Hilbert-Schmidt Operators Research Articles

Stabilizing Discontinuous Galerkin Methods Using Dafermos’ Entropy Rate Criterion: II—Systems of Conservation Laws and Entropy Inequality Predictors

A novel approach for the stabilization of the Discontinuous Galerkin method based on the Dafermos entropy rate crition is presented. First, estimates for the maximal possible entropy dissipation rate of a weak solution are derived. Second, families of conservative Hilbert–Schmidt operators are identified to dissipate entropy. Steering these operators using the bounds on the entropy dissipation results in high-order accurate shock-capturing DG schemes for the one-dimensional Euler equations, satisfying the entropy rate criterion and an entropy inequality. Other testcases include the one-dimensional Buckley–Leverett equation.

Linearized Boltzmann collision operator for a mixture of monatomic and polyatomic chemically reacting species

At higher altitudes near space shuttles moving at hypersonic speed the air is excited to high temperatures. Then not only mechanical collisions are affecting the gas flow, but also chemical reactions have an impact on such hypersonic flows. In this work we insert chemical reactions, in form of dissociations and associations, in a model for a mixture of mono- and polyatomic (non-reacting) species. More general chemical reactions, e.g., bimolecular ones, can be obtained by instant combinations of the considered reactions. Polyatomicity is here modelled by a continuous internal energy variable and the evolution of the gas is described by a Boltzmann equation. In the Chapman-Enskog process—and related half-space problems—the linearized Boltzmann collision operator plays a central role. Here we extend some important properties of the linearized operator to the considered model with chemical reactions. A compactness result, that the linearized operator can be decomposed into a sum of a positive multiplication operator—the collision frequency—and a compact integral operator, is obtained. The terms of the integral operator are shown to be (at least) uniform limits of Hilbert-Schmidt integral operators and, thereby, compact operators. Self-adjointness of the linearized operator follows as a direct consequence. Also, bounds on—including coercivity of—the collision frequency is obtained for hard sphere, as well as hard potentials with cutoff, like models. As consequence, Fredholmness as well as the domain of the linearized operator are obtained.

Learning Nonlinear Systems via Volterra Series and Hilbert-Schmidt Operators

Learning Nonlinear Systems via Volterra Series and Hilbert-Schmidt Operators

التكرار في فضاء مؤثرات هلبرت-شميدت

في هذا البحث, تم برهان بانه اذا كانت شبه الزمرة-C0 فوضوية وزائدة او فائقة الاختلاط فان الضرب من جهة اليسار المرتبط بشبه الزمرة-C0 في فضاء مؤثرات هلبرت-شميدت متكرر اذا وفقط اذا كان مفرط الدوران. كذلك, تم بيان انه في ظل بعض الشروط يكون تكرار شبه الزمرة-C0 وتكرار الضرب لشبه الزمرة-C0 من جهة اليسار المرتبط بها في فضاء مؤثرات هلبرت-شميدت متكافئين. علاوة على ذلك, بعض الشروط الكافية للتكرار وزيادة الدوران للضرب من جهة اليسار لشبه الزمرة -C0 تم عرضها بالاعتماد على المجموعات الجزئية المتشعبة.

On the M 2,A,A -numerical range and the M 2,A,A -maximal numerical range of the basic elementary operator M 2,B,C

Let A be a positive bounded operator acting on a complex Hilbert space H . For two bounded operators B and C on H , we denote by M 2 , B , C the basic elementary operator on the class of Hilbert–Schmidt operators C 2 ( H ) , i.e. M 2 , B , C ( X ) = BXC for all X ∈ C 2 ( H ) . In this paper, we investigate the A -numerical range W A ( M 2 , B , ( C ♯ A ) ∗ ) and the A -maximal numerical range W max A ( M 2 , B , ( C ♯ A ) ∗ ) , where A = M 2 , A , A , C ♯ A is the reduced solution of the equation AX = C ∗ A and C ∗ is the adjoint of C. Within this framework, we show, under some A-hyponormality conditions, the following two equalities: W A ( M 2 , B , ( C ♯ A ) ∗ ) ¯ = co ( W A ( B ) ¯ ⋅ W A ( C ) ¯ ) and W max A ( M 2 , B , ( C ♯ A ) ∗ ) = co ( W max A ( B ) ⋅ W max A ( C ) ) , where W A ( ⋅ ) , W max A ( ⋅ ) and co ( ⋅ ) denote respectively the A-numerical range, the A-maximal numerical range and the convex hull. Here, the bar stands for the closure. The first equality allows us to establish that ‖ M 2 , B , ( C ♯ A ) ∗ ‖ A = ‖ B ‖ A ‖ C ‖ A , where ‖ ⋅ ‖ A and ‖ ⋅ ‖ A designate the A -operator seminorm and the A-operator seminorm, respectively.

