Infinite-dimensional Hilbert Space Research Articles

Coupled systems of subdifferential type with integral perturbation and fractional differential equations

This paper is mainly devoted to study a class of first-order differential inclusions governed by time-dependent subdifferential operators involving an integral perturbation. Employing then the constructive method used there, we also handle the associated second-order differential inclusion.
 Our final topic, accomplished in infinite-dimensional Hilbert spaces, is to develop some variants related to coupled systems by such differential inclusions and fractional differential equations.

Banach-valued Bloch-type functions on the unit ball of a Hilbert space and weak spaces of Bloch-type

In this article, we study the space $\mathcal B_\mu(B_X,Y)$ of $Y$-valued Bloch-type functions on the unit ball $B_X$ of an infinite dimensional Hilbert space $X$ with $\mu$ is a normal weight on $B_X$ and $Y$ is a Banach space. We also investigate the characterizations of the space $\mathcal{WB}_\mu(B_X)$ of $Y$-valued, locally bounded, weakly holomorphic functions associated with the Bloch-type space $\mathcal B_\mu(B_X)$ of scalar-valued functions in the sense that $f\in \mathcal{WB}_\mu(B_X)$ if $w\circ f \in \mathcal B_\mu(B_X)$ for every $w \in \mathcal W,$ a separating subspace of the dual $Y'$ of $Y.$

The Strong Converse Exponent of Discriminating Infinite-Dimensional Quantum States

The sandwiched Rényi divergences of two finite-dimensional density operators quantify their asymptotic distinguishability in the strong converse domain. This establishes the sandwiched Rényi divergences as the operationally relevant ones among the infinitely many quantum extensions of the classical Rényi divergences for Rényi parameter alpha >1. The known proof of this goes by showing that the sandwiched Rényi divergence coincides with the regularized measured Rényi divergence, which in turn is proved by asymptotic pinching, a fundamentally finite-dimensional technique. Thus, while the notion of the sandwiched Rényi divergences was extended recently to density operators on an infinite-dimensional Hilbert space (in fact, even for states of an arbitrary von Neumann algebra), these quantities were so far lacking an operational interpretation similar to the finite-dimensional case, and it has also been open whether they coincide with the regularized measured Rényi divergences. In this paper we fill this gap by answering both questions in the positive for density operators on an infinite-dimensional Hilbert space, using a simple finite-dimensional approximation technique. We also initiate the study of the sandwiched Rényi divergences, and the related problem of the strong converse exponent, for pairs of positive semi-definite operators that are not necessarily trace-class (this corresponds to considering weights in a general von Neumann algebra setting). This is motivated by the need to define conditional Rényi entropies in the infinite-dimensional setting, while it might also be interesting from the purely mathematical point of view of extending the concept of Rényi (and other) divergences to settings beyond the standard one of positive trace-class operators (positive normal functionals in the von Neumann algebra setting). In this spirit, we also discuss the definition and some properties of the more general family of Rényi (alpha ,z)-divergences of positive semi-definite operators on an infinite-dimensional separable Hilbert space.

Local Potentials Reconstructed within Linearly Independent Product Basis Sets of Increasing Size.

Given a matrix representation of a local potential v(r) within a one-electron basis set of functions that form linearly independent products (LIP), it is possible to construct a well-defined local potential that is equivalent to v(r) within that basis set and has the form of an expansion in basis function products. Recently, we showed that for exchange-correlation potentials vXC(r) defined on the infinite-dimensional Hilbert space, the potentials reconstructed from matrices of vXC(r) within minimal LIP basis sets of occupied Kohn-Sham orbitals bear only qualitative resemblance to the originals. Here, we show that if the LIP basis set is enlarged by including low-lying virtual Kohn-Sham orbitals, the agreement between and vXC(r) improves to the extent that the basis function products are appropriate as a basis for vXC(r). These findings validate the LIP technology as a rigorous potential reconstruction method.

Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space

In this work we investigate Stinespring dilations of quantum-dynamical semigroups, which are known to exist by means of a constructive proof given by Davies in the early 70s. We show that if the semigroup describes an open system, that is, if it does not consist of only unitary channels, then the evolution of the dilated closed system has to be generated by an unbounded Hamiltonian; subsequently the environment has to correspond to an infinite-dimensional Hilbert space, regardless of the original system. Moreover, we prove that the second derivative of Stinespring dilations with a bounded total Hamiltonian yields the dissipative part of some quantum-dynamical semigroup — and vice versa. In particular this characterizes the generators of quantum-dynamical semigroups via Stinespring dilations.

Flows in Infinite-Dimensional Phase Space Equipped with a Finitely-Additive Invariant Measure

Finitely-additive measures invariant to the action of some groups on a separable infinitedimensional real Hilbert space are constructed. The invariantness of a measure is studied with respect to the group of shifts on a vector of Hilbert space, the orthogonal group and some groups of symplectomorphisms of the Hilbert space equipped with the shift-invariant symplectic form. A considered invariant measure is locally finite, σ finite, but it is not countably additive. The analog of the ergodic decomposition of invariant finitely additivemeasures with respect to some groups are obtained. The set of measures that are invariant with respect to a group is parametrized using the obtained decomposition. The paper describes the spaces of complex-valued functions which are quadratically integrable with respect to constructed invariant measures. This space is used to define the Koopman unitary representation of the group of transformations of the Hilbert space. To define the strong continuity subspaces of a Koopman group, we analyze the spectral properties of its generator.

Generalization of numerical range of polynomial operator matrices

Suppose that is a polynomial matrix operator where for , are complex matrix and let be a complex variable. For an Hermitian matrix , we define the -numerical range of polynomial matrix of as , where . In this paper we study and our emphasis is on the geometrical properties of . We consider the location of in the complex plane and a theorem concerning the boundary of is also obtained. Possible generalazations of our results including their extensions to bounded linerar operators on an infinite dimensional Hilbert space are described.

On the structure group of an infinite dimensional JB-algebra

We extend several results for the structure group of a real Jordan algebra V, to the setting of infinite dimensional JB-algebras. We prove that the structure group Str(V), the cone preserving group G(Ω) and the automorphism group Aut(V) of the algebra V are embedded Banach-Lie groups of GL(V), and that each of the inclusions Aut(V)⊂G(Ω)⊂Str(V) are of embedded Banach-Lie subgroups. We give a full description of the components of Str(V) via cones, isotopes and central projections. We apply these results to V=B(H)sa the special JB-algebra of self-adjoint operators on an infinite dimensional complex Hilbert space, describing the groups Str(V),G(Ω),Aut(V), their Banach-Lie algebras and their connected components. We show that the action of the unitary group of H on Aut(V) has smooth local cross sections, thus Aut(V) is a smooth principal bundle over the unitary group, with structure group S1.

Exponentially m-isometric operators on Hilbert spaces

For a positive integer m, a bounded linear operator T on a Hilbert space is called an exponentially m-isometric operator if ∑k=0m(−1)m−k(mk)ekT⁎ekT=0. For 1≤n≤m, skew-n-selfadjoint operators, nilpotent operators of order less than or equal to [m+12], the greatest integer not greater than m+12, and 2πi multiples of idempotents are main examples of such operators. We establish a decomposition theorem for strict exponentially m-isometric operators with finite spectrum and prove that they are exponentially isometric m-Jordan. Finally, the dynamics of this operator will be considered. We will show that there is no N-supercyclic exponentially m-isometric operator on an infinite-dimensional Hilbert space.

Commutant hypercyclicity of Hilbert space operators

An operator T on a Hilbert space H is commutant hypercyclic if there is a vector x in H such that the set {Sx : TS = ST} is dense in H. We prove that operators on finite dimensional Hilbert space, a rich class of weighted shift operators, isometries, exponentially isometries and idempotents are all commutant hypercyclic. Then we discuss on commutant hypercyclicity of 2 ? 2 operator matrices. Moreover, for each integer number n ? 2, we give a commutant hypercyclic nilpotent operator of order n on an infinite dimensional Hilbert space. Finally, we study commutant transitivity of operators and give necessary and sufficient conditions for a vector to be a commutant hypercyclic vector.

