Infinite-dimensional Separable Hilbert Space Research Articles

The semi-Fredholmness and property (ω) for 2×2 upper triangular operator matrices

Let be a upper triangular operator matrix acting on , where and are complex infinite dimensional separable Hilbert spaces. In this paper, we study the semi-Fredholmness for for some invertible , and using the conclusions obtained, we characterize the equivalent conditions such that for any invertible .

Master Bellman equation in the Wasserstein space: Uniqueness of viscosity solutions

We study the Bellman equation in the Wasserstein space arising in the study of mean field control problems, namely stochastic optimal control problems for McKean-Vlasov diffusion processes. Using the standard notion of viscosity solution à la Crandall-Lions extended to our Wasserstein setting, we prove a comparison result under general conditions on the drift and reward coefficients, which coupled with the dynamic programming principle, implies that the value function is the unique viscosity solution of the Master Bellman equation. This is the first uniqueness result in such a second-order context. The classical arguments used in the standard cases of equations in finite-dimensional spaces or in infinite-dimensional separable Hilbert spaces do not extend to the present framework, due to the awkward nature of the underlying Wasserstein space. The adopted strategy is based on finite-dimensional approximations of the value function obtained in terms of the related cooperative n n -player game, and on the construction of a smooth gauge-type function, built starting from a regularization of a sharp estimate of the Wasserstein metric; such a gauge-type function is used to generate maxima/minima through a suitable extension of the Borwein-Preiss generalization of Ekeland’s variational principle on the Wasserstein space.

Parallel Sum of Bounded Operators with Closed Ranges

Let H be a separable infinite dimensional complex Hilbert space and B(H) be the set of all bounded linear operators on H. In this paper, we present several conditions under which the distributive law of the parallel sum is valid. It is proved that the parallel sum for positive operators with closed ranges is continued at 0. For A,B∈B(H) with closed ranges, it is proved that A≤¯B if and only if A and B−A are parallel summable with the parallel sum A:(B−A)=0, where the symbol “≤¯” denotes the minus partial order.

Nonstationary fractionally integrated functional time series

We study a functional version of nonstationary fractionally integrated time series, covering the functional unit root as a special case. The time series taking values in an infinite-dimensional separable Hilbert space are projected onto a finite number of sub-spaces, the level of nonstationarity allowed to vary over them. Under regularity conditions, we derive a weak convergence result for the projection of the fractionally integrated functional process onto the asymptotically dominant sub-space, which retains most of the sample information carried by the original functional time series. Through the classic functional principal component analysis of the sample variance operator, we obtain the eigenvalues and eigenfunctions which span a sample version of the dominant sub-space. Furthermore, we introduce a simple ratio criterion to consistently estimate the dimension of the dominant sub-space, and use a semiparametric local Whittle method to estimate the memory parameter. Monte-Carlo simulation studies are given to examine the finite-sample performance of the developed techniques.

A Note on the Range of a Derivation

Let H be a separable infinite dimensional complex Hilbert space, and let L(H) denote the algebra of all bounded linear operators on H into itself. Given A, B ∈ L(H), define the generalized derivation δA, B ∈ L(L(H)) by δA, B(X) = AX - XB. An operator A ∈ L(H) is P-symmetric if AT = TA implies AT* = T* A for all T ∈ C1(H) (trace class operators). In this paper, we give a generalization of P-symmetric operators. We initiate the study of the pairs (A, B) of operators A, B ∈ L(H) such that R(δA, B) W* = R(δA, B) W*, where R(δA, B) W* denotes the ultraweak closure of the range of δA, B. Such pairs of operators are called generalized P-symmetric. We establish a characterization of those pairs of operators. Related properties of P-symmetric operators are also given.

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.

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.

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.

Perturbations of spectra and property (R) for upper triangular operator matrices

Let H and K be two complex infinite dimensional separable Hilbert spaces. For T∈B(H), T is said to satisfy property (R) if the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper, we mainly give the sufficient and necessary conditions for the 2×2 upper triangular operator matrices such that they satisfy property (R) using the features of the elements on the diagonal.

