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Concept# Hilbert–Schmidt operator

Summary

In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
|A|^2_{\operatorname{HS}} \ \stackrel{\text{def}}{=}\ \sum_{i \in I} |Ae_i|^2_H,
where {e_i: i \in I} is an orthonormal basis. The index set I need not be countable. However, the sum on the right must contain at most countably many non-zero terms, to have meaning. This definition is independent of the choice of the orthonormal basis.
In finite-dimensional Euclidean space, the Hilbert–Schmidt norm |\cdot|_\text{HS} is identical to the Frobenius norm.
||·||HS is well defined
The Hilbert–Schmidt norm does not depend on the choice of orthonormal basis. Indeed, if {e_i}*{i\in I} and {f_j}*{j\in I} are such bases, then

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Jean Louis-Alexandre Fornerod, Hoài-Minh Nguyên

Cakoni and Nguyen recently proposed very general conditions on the coefficients of Maxwell equations for which they established the discreten ess of the set of eigenvalues of the transmission problem and studied their locations. In this paper, we establish the completeness of the generalized eigenfunctions and derive an optimal upper bound for the counting function under these conditions, assuming additionally that the coefficients are twice continuously differentiable. The approach is based on the spectral theory of Hilbert-Schmidt operators.

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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, expoiting recent advances in the spectral analysis of functional time series. This allows us to quantify the inherent ill-posedness, and to motivate a Tikhonov regularisation technique in the frequency domain. Our main result is 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 Brillinger-type mixing conditions. It is seen that, just as the covariance operator eigenstructure plays a central role in the independent case, so does the spectral density operator's eigenstructure in the dependent case. 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.

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