in the previous paper concerning the classification of irreducible von Neumann algebras into three categories. In the second part of the paper, we consider an elementary relativistic system within Wigner’s approach defined as a locally-faithful irreducible strongly-continuous unitary representation of the Poincaré group in a quaternionic Hilbert space. We prove that, if the squared-mass operator is non-negative, the system admits a natural, Poincaré invariant and unique up to sign, complex structure which commutes with the whole algebra of observables generated by the representation itself. This complex structure leads to a physically equivalent reformulation of the theory in a complex Hilbert space. Within this complex formulation, differently from what happens in the quaternionic one, all self-adjoint operators represent observables in agreement with Solèr’s thesis, the standard quantum version of Noether theorem may be formulated and the notion of composite system may be given in terms of tensor product of elementary systems. In the third part of the paper, we focus on the physical hypotheses adopted to define a quantum elementary relativistic system relaxing them on the one hand, and making our model physically more general on the other hand. We use a physically more accurate notion of irreducibility regarding the algebra of observables only, we describe the symmetries in terms of automorphisms of the restricted lattice of elementary propositions of the quantum system and we adopt a notion of continuity referred to the states viewed as probability measures on the elementary propositions. Also in this case, the final result proves that there exists a unique (up to sign) Poincaré invariant complex structure making the theory complex and completely fitting into Solèr’s picture. The overall conclusion is that relativistic elementary systems are naturally and better described in complex Hilbert spaces even if starting from a real or quaternionic Hilbert space formulation and this complex description is uniquely fixed by physics.

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

In recently years, frames in Krein spaces had been considered. The paper presents a family of generators for a Krein space by their frames. These generators are dual frames and operator dual frames corresponding to a given frame in a Krein space. We characterize all generalized dual frames of a primary frame. Also, approximately dual frames in a Krein space are introduced and, we study the relation between approximately dual frames and operator duals in a Krein space. Finally, perturbation of frames in this space is considered.

We give a comprehensive introduction to a general modular frame construction in Hilbert C*-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C*-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications to classical frame theory are indicated. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. In particular, the existence and uniqueness of the closest (normalized) tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closest tight frames often contains a multiple of its symmetric orthogonalization.

In this paper, we introduced and characterized the controlled $g$-duals of a frame in a separable Hilbert space $\mathcal{H}$ . Afterwards, we obtained new $C$-controlled $g$-dual frames from the given $C$-controlled $g$-dual frames. In addition, the approximation for controlled $g$-dual frames was defined and some of their properties were investigated. Finally, we characterized the relationship between approximately $C$-controlled dual and $C$-controlled $g$-dual.

Dual frames are generalized Riesz bases which have potential applications in signal processing. In this paper, the construction of dual frames of matrix-valued wave packet systems in the matrix-valued function space [Formula: see text] from dual pairs of atomic wave packet frames in [Formula: see text] is studied. A class of matrix-valued dual generators from its associated dual pair of atomic wave packets has been obtained. We provide a characterization of matrix-valued dual window functions in terms of orthogonality of wave packet Bessel sequences. A perturbation result with respect to window functions for matrix-valued dual frames is given. It is well known that an orthogonal Parseval Hilbert frame for a Hilbert space turned out be an orthonormal basis for the space, however, this is not true for an orthogonal matrix-valued wave packet Parseval frame for the underlying matrix-valued function space. We give a type of matrix-valued orthonormal basis associated with an orthonormal basis of [Formula: see text].

In this paper, we introduce and study the frames in separable quaternionic Hilbert spaces. Results on the existence of frames in quaternionic Hilbert spaces have been given. Also, a characterization of frame in quaternionic Hilbert spaces in terms of frame operator is given. Finally, a Paley-Wiener type perturbation result for frames in quaternionic Hilbert space has been obtained.

Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space \(V_q(\Phi, \Lambda)\) modeling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wide-band communication. In particular, the space \(V_q(\Phi, \Lambda)\) is generated by a family of well-localized molecules \(\Phi\) of similar size located on a relatively separated set \(\Lambda\) using \(\ell^q\) coefficients, and hence is locally finitely generated. Moreover that space \(V_q(\Phi, \Lambda)\) includes finitely generated shift-invariant spaces, spaces of non-uniform splines, and the twisted shift-invariant space in Gabor (Wilson) system as its special cases. Use the well-localization property of the generator \(\Phi\), we show that if the generator \(\Phi\) is a frame for the space \(V_2(\Phi, \Lambda)\) and has polynomial (sub-exponential) decay, then its canonical dual (tight) frame has the same polynomial (sub-exponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator \(\Phi\) for the space \(V_q(\Phi, \Lambda)\) with \(q\ne 2\), and of the polynomial (sub-exponential) decay property of the mask associated with a refinable function that has polynomial (sub-exponential) decay.

A relation between the output regulation and the observer design for nonlinear systems

Bases play a prominent role in the analysis of vector spaces, as well in the finite-dimensional as in the infinite-dimensional case. The idea is the same in both cases, namely to consider a family of elements such that all vectors in the considered space can be expressed in a unique way as a linear combination of these elements. In the infinite-dimensional case the situation is complicated: we are forced to work with infinite series, and different concepts of a basis are possible, depending on how we want the series to converge. For example, are we asking for the series to converge with respect to a fixed order of the elements (conditional convergence) or do we want it to converge regardless of how the elements are ordered (unconditional convergence)? We define the relevant types of bases in general Banach spaces in Section 3.1, but besides this we mainly consider Hilbert spaces. In Section 3.4 we discuss the most important properties of orthonormal bases in Hilbert spaces; we expect the reader to have some basic knowledge about this subject. A slight (but useful) modification leads to the definition of Riesz bases, which are treated in detail in Section 3.6. Orthonormal bases and Riesz bases satisfy the so-called Bessel inequality, which is the key to the observation that they deliver unconditionally convergent expansions and can be ordered in an arbitrary way. Sequences satisfying the Bessel inequality are therefore discussed already in Section 3.2.

IntroductionThe dynamic behavior analysis of nonlinear physical systems plays a critical role in understanding complex processes across various domains, including education, where interactive simulations of such systems can enhance conceptual learning. Traditional modeling techniques for nonlinear systems often fail to capture their high-dimensional, multi-scale, and chaotic nature due to oversimplified assumptions or reliance on linear approximations.MethodsIn this study, we present a novel framework leveraging computer vision and advanced neural architectures to analyze the dynamic behaviors of nonlinear physical systems. The proposed Physics-Informed Nonlinear Dynamics Network (PNDN) integrates data-driven embeddings with physics-based constraints, offering a robust solution for capturing intricate dynamics and ensuring adherence to physical principles.ResultsExperimental results highlight the model’s superior performance in reconstructing and predicting nonlinear system behaviors under diverse conditions, establishing its utility for real-time educational simulations.DiscussionThis approach bridges the gap between computational modeling and educational innovation, providing learners with interactive tools to explore complex physical phenomena.

Using a left multiplication defined on a right quaternionic Hilbert space, we shall demonstrate that various classes of coherent states such as the canonical coherent states, pure squeezed states which are obtained by the sole action of the squeeze operator on the vacuum state, fermionic coherent states can be defined with all the desired properties on a right quaternionic Hilbert space. Furthermore, we shall also demonstrate that squeezed states, which are obtained by the action of the squeeze operator on canonical coherent states, in other words, they are obtained by the action of the squeeze operator followed by the action of the displacement operator on the vacuum state, can be defined on the same Hilbert space, but the noncommutativity of quaternions prevents us in getting the desired results.

Some results on controlled frames in hilbert spaces

Numerically and spectrally optimal dual frames in Hilbert spaces