The spectral picture and joint spectral radius of the generalized spherical Aluthge transform

For an arbitrary commuting d–tuple T of Hilbert space operators, we fully determine the spectral picture of the generalized spherical Aluthge transform Δt(T) and we prove that the spectral radius of T can be calculated from the norms of the iterates of Δt(T). Let T≡(T1,…,Td) be a commuting d–tuple of bounded operators acting on an infinite dimensional separable Hilbert space, let P:=T1⁎T1+⋯+Td⁎Td, and let(T1⋮Td)=(V1⋮Vd)P be the canonical polar decomposition, with (V1,…,Vd) a (joint) partial isometry and⋂i=1dkerTi=⋂i=1dkerVi=kerP. For 0≤t≤1, we define the generalized spherical Aluthge transform of T byΔt(T):=(PtV1P1−t,…,PtVdP1−t). We also let ‖T‖2:=‖P‖. We first determine the spectral picture of Δt(T) in terms of the spectral picture of T; in particular, we prove that, for any 0≤t≤1, Δt(T) and T have the same Taylor spectrum, the same Taylor essential spectrum, the same Fredholm index, and the same Harte spectrum. We then study the joint spectral radius rT(T), and prove that rT(T)=limn⁡‖Δt(n)(T)‖2(0<t<1), where Δt(n) denotes the n–th iterate of Δt. For d=t=1, we give an example where the above formula fails.

Stochastic optimal control in infinite dimensions with state constraints

We consider the state-constrained stochastic optimal control problem in infinite-dimensional separable Hilbert spaces, where the state process is driven by the Q-Wiener process and the (possibly unbounded) linear operator. By applying the stochastic target theory and the backward reachability approach, we show that the original (possibly discontinuous) value function can be represented by the zero-level set of the auxiliary (continuous) value function. The auxiliary value function is obtained from the penalized unconstrained stochastic control problem (in infinite dimensions) that includes an additional control variable as a consequence of the (infinite-dimensional) martingale representation theorem. We then prove that the auxiliary value function is a unique (continuous) viscosity solution to the associated Hamilton–Jacobi–Bellman (HJB) equation in infinite dimensions. Note that the viscosity analysis developed in our paper generalizes that presented in the existing literature, since the corresponding infinite-dimensional HJB equation includes an additional operator-valued control variable in the Hamiltonian maximization and depends on an additional initial state variable.

Linear expand-contract plasticity of ellipsoids revisited

This work is aimed to describe linearly expand-contract plastic ellipsoids given via quadratic form of a bounded positively defined self-adjoint operator in terms of its spectrum.Let $Y$ be a metric space and $F\colon Y\to Y$ be a map. $F$ is called non-expansive if it does not increase distance between points of the space $Y$. We say that a subset $M$ of a normed space $X$ is linearly expand-contract plastic (briefly an LEC-plastic) if every linear operator $T\colon X \to X$ whose restriction on $M$ is a non-expansive bijection from $M$ onto $M$ is an isometry on $M$.In the paper, we consider a fixed separable infinite-dimensional Hilbert space $H$. We define an ellipsoid in $H$ as a set of the following form $E =\left\{x \in H\colon \left\langle x, Ax \right\rangle \le 1 \right\}$ where $A$ is a self-adjoint operator for which the following holds: $\inf_{\|x\|=1} \left\langle Ax,x\right\rangle &gt;0$ and $\sup_{\|x\|=1} \left\langle Ax,x\right\rangle &lt; \infty$.We provide an example which demonstrates that if the spectrum of the generating operator $A$ has a non empty continuous part, then such ellipsoid is not linearly expand-contract plastic.In this work, we also proof that an ellipsoid is linearly expand-contract plastic if and only if the spectrum of the generating operator $A$ has empty continuous part and every subset of eigenvalues of the operator $A$ that consists of more than one element either has a maximum of finite multiplicity or has a minimum of finite multiplicity.