Compactness property of the linearized Boltzmann collision operator for a multicomponent polyatomic gas

The linearized Boltzmann collision operator is fundamental in many studies of the Boltzmann equation and its main properties are of substantial importance. The decomposition into a sum of a positive multiplication operator, the collision frequency, and an integral operator is trivial. Compactness of the integral operator for monatomic single species is a classical result, while corresponding results for monatomic mixtures and polyatomic single species are more recently obtained. This work concerns the compactness of the operator for a multicomponent mixture of polyatomic species, where the polyatomicity is modeled by a discrete internal energy variable. With a probabilistic formulation of the collision operator as a starting point, compactness is obtained by proving that the integral operator is a sum of Hilbert-Schmidt integral operators and operators, which are uniform limits of Hilbert-Schmidt integral operators, under some assumptions on the collision kernel. The assumptions are essentially generalizations of the Grad's assumptions for monatomic single species. Self-adjointness of the linearized collision operator follows. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model. Then it follows that the linearized collision operator is a Fredholm operator, and its domain is also obtained.

Formula omitted]-isometries and their harmonious applications to Hilbert-Schmidt operators

Numerous works have been dedicated to the topic of m-isometries, including [2–6,14,18–20,27,47,48]. In this article, we introduce the concept of (m,NA)-isometry, where A is a non-zero operator and m is a positive integer, as an extension of the m-isometry class created by J. Alger and M. Stankus in the 1980s. We present some algebraic and spectral characteristics of (m,NA)-isometries. Additionally, we investigate the product of an (m,NA)-isometry by an (n,NB)-isometry, which enhances and broadens the previous work of Gu et al. on m-isometries [40]. Finally, we apply our main findings to elementary operators defined on the Hilbert-Schmidt class, which can be identified with a tensor product. This provides a new, less complicated, and non-combinatorial proof of Theorem 2.10 of [30].

LRD spectral analysis of multifractional functional time series on manifolds

This paper addresses the estimation of the second-order structure of a manifold cross-time random field (RF) displaying spatially varying Long Range Dependence (LRD), adopting the functional time series framework introduced in Ruiz-Medina (Fract Calc Appl Anal 25:1426–1458, 2022). Conditions for the asymptotic unbiasedness of the integrated periodogram operator in the Hilbert–Schmidt operator norm are derived beyond structural assumptions. Weak-consistent estimation of the long-memory operator is achieved under a semiparametric functional spectral framework in the Gaussian context. The case where the projected manifold process can display Short Range Dependence (SRD) and LRD at different manifold scales is also analyzed. The performance of both estimation procedures is illustrated in the simulation study, in the context of multifractionally integrated spherical functional autoregressive–moving average (SPHARMA(p,q)) processes.

On linear stochastic flows

We study the existence of the stochastic flow associated to a linear stochastic evolution equation d ⁡ X = A X d ⁡ t + ∑ k B k X d ⁡ W k , \begin{equation*} \operatorname {d} X= AX\operatorname {d} t +\sum _{k} B_k X\operatorname {d} W_k, \end{equation*} on a Hilbert space. Our first result covers the case where A A is the generator of a C 0 C_0 -semigroup, and ( B k ) (B_k) is a sequence of bounded linear operators such that ∑ k ‖ B k ‖ > + ∞ \sum _k\|B_k\|>+\infty . We also provide sufficient conditions for the existence of stochastic flows in the Schatten classes beyond the space of Hilbert–Schmidt operators. Some new results and examples concerning the so-called commutative case are presented as well.

