The problem of regression under Gaussian assumptions is treated generally. The relationship between Bayesian prediction, regularization and smoothing is elucidated. The ideal regression is the posterior mean and its computation scales as O(n3), where n is the sample size. We show that the optimal m-dimensional linear model under a given prior is spanned by the first m eigenfunctions of a covari...

It is shown that if a paranormal contraction T has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator Q = T 2∗T 2 − 2T ∗T + I also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator D is a proper contraction and its itself-commutator is a trace-class strict contr...

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such exte...

Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such exte...

In applications, many states given for a system can be expressed by orthonormal elements, called “state elements”, taken in separable Hilbert space (called space”). The exact nature of the depends on system; example, state position and momentum is square-integrable functions. symmetries quantum represented class unitary operators that act space. ladder have effect lowering or raising energy sta...

We define Schatten classes of adjointable operators on Hilbert modules over abelian C⁎-algebras. Many key features carry from the space case. In particular, form two-sided ideals compact and are equipped with a Banach norm C⁎-valued trace expected properties. For trivial bundles, we show that our Schatten-class can be identified bijectively Schatten-norm–continuous maps base into fiber, fiberwi...

In his 1953 paper [Matem. Sbornik 33 (1953), 597 – 626] Mark Krein presented an example of a symmetric rank one perturbation of a selfadjoint operator such that for all values of the spectral parameter in the interior of the spectrum, the difference of the corresponding spectral projections is not trace class. In the present note it is shown that in the case in question this difference has simp...

Josza’s definition of fidelity [10] for a pair of (mixed) quantum states is studied in the context of two types of operator algebras. The first setting is mainly algebraic in that it involves unital C∗-algebras A that possess a faithful trace functional τ . In this context, the role of quantum states (that is, density operators) in the classical quantum-mechanical framework is assumed by positi...

We show that the eigenvalues of the first order partial differential equation derived by quasi-classical approximation of the Schrödinger equation can be computed from the trace of a classical operator. The derived trace formula is different from the Gutzwiller trace formula. The Fredholm determinant of the new operator is an entire function of the complex energy plane in contrast to the semi-c...

We derive a new pointwise characterization of the subdifferential of the total variation (TV) functional. It involves a full trace operator which maps certain L vectorfields to integrable functions with respect to the total variation measure of the derivative of a bounded variation function. This full trace operator extents a notion of normal trace, frequently used, for example, to characterize...