Hilbert Space Research Articles

Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

Characteristic kernels on Hilbert spaces, Banach spaces, and on sets of measures

On the extension of singular linear infinite-dimensional Hamiltonian flows

We study the phenomenon of phase trajectories of a Hamiltonian system going to infinity in a finite time, the phase space of which is a separable Hilbert space. It is shown that if the Hamiltonian is a densely defined quadratic form on the phase space, which is not majorized either from below or from above by the quadratic form of the Hilbert norm, then the phase trajectories allow going to infinity in a finite time. To describe the phase flow of such Hamiltonian systems, an extended phase space is introduced, which is a locally convex space to which the Hamiltonian function, trajectories of the Hamiltonian system, and the symplectic form defined on the original Hilbert space can be extended. Flowinvariant measures on extended space are also studied. The properties of the Koopman unitary representation of the extended phase flow in the Hilbert space of functions that are quadratically integrable with respect to an invariant measure are investigated.

Complementation and Lebesgue‐type decompositions of linear operators and relations

AbstractIn this paper, a new general approach is developed to construct and study Lebesgue‐type decompositions of linear operators or relations in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue‐type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of . The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue‐type decomposition for pairs of quadratic forms. The approach built in this paper uses so‐called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.

Generalized 𝑞-Fock spaces and structural identities

Using q q -calculus we study a family of reproducing kernel Hilbert spaces which interpolate between the Hardy space and the Fock space. We give characterizations of these spaces in terms of classical operators such as integration and backward-shift operators, and their q q -calculus counterparts. Furthermore, these new spaces allow us to study intertwining operators between classic backward-shift operators and the q-Jackson derivative.

Multi-Pursuer and One-Evader Evasion Differential Game with Integral Constraints for an Infinite System of Binary Differential Equations

In the Hilbert space l2, a differential evasion game involving multiple pursuers is considered. Integral constraints are imposed on player control functions. The pursuers are tasked with bringing the state of a system back to the origin of l2, while the evader simultaneously tries to avoid it. It is assumed that the energy of the evader is greater than the total energy of the pursuers. In this paper, we contribute to the solution of the differential evasion game with multiple pursuers by building an exact strategy for the evader.

A new optimization approach to solving split equality problems in Hilbert spaces

A new optimization approach to solving split equality problems in Hilbert spaces

Ideal categories of the normed algebra of finite rank bounded operators on a Hilbert space

In this paper, we introduce the normal category [Formula: see text] [[Formula: see text]] of principal left [right] ideals of the normed algebra [Formula: see text] of all finite rank bounded operators on a Hilbert space [Formula: see text] and show that [Formula: see text] and [Formula: see text] are isomorphic using Hilbert space duality. We also prove that the semigroup of all normal cones in [Formula: see text] is isomorphic to the semigroup of all finite rank operators on [Formula: see text]. Further, we construct bounded normal cones in [Formula: see text] such that the set of all bounded normal cones in [Formula: see text] forms a normed algebra isomorphic to the normed algebra [Formula: see text].

NSimplex Zen : A Novel Dimensionality Reduction for Euclidean and Hilbert Spaces

Dimensionality reduction techniques map values from a high dimensional space to one with a lower dimension. The result is a space which requires less physical memory and has a faster distance calculation. These techniques are widely used where required properties of the reduced-dimension space give an acceptable accuracy with respect to the original space. Many such transforms have been described. They have been classified in two main groups: linear and topological . Linear methods such as Principal Component Analysis (PCA) and Random Projection (RP) define matrix-based transforms into a lower dimension of Euclidean space. Topological methods such as Multidimensional Scaling (MDS) attempt to preserve higher-level aspects such as the nearest-neighbour relation, and some may be applied to non-Euclidean spaces. Here, we introduce nSimplex Zen , a novel topological method of reducing dimensionality. Like MDS, it relies only upon pairwise distances measured in the original space. The use of distances, rather than coordinates, allows the technique to be applied to both Euclidean and other Hilbert spaces, including those governed by Cosine, Jensen–Shannon and Quadratic Form distances. We show that in almost all cases, due to geometric properties of high-dimensional spaces, our new technique gives better properties than others, especially with reduction to very low dimensions.

