Frames for source recovery from non-uniform dynamical samples

We exhibit a family of *-isomorphisms mapping the CAR algebra onto its even subalgebra. In 1970 E. Stormer proved that the even subalgebra of the CAR algebra over an infinite dimensional separable Hilbert space is UHF of type 2 ∞ , hence *-isomorphic to the CAR algebra itself (1). But it seems to be unknown that such isomorphisms have a nice and sim- ple realization in terms of Bogoliubov endomorphisms with "statistical dimension" √ 2. Bogoliubov endomorphisms are conveniently described using Araki's "selfdual" CAR algebra formalism (2). Let K be an infinite dimensional separable complex Hilbert space, equipped with a complex conjugation k 7→ k ∗ , and let C(K) denote the unique (simple) C*-algebra generated by 1 and the elements of K, subject to the anticommutation relation k ∗ k ' + kk ∗ = h k, ki 1, k, k ' ∈ K.

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.

In this work, we prove that any element in the tensor product of separable infinite-dimensional Hilbert spaces can be expressed as a matrix product state (MPS) of possibly infinite bond dimension. The proof is based on the singular value decomposition of compact operators and the connection between tensor products and Hilbert–Schmidt operators via the Schmidt decomposition in infinite-dimensional separable Hilbert spaces. The construction of infinite-dimensional MPS (idMPS) is analogous to the well-known finite-dimensional construction in terms of singular value decompositions of matrices. The infinite matrices in idMPS give rise to operators acting on (possibly infinite-dimensional) auxiliary Hilbert spaces. As an example we explicitly construct an MPS representation for certain eigenstates of a chain of three coupled harmonic oscillators.

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega $ but also in $\int \omega \,d\omega :=\omega ^{2} -\langle \omega \rangle $ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega ^{2} -\langle \omega \rangle $ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

In this paper, we study the existence of classical and generalized solutions for nonlinear differential inclusions <TEX>$x^{\prime}(t){\in}F(t,x(t))$</TEX> in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condensing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper.

In this note we investigate Weyl's theorem for �-paranormal operators on a separable infinite dimensional Hilbert space. We prove that if T is a �-paranormal operator satisfying Property (E) - (TI)HT({�}) is closed for each � 2 C, where HT({�}) is a local spectral subspace T, then Weyl's theorem holds for T. Let H denote an infinite dimensional separable Hilbert space. Let B(H) and K(H) denote the algebra bounded linear operators and the ideal compact operators on H, respectively. If T 2 B(H) write N(T) and R(T) for the null space and range T; �(T) := dimN(T); �(T) := dimN(T � ); �(T) for the spectrum T; �ap(T) for the approximate point spectrum T; �0(T) for the set eigenvalues T. An operator T 2 B(H) is called Fredholm if it has closed range with finite dimensional null space and its range finite co-dimension. The index a Fredholm operator T 2 B(H) is given by ind(T) := �(T) − �(T). An operator T 2 B(H) is called Weyl if it is Fredholm index zero. An operator T 2 B(H) is called Browder if it is Fredholm of finite ascent and descent: equivalently ((11, Theorem 7.9.3)) if T is Fredholm and T − �I is invertible for sufficiently small� 6 0 in C. The essential spectrume(T), the Weyl spectrum !(T) and the Browder spectrumb(T) T 2 B(H) are defined by ((10), (11), (12))

Let <TEX>$\mathcal{H}$</TEX> be a complex separable infinite dimensional Hilbert space. In this paper, a necessary and sufficient condition is given for an operator T on <TEX>$\mathcal{H}$</TEX> to satisfy that <TEX>$f(T)$</TEX> obeys generalized Weyl's theorem for each function <TEX>$f$</TEX> analytic on some neighborhood of <TEX>${\sigma}(T)$</TEX>. Also we investigate the stability of generalized Weyl's theorem under (small) compact perturbations.

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic probability which is positive on non-empty open sets, we conclude that there is a C0-residual subset of cocycles within which, for almost every x, either the Oseledets–Ruelle's decomposition along the orbit of x is dominated or all the Lyapunov exponents are equal to -∞.

The Invariant Subset Problem on the Hilbert space is to know whether there exists a bounded linear operator T on a separable infinite-dimensional Hilbert space H such that the orbit {Tnx; n ≥ 0} of every non-zero vector x ∈ H under the action of T is dense in H. We show that there exists a bounded linear operator T on a complex separable infinite-dimensional Hilbert space H and a unitary operator V on H, such that the following property holds true: for every non-zero vector x ∈ H, either x or V x has a dense orbit under the action of T. As a consequence, we obtain in particular that there exists a minimal action of the free semi-group with two generators F+ 2 on a complex separable infinite-dimensional Hilbert space H. The proof involves Read’s type operators on the Hilbert space, and we show in particular that these operators — which were potential counterexamples to the Invariant Subspace Problem on the Hilbert space — do have non-trivial invariant closed subspaces.

For a pair of linear bounded operators $T$ and $A$ on a complex Banach space $X$, if $T$ commutes with $A,$ then the orbits $\{A^n TA^{-n}\}$ of $T$ under $A$ are uniformly bounded. The study of the converse implication was started in the 1970s by J. A. Deddens. In this paper, we present a new approach to this type of question using two localization theorems; one is an operator version of a theorem of tauberian type given by Katznelson-Tzafriri and the second one is on power-bounded operators by Gelfand-Hille. This improves former results of Deddens-Stampfli-Williams.

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.

Resumen

Recently, Geher and Semrl have characterized the general form of surjective isometries of the set of all projections on an infinite-dimensional separable Hilbert space using unitaries and antiunitaries. In this paper, we study the surjective L2-isometries of the projection lattice of an infinite dimensional Hilbert space and show that every such isometry can also be described by unitaries and antiunitaries.

We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded operators are determined by unitarily equivalent classes of the operator ranges and the nullity of the original bounded operators giving graphs. We construct several non-isomorphic examples of two subspace systems in an infinite-dimensional Hilbert space. Even if we study n subspace systems for $$n \ge 3$$ , we can use the analysis of any two subspaces of the n subspaces.

We study the relative position of three subspaces in a separable infinite-dimensional Hilbert space. In the finite-dimensional case, Brenner described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space. We also give a partial result which gives a condition on a system to have a (dense) decomposition containing a pentagon.