Representation of operators using fusion frames

To solve operator equations numerically, matrix representations are employing bases or more recently frames. For finding the numerical solution of operator equations a decomposition in subspaces is needed in many applications. To combine those two approaches, it is necessary to extend the known methods of matrix representation to the utilization of fusion frames. In this paper, we investigate this representation of operators on a Hilbert space H with Bessel fusion sequences, fusion frames and fusion Riesz bases. Fusion frames can be considered as a frame-like family of subspaces. Taking the particular property of the duality of fusion frames into account, this allows us to define a matrix representation in a canonical as well as an alternate way, the later being more efficient and well behaved in respect to inversion. We will give the basic definitions and show some structural results, like that the functions assigning the alternate representation to an operator is an algebra homomorphism. We give formulas for pseudo-inverses and the inverses (if existing) of such matrix representations. We apply this idea to Schatten p-class operators. Consequently, we show that tensor products of fusion frames are fusion frames in the space of Hilbert-Schmidt operators. We will show how this can be used for the solution of operator equations and link our approach to the additive Schwarz algorithm. Consequently, we propose some methods for solving an operator equation by iterative methods on the subspaces. Moreover, we implement the alternate Schwarz algorithms employing our perspective and provide small proof-of-concept numerical experiments. Finally, we show the application of this concept to overlapped convolution and the non-standard wavelet representation of operators.

The Hilbert–Schmidt property for semiclassical Fourier integral operators with operator symbol

In this paper, we define a particular class of semiclassical Fourier integral operators with operator symbol ([Formula: see text]-FIOs for short). We study the [Formula: see text]-boundedness and [Formula: see text]-compactness of [Formula: see text]-FIOs. Mainly, we prove that class of [Formula: see text]-FIOs denotes Hilbert–Schmidt operators if the order function belongs to [Formula: see text].

Radonification of a cylindrical Lévy process

Abstract In this work, we present a direct proof about radonification of a cylindrical Lévy process. The radonification technique has been very useful to define a genuine stochastic process starting from a cylindrical process; this is possible thanks to the Hilbert–Schmidt operators. With this work, we want to propose a self-contained simple proof to those who are not familiar with this method and also present our result which is to apply the radonification method to the case of a cylindrical Lévy process.

Cesàro-type operators associated with Borel measures on the unit disc acting on some Hilbert spaces of analytic functions

Given a complex Borel measure μ on the unit disc D={z∈C:|z|<1}, we consider the Cesàro-type operator Cμ defined on the space Hol(D) of all analytic functions in D as follows:If f∈Hol(D), f(z)=∑n=0∞anzn (z∈D), then Cμ(f)(z)=∑n=0∞μn(∑k=0nak)zn, (z∈D), where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn=∫Dwndμ(w).We study the action of the operators Cμ on some Hilbert spaces of analytic function in D, namely, the Hardy space H2 and the weighted Bergman spaces Aα2 (α>−1). Among other results, we prove that, if we set Fμ(z)=∑n=0∞μnzn (z∈D), then Cμ is bounded on H2 or on Aα2 if and only if Fμ belongs to the mean Lipschitz space Λ1/22. We prove also that Cμ is a Hilbert-Schmidt operator on H2 if and only if Fμ belongs to the Dirichlet space D, and that Cμ is a Hilbert-Schmidt operator on Aα2 if and only if Fμ belongs to the Dirichlet-type space D−1−α2.

Pseudo-differential operators, Wigner transform and Weyl transform on the affine Poincaré group

In this paper, we study harmonic analysis on the affine Poincaré group Paff, which is a non-unimodular group, and obtain pseudo-differential operators with operator valued symbols. More precisely, we study the boundedness properties of pseudo-differential operators on Paff. We also provide a necessary and sufficient condition on the operator-valued symbols such that the corresponding pseudo-differential operators are in the class of Hilbert–Schmidt operators. Consequently, we obtain a characterization of the trace class pseudo-differential operators on the Poincaré affine group Paff, and provide a trace formula for these trace class operators. Finally, we study the Wigner transform, and Weyl transform associated with the operator valued symbol on the Poincaré affine group Paff.

A Numerical Radius Inequality for Hilbert–Schmidt Operators

A Numerical Radius Inequality for Hilbert–Schmidt Operators

Linearized Boltzmann collision operator: II. Polyatomic molecules modeled by a continuous internal energy variable

The linearized collision operator of the Boltzmann equation for single species can be written as a sum of a positive multiplication operator, the collision frequency, and a compact integral operator. This classical result has more recently, been extended to multi-component mixtures and polyatomic single species with the polyatomicity modeled by a discrete internal energy variable. In this work we prove compactness of the integral operator for polyatomic single species, with the polyatomicity modeled by a continuous internal energy variable, and the number of internal degrees of freedom greater or equal to two. The terms of the integral operator are shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators. Self-adjointness of the linearized collision operator follows. Coercivity of the collision frequency are shown for hard-sphere like and hard potential with cut-off like models, implying Fredholmness of the linearized collision operator.

