Search results for "Operator theory"
showing 10 items of 95 documents
Automatic continuity of generalized local linear operators
1980
In this note, we present a general automatic continuity theory for linear mappings between certain topological vector spaces. The theory applies, in particular, to local operators between spaces of functions and distributions, to algebraic homomorphisms between certain topological algebras, and to linear mappings intertwining generalized scalar operators.
On spectra of geometric operators on open manifolds and differentiable groupoids
2001
We use a pseudodifferential calculus on differentiable groupoids to obtain new analytical results on geometric operators on certain noncompact Riemannian manifolds. The first step is to establish that the geometric operators belong to a pseudodifferential calculus on an associated differentiable groupoid. This then leads to Fredholmness criteria for geometric operators on suitable noncompact manifolds, as well as to an inductive procedure to compute their essential spectra. As an application, we answer a question of Melrose on the essential spectrum of the Laplace operator on manifolds with multicylindrical ends.
Schaefer–Krasnoselskii fixed point theorems using a usual measure of weak noncompactness
2012
Abstract We present some extension of a well-known fixed point theorem due to Burton and Kirk [T.A. Burton, C. Kirk, A fixed point theorem of Krasnoselskii–Schaefer type, Math. Nachr. 189 (1998) 423–431] for the sum of two nonlinear operators one of them compact and the other one a strict contraction. The novelty of our results is that the involved operators need not to be weakly continuous. Finally, an example is given to illustrate our results.
Ergodic properties of operators in some semi-Hilbertian spaces
2012
This article deals with linear operators T on a complex Hilbert space ℋ, which are bounded with respect to the seminorm induced by a positive operator A on ℋ. The A-adjoint and A 1/2-adjoint of T are considered to obtain some