Judgement of two Weyl type theorems for bounded linear operators

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ? B(H) is said to satisfy property (UW?) if ?a(T)\?ea(T) = ?(T), where ?a(T) and ?ea(T) denote the approximate point spectrum and the essential approximate point spectrum of T respectively, ?(T) denotes the set of all poles of T. T ? B(H) satisfiesa-Weyl?s theorem if ?a(T)\?ea(T) = ?a 00(T), where ?a 00(T) = {? ? iso?a(T) : 0 < n(T ??I) < ?}. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property (UW?) and a-Weyl?s theorem by topological uniform descent. In addition, the property (UW?) and a-Weyl?s theorem under perturbations are also discussed.

A novel class of forward-backward explicit iterative algorithms using inertial techniques to solve variational inequality problems with quasi-monotone operators

<abstract><p>The theory of variational inequalities is an important tool in physics, engineering, finance, and optimization theory. The projection algorithm and its variants are useful tools for determining the approximate solution to the variational inequality problem. This paper introduces three distinct extragradient algorithms for dealing with variational inequality problems involving quasi-monotone and semistrictly quasi-monotone operators in infinite-dimensional real Hilbert spaces. This problem is a general mathematical model that incorporates a set of applied mathematical models as an example, such as equilibrium models, optimization problems, fixed point problems, saddle point problems, and Nash equilibrium point problems. The proposed algorithms employ both fixed and variable stepsize rules that are iteratively transformed based on previous iterations. These algorithms are based on the fact that no prior knowledge of the Lipschitz constant or any line-search framework is required. To demonstrate the convergence of the proposed algorithms, some simple conditions are used. Numerous experiments have been conducted to highlight the numerical capabilities of algorithms.</p></abstract>

Some effect of drift of the generalized Brownian motion process: Existence of the operator-valued generalized Feynman integral

In this paper an analytic operator-valued generalized Feynman integral was studied on a very general Wiener space Ca,b[0, T]. The general Wiener space Ca,b[0, T] is a function space which is induced by the generalized Brownian motion process associated with continuous functions a and b. The structure of the analytic operator-valued generalized Feynman integral is suggested and the existence of the analytic operator-valued generalized Feynman integral is investigated as an operator from L1(R, ??,a) to L?(R) where ??,a is a ?-finite measure on R given by d??,a = exp{?Var(a)u2}du, where ? > 0 and Var(a) denotes the total variation of the mean function a of the generalized Brownian motion process. It turns out in this paper that the analytic operator-valued generalized Feynman integrals of functionals defined by the stochastic Fourier-Stieltjes transform of complex measures on the infinite dimensional Hilbert space C?a,b[0, T] are elements of the linear space ? ?>0 L(L1(R, ??,a), L?(R)).

Decompositions of matrices by using commutators

We will use commutators to provide decompositions of 3×3 matrices as sums whose terms satisfy some polynomial identities, and we apply them to bounded linear operators and endomorphisms of free modules of infinite rank. In particular it is proved that every bounded operator of an infinite-dimensional complex Hilbert space is a sum of four automorphisms of order 3 and that every simple ring that is obtained as a quotient of the endomorphism ring of an infinite-dimensional vector space modulo its maximal ideal is a sum of three nilpotent subrings.

Unitless Frobenius Quantales

It is often stated that Frobenius quantales are necessarily unital. By taking negation as a primitive operation, we can define Frobenius quantales that may not have a unit. We develop the elementary theory of these structures and show, in particular, how to define nuclei whose quotients are Frobenius quantales. This yields a phase semantics and a representation theorem via phase quantales. Important examples of these structures arise from Raney’s notion of tight Galois connection: tight endomaps of a complete lattice always form a Girard quantale which is unital if and only if the lattice is completely distributive. We give a characterisation and an enumeration of tight endomaps of the diamond lattices \(M_n\) and exemplify the Frobenius structure on these maps. By means of phase semantics, we exhibit analogous examples built up from trace class operators on an infinite dimensional Hilbert space. Finally, we argue that units cannot be properly added to Frobenius quantales: every possible extention to a unital quantale fails to preserve negations.