Linearized Boltzmann Collision Operator: I. Polyatomic Molecules Modeled by a Discrete Internal Energy Variable and Multicomponent Mixtures

The linearized Boltzmann collision operator appears in many important applications of the Boltzmann equation. Therefore, knowing its main properties is of great interest. This work extends some classical results for the linearized Boltzmann collision operator for monatomic single species to the case of polyatomic single species, while also reviewing corresponding results for multicomponent mixtures of monatomic species. The polyatomicity is modeled by a discrete internal energy variable, that can take a finite number of (given) different values. Results concerning the linearized Boltzmann collision operator being a nonnegative symmetric operator with a finite-dimensional kernel are reviewed.A compactness result, saying that the linearized operator can be decomposed into a sum of a positive multiplication operator, the collision frequency, and a compact operator, bringing e.g., self-adjointness, is extended from the classical result for monatomic single species, under reasonable assumptions on the collision kernel. With a probabilistic formulation of the collision operator as a starting point, the compactness property is shown by a splitting, such that the terms can be shown to be, or be the uniform limit of, Hilbert-Schmidt integral operators and as such being compact operators. Moreover, bounds on - including coercivity of - the collision frequency are obtained for a hard sphere like model, from which Fredholmness of the linearized collision operator follows, as well as its domain.

The completion of covariance kernels

We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I×I to a covariance kernel on the entire domain I×I. For a broad class of domains Ω called serrated domains, we are able to present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We interpret the canonical completion via the graphical model structure it induces on the associated Gaussian process. Furthermore, we show how the estimation of the canonical completion reduces to the solution of a system of linear statistical inverse problems in the space of Hilbert–Schmidt operators, and derive rates of convergence. We conclude by providing extensions of our theory to more general forms of domains, and by demonstrating how our results can be used to construct covariance estimators from sample path fragments of the associated stochastic process. Our results are illustrated numerically by way of a simulation study and a real example.

Pairwise learning problems with regularization networks and Nyström subsampling approach

Pairwise learning usually refers to the learning problem that works with pairs of training samples, such as ranking, similarity and metric learning, and AUC maximization. To overcome the challenge of pairwise learning in the large scale computation, this paper introduces Nyström sampling approach to the coefficient-based regularized pairwise algorithm in the context of kernel networks. Our theorems establish that the obtained Nyström estimator achieves the minimax error over all estimators using the whole data provided that the subsampling level is not too small. We derive the function relation between the subsampling level and regularization parameter that guarantees computation cost reduction and asymptotic behaviors’ optimality simultaneously. The Nyström coefficient-based pairwise learning method does not require the kernel to be symmetric or positive semi-definite, which provides more flexibility and adaptivity in the learning process. We apply the method to the bipartite ranking problem, which improves the state-of-the-art theoretical results in previous works. By developing probability inequalities for U-statistics on Hilbert–Schmidt operators, we provide new mathematical tools for handling pairs of examples involved in pairwise learning.

Power-norms based on Hilbert $$C^*$$-modules

Suppose that $${\mathscr {E}}$$ and $${\mathscr {F}}$$ are Hilbert $$C^*$$ -modules. We present a power-norm $$\left( \left\| \cdot \right\| ^{{\mathscr {E}}}_n:n\in {\mathbb {N}}\right) $$ based on $${\mathscr {E}}$$ and obtain some of its fundamental properties. We introduce a new definition of the absolutely (2, 2)-summing operators from $${\mathscr {E}}$$ to $${\mathscr {F}}$$ , and denote the set of such operators by $${\tilde{\Pi }}_2({\mathscr {E}},{\mathscr {F}})$$ with the convention $${\tilde{\Pi }}_2({\mathscr {E}})={\tilde{\Pi }}_2({\mathscr {E}},{\mathscr {E}})$$ . It is known that the class of all Hilbert–Schmidt operators on a Hilbert space $${\mathscr {H}}$$ is the same as the space $${\tilde{\Pi }}_2({\mathscr {H}})$$ . We show that the class of Hilbert–Schmidt operators introduced by Frank and Larson coincides with the space $${\tilde{\Pi }}_2({\mathscr {E}})$$ for a countably generated Hilbert $$C^*$$ -module $${\mathscr {E}}$$ over a unital commutative $$C^*$$ -algebra. These results motivate us to investigate the properties of the space $${\tilde{\Pi }}_2({\mathscr {E}},{\mathscr {F}})$$ .