Space Of Hilbert-Schmidt Operators Research Articles

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.

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.

Methodology and Convergence Rates for Functional Time Series Regression

The functional linear model extends the notion of linear regression to the case where the response and covariates are iid elements of an infinite dimensional Hilbert space. The unknown to be estimated is a Hilbert-Schmidt operator, whose inverse is by definition unbounded, rendering the problem of inference ill-posed. In this paper, we consider the more general context where the sample of response/covariate pairs forms a weakly dependent stationary process in the respective product Hilbert space: simply stated, the case where we have a regression between functional time series. We consider a general framework of potentially nonlinear processes, exploiting recent advances in the spectral analysis of time series. Our main result is the establishment of the rate of convergence for the corresponding estimators of the regression coefficients, the latter forming a summable sequence in the space of Hilbert-Schmidt operators. In a sense, our main result can be seen as a generalisation of the classical functional linear model rates, to the case of time series, and rests only upon cumulant mixing conditions. While the analysis becomes considerably more involved in the dependent case, the rates are strikingly comparable to those of the i.i.d. case, but at the expense of an additional factor caused by the necessity to estimate the spectral density operator at a nonparametric rate, as opposed to the parametric rate for covariance operator estimation.

Closable Multipliers

Let (X, μ) and (Y, ν) be standard measure spaces. A function $${\varphi\in L^\infty(X\times Y,\mu\times\nu)}$$ is called a (measurable) Schur multiplier if the map S φ , defined on the space of Hilbert-Schmidt operators from L 2(X, μ) to L 2(Y, ν) by multiplying their integral kernels by φ, is bounded in the operator norm. The paper studies measurable functions φ for which S φ is closable in the norm topology or in the weak* topology. We obtain a characterisation of w*-closable multipliers and relate the question about norm closability to the theory of operator synthesis. We also study multipliers of two special types: if φ is of Toeplitz type, that is, if φ(x, y) = f(x − y), $${x,y\in G}$$ , where G is a locally compact abelian group, then the closability of φ is related to the local inclusion of f in the Fourier algebra A(G) of G. If φ is a divided difference, that is, a function of the form (f(x) − f(y))/(x − y), then its closability is related to the “operator smoothness” of the function f. A number of examples of non-closable, norm closable and w*-closable multipliers are presented.

Représentation autorégressive de l'opérateur de covariance empirique d'un ARH(1). Applications

Given an order 1 autoregressive process with values in a separable Hilbert space we show that its empirical covariance operator, say Cn, is the empirical mean of an autoregressive process which takes its values in the space of Hilbert-Schmidt operators with a noise which is an Hilbertian martingale difference. This allows to obtain precise results concerning the asymptotic behaviour of C n and its eigenvalues.

Convergence of Galerkin approximations for operator Riccati equations—A nonlinear evolution equation approach

We develop an approximation and convergence theory for Galerkin approximations to infinite dimensional operator Riccati differential equations formulated in the space of Hilbert-Schmidt operators on a separable Hilbert space. We treat the Riccati equation as a nonlinear evolution equation with dynamics described by a nonlinear monotone perturbation of a strongly coercive linear operator. We prove a generic approximation result for quasi-autonomous nonlinear evolution systems involving accretive operators which we then use to demonstrate the Hilbert-Schmidt norm convergence of Galerkin approximations to the solution of the Riccati equation. We illustrate the application of our results in the context of a linear quadratic optimal control problem for a one dimensional heat equation.

On quantum holonomy for mixed states

There is a natural connection and parallel transport on the Hilbert tensor product ℋ⊗ℋ (or, equivalently, the space of Hilbert-Schmidt operators), the elements of which represent density matrices in ℋ up to unitary operators. We postulate a time evolution equation, which leads to this connection after extracting a proper ‘dynamical’ unitary phase. As an example, we compute the holonomy of a loop of temperature states for the spin in a rotating magnetic field.