Nearly invariant subspaces for shift semigroups

Let {T(t)}t⩾0 be a C0-semigroup on an infinite-dimensional separable Hilbert space; a suitable definition of near {T(t)*}t⩾0 invariance of a subspace is presented in this paper. A series of prototypical examples for minimal nearly {S(t)*}t⩾0 invariant subspaces for the shift semigroup {S(t)}t⩾0 on L2(0, ∞) are demonstrated, which have close links with near $$T_\theta^*$$ invariance on Hardy spaces of the unit disk for an inner function θ. Especially, the corresponding subspaces on Hardy spaces of the right half-plane and the unit disk are related to model spaces. This work further includes a discussion on the structure of the closure of certain subspaces related to model spaces in Hardy spaces.

A representation of compact C-normal operators

Let C be a conjugation on a complex separable Hilbert space H. A bounded linear operator T is said to be C-normal if . In this paper, first, we give a representation of C-normal operators on finite dimensional Hilbert space and later extend it to compact C-normal operators on infinite-dimensional separable Hilbert spaces. In the end, we discuss the eigenvalue problem for C-normal operators and show that every compact C-normal operator has a solution for the eigenvalue problem.

On Hereditarily Codiskcyclic Operators

Many codiskcyclic operators on infinite-dimensional separable Hilbert space do not satisfy the criterion of codiskcyclic operators. In this paper, a kind of codiskcyclic operators satisfying the criterion has been characterized, the equivalence between them has been discussed and the class of codiskcyclic operators satisfying their direct summand is codiskcyclic. Finally, this kind of operators is used to prove that every codiskcyclic operator satisfies the criterion if the general kernel is dense in the space.

CFI upper triangular operator matrices

Let and be complex infinite dimensional separable Hilbert spaces. For given operators and , we provide necessary and sufficient conditions which make a upper triangular operator matrix a CFI operator for some (or every) .

On the complete metrisability of spaces of contractive semigroups

The space of unitary C_{0}-semigroups on a separable infinite-dimensional Hilbert space, when viewed under the topology of uniform weak operator convergence on compact subsets of {mathbb {R}}_{+}, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [4]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive C_{0}-semigroups constitute Polish spaces and thus positively addresses the open problem.

On simplicity of Lie algebras of compact operators: A direct approach

We investigate an algebraic variant of the Wojtyński problem on the simplicity of Lie algebras of compact operators on a separable infinite-dimensional complex Hilbert space. We prove the non-simplicity of Lie algebras of compact operators under a mild softness condition using the notion of soft-edged operator ideals introduced by Kaftal and Weiss. We believe our study offers a new perspective on the investigation of the simplicity of Lie algebras by relating it to the study of operator ideals.

On the space of Schwartz operators in the symmetric Fock space and its dual

A long-standing problem that arises when constructing the mathematical apparatus of quantum mechanics is the need to work with unbounded operators. Since the space of nuclear operators is preconjugate for the algebra of all bounded operators, we can consider the states of a quantum system as nuclear operators, and consider bounded operators as observables. In this case, taking the trace for the product of a nuclear operator (a quantum state) and a bounded operator (a quantum observable) gives the average value of a quantum observable in a fixed state. The existence of such an average for unbounded operators is not guaranteed. If we want to define a space of observables that includes such naturally occurring unbounded operators as the position and momentum operators, for which average values are always determined, we should consider a space of states smaller than all nuclear operators. Recently, this approach has been mathematically correct implemented in the Hilbert space H = L^2(R^N). The so-called space of Schwarz operators, equipped with a system of semi-norms and being a Frechet space, was chosen as the space of states. Schwarz operators appeared to be integral operators whose kernels are functions belonging to the usual Schwarz space. The dual space to the space of Schwarz operators should be considered as the space of quantum observables and it really includes such standard observables as polynomials from the products of the position and momentum operators. In the present work we transfer this approach to the symmetric Fock space H = F(H) over an infinite dimensional separable Hilbert space H. We introduce the space of Schwartz operators in F(H) over an infinite dimensional separable space H and investigate which of the standard operators of quantum white noise belong to the space dual to the space of Schwartz operators.