Minimal periods for semilinear parabolic equations

AbstractWe show that, if $$-A$$ - A generates a bounded holomorphic semigroup in a Banach space X, $$\alpha \in [0,1)$$ α ∈ [ 0 , 1 ) , and $$f:D(A)\rightarrow X$$ f : D ( A ) → X satisfies $$\Vert f(x)-f(y)\Vert \le L\Vert A^\alpha (x-y)\Vert $$ ‖ f ( x ) - f ( y ) ‖ ≤ L ‖ A α ( x - y ) ‖ , then a non-constant T-periodic solution of the equation $${\dot{u}}+Au=f(u)$$ u ˙ + A u = f ( u ) satisfies $$LT^{1-\alpha }\ge K_\alpha $$ L T 1 - α ≥ K α where $$K_\alpha >0$$ K α > 0 is a constant depending on $$\alpha $$ α and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators $$A\ge 0$$ A ≥ 0 in a Hilbert space. For the latter case, we obtain - with a conceptually new proof - the optimal constant $$K_\alpha $$ K α , which only depends on $$\alpha $$ α , and we also include the case $$\alpha =1$$ α = 1 . In Hilbert spaces H and for $$\alpha =0$$ α = 0 , we present a similar result with optimal constant where Au in the equation is replaced by a possibly unbounded gradient term $$\nabla _H{\mathscr {E}}(u)$$ ∇ H E ( u ) . This is inspired by applications with bounded gradient terms in a paper by Mawhin and Walter.

A reconsideration on convergence of the extra-gradient method for solving quasimonotone variational inequalities

ABSTRACT In this paper, we continue to investigate convergence of the extra-gradient method for solving quasimonotone variational inequalities in Hilbert spaces. Under some suitable conditions, we prove that the iterative sequence generated by the extra-gradient method converges weakly to a solution of quasimonotone variational inequalities. Our result corrects and improves the main result in Salahuddin [The extra-gradient method for quasimonotone variational inequalities. Optimization. 2022;71:2519–2528].

On Kitaev’s determinant formula

A sufficient condition is established under which det ( A B A − 1 B − 1 ) = 1 \det (ABA^{-1}B^{-1})=1 for a pair of bounded, invertible operators A , B A,B on a Hilbert space.

The spectral form of the functional model for maximally dissipative operators: A Lagrange identity approach

This paper is a contribution to the theory of functional models. In particular, it develops the so-called spectral form of the functional model where the selfadjoint dilation of the operator is represented as the operator of multiplication by an independent variable in some auxiliary vector-valued function space. With the help of a Lagrange identity, in the present version the relationship between this auxiliary space and the original Hilbert space will be explicit. A simple example is provided.

Toward a Full Configurational Accuracy Calculation of an Arbitrary Molecule via Fragment Embedding and a Stochastic Solver.

We demonstrate the feasibility of using a stochastic solver, full configuration interaction quantum Monte Carlo with the initiator approximation (i-FCIQMC), to converge fragment embedding calculations, namely bootstrap embedding (BE). We first propose and test a general protocol for converging BE-i-FCIQMC calculations and then suggest how the quality of the calculation compares against that of deterministic BE-FCI using different numbers of walkers. We then demonstrate that BE-i-FCIQMC can perform as well as BE-FCI in the large walker limit and how different factors, including the size of the Hilbert space of the fragments, the number of walkers, and the nature of the chemical system, affect the achievable matching error. We finally perform BE-FCI calculations in realistic systems like benzene and cyclohexane using a double-ζ basis set. This work demonstrates the potential of performing FCI quality calculations in realistic systems using BE.

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Tseng’s extragradient method with double projection for solving pseudomonotone variational inequality problems in Hilbert spaces

Maximum Principle for Mean Field Type Control Problems with General Volatility Functions

In this paper, we study the maximum principle of mean field type control problems when the volatility function depends on the state and its measure and also the control, by using our recently developed method in [Bensoussan, A., Huang, Z. and Yam, S. C. P. [2023] Control theory on Wasserstein space: A new approach to optimality conditions, Ann. Math. Sci. Appl.; Bensoussan, A., Tai, H. M. and Yam, S. C. P. [2023] Mean field type control problems, some Hilbert-space-valued FBSDEs, and related equations, preprint (2023), arXiv:2305.04019; Bensoussan, A. and Yam, S. C. P. [2019] Control problem on space of random variables and master equation, ESAIM Control Optim. Calc. Var. 25, 10]. Our method is to embed the mean field type control problem into a Hilbert space to bypass the evolution in the Wasserstein space. We here give a necessary condition and a sufficient condition for these control problems in Hilbert spaces, and we also derive a system of forward–backward stochastic differential equations.