On the commutant of B(H) in its ultrapower

Let $${\cal B}\left( H \right)$$ be the algebra of bounded linear operators on a separable infinite-dimensional Hilbert space H. In 2004 Kirchberg asked whether the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial. In [13] the authors have shown that under the Continuum Hypothesis the commutant of $${\cal B}\left( H \right)$$ in its ultrapower depends on the choice of the ultrafilter. We here give a combinatorial characterization of the class of non-principal ultrafilters for which this commutant is non-trivial, answering Question 5.2 of [13]. This reduces Kirchberg’s question to a purely set-theoretic question: Can the existence of non-flat ultrafilters be proven in ZFC? In addition, we introduce the notion of quasi P-points and show that for such ultrafilters, and for ultrafilters satisfying the three functions property, the relative commutant of $${\cal B}\left( H \right)$$ in its ultrapower is trivial.

Codisk-cyclic Mixing Operators

Let be an infinite dimensional separable complex Hilbert space, and be a bounded linear operator. is called codisk mixing operator, - mixing operator, if for any non-empty open subsets of , there are and such that for all . In this paper, we studied a necessarily and sufficiently conditions of - mixing operators, and discused the direct sum of two - mixing operators.

A d-dimensional analyst’s travelling salesman theorem for subsets of Hilbert space

We are interested in quantitative rectifiability results for subsets of infinite dimensional Hilbert space H. We prove a version of Azzam and Schul’s d-dimensional Analyst’s Travelling Salesman Theorem in this setting by showing for any lower d-regular set \(E \subseteq H\) that $$\begin{aligned} \textrm{diam}(E)^d + \beta ^d(E) \sim \mathscr {H}^d(E) + \text {Error}, \end{aligned}$$where \(\beta ^d(E)\) give a measure of the curvature of E and the error term is related to the theory of uniform rectifiability (a quantitative version of rectifiability introduced by David and Semmes). To do this, we show how to modify the Reifenberg Parametrization Theorem of David and Toro so that it holds in Hilbert space. As a corollary, we show that a set \(E \subseteq H\) is uniformly rectifiable if and only if it satisfies the so-called Bilateral Weak Geometric Lemma, meaning that E is bi-laterally well approximated by planes at most scales and locations.

Quantum adiabatic theorem for unbounded Hamiltonians with a cutoff and its application to superconducting circuits.

We present a new quantum adiabatic theorem that allows one to rigorously bound the adiabatic timescale for a variety of systems, including those described by originally unbounded Hamiltonians that are made finite-dimensional by a cutoff. Our bound is geared towards the qubit approximation of superconducting circuits and presents a sufficient condition for remaining within the [Formula: see text]-dimensional qubit subspace of a circuit model of [Formula: see text] qubits. The novelty of this adiabatic theorem is that, unlike previous rigorous results, it does not contain [Formula: see text] as a factor in the adiabatic timescale, and it allows one to obtain an expression for the adiabatic timescale independent of the cutoff of the infinite-dimensional Hilbert space of the circuit Hamiltonian. As an application, we present an explicit dependence of this timescale on circuit parameters for a superconducting flux qubit and demonstrate that leakage out of the qubit subspace is inevitable as the tunnelling barrier is raised towards the end of a quantum anneal. We also discuss a method of obtaining a [Formula: see text] effective Hamiltonian that best approximates the true dynamics induced by slowly changing circuit control parameters. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.

On a question of Kazhdan and Yom Din

Studying a question of Kazhdan and Yom Din we first prove some fixed point theorems for linear actions of a discrete countable group G on a dual Banach space V*, and then show that the answer to the question in full generality is in the negative. In an appendix it is shown that the answer is negative also for the Banach space \({\cal B}\left( H \right)\) of bounded linear operators on an infinite-dimensional Hilbert space.