Dynamics of the atom-polyatomic molecule conversion with quantum decoherence

Abstract Abstract. Based on the mean-field approximation and the two-mode Bose-Hubbard model, we investigate the dynamical properties of atom-polyatomic molecule conversion systems under dissipation and dephasing. For such a many-body system problem, we partition phase space based on the states of fixed points, study the movement of fixed points and the transition to attractors in an open system. Additionally, for all the atoms that constitute a molecule, depending on whether the species of these atoms are identical or not, we introduce the innovative concepts of unbiased and biased systems. We discover that in biased conversion systems under particle losses, the evolution trajectories are distributed on a three-dimensional Bloch sphere with an internal structure. There exists a significant difference from the two-dimensional Hilbert space of an unbiased system. Finally, we discuss the sweeping rate of the time-varying magnetic field applied externally to the conversion system, and analyze a feasible scheme for the quantum coherent synthesis of triatomic molecules.

On a class of evolution problems driven by maximal monotone operators with integral perturbation

The present paper is dedicated to the study of a first-order differential inclusion driven by time and state-dependent maximal monotone operators with integral perturbation, in the context of Hilbert spaces. Based on a fixed point method, we derive a new existence theorem for this class of differential inclusions. Then, we investigate an optimal control problem subject to such a class, by considering control maps acting in the state of the operators and the integral perturbation.

Measure induced Hankel and Toeplitz type operators on weighted Dirichlet spaces

For a complex Borel measure μ \mu on the open unit disk, and for a weighted Dirichlet space H s \mathcal {H}_s with 0 > s > 1 0>s>1 , we characterize the boundedness of the measure induced Hankel type operator H μ , s : H s → H s ¯ H_{\mu ,s}: \mathcal {H}_s \to \overline {\mathcal {H}_s} , extending the results of Xiao [Bull. Austral. Math. Soc. 62 (2000), pp. 135–140] for the classical Hardy space H 2 = H 1 H^2=\mathcal {H}_1 , and of Arcozzi, Rochberg, Sawyer, and Wick [J. Lond. Math. Soc. (2) 83 (2011), pp. 1–18] for the classical Dirichlet space D = H 0 \mathcal {D}= \mathcal {H}_0 . Our approach relies on some recent results about weak products of complete Nevanlinna-Pick reproducing kernel Hilbert spaces. We also include some related results on Hankel measures, Carleson measures, and Toeplitz type operators on weighted Dirichlet spaces H s \mathcal {H}_s , 0 > s > 1 0>s>1 .

A collection of seminorms linking the A-numerical radius and the operator A-seminorm

We investigate a novel operator seminorm, QA,mλ,f, for an A-bounded operator Q, where A is a positive operator on a complex Hilbert space (K,⟨·,·⟩). This seminorm is defined using a continuous increasing and bijective function f:R+⟶R+ and an interpolational path mλ of the symmetric mean m. Specifically, QA,mλ,f=supf−1fQy,yAmλfQyA:y∈K,yA=1, where f−1 represents the reciprocal function of f, and ⟨·,·⟩A and ·A denote the semi-inner product and seminorm, respectively, induced by A on K. We explore various bounds and relationships associated with this new concept, establishing connections with existing literature.

On the singularities of distance functions in Hilbert spaces

AbstractFor a given closed nonempty subset E of a Hilbert space H, the singular set $$\Sigma _E$$ Σ E consists of the points in $$H\setminus E$$ H \ E where the distance function $$d_E$$ d E is not Fréchet differentiable. It is known that $$\Sigma _E$$ Σ E is a weak deformation retract of the open set $$\mathcal {G}_E=\{x\in H: d_{\overline{{\text {co}}}\,E}(x)< d_E(x)\}$$ G E = { x ∈ H : d co ¯ E ( x ) < d E ( x ) } . This short paper sheds light on the relationship between the connected components of the three sets $$\Sigma _E\subset \mathcal {G}_E\subseteq H{\setminus } E$$ Σ E ⊂ G E ⊆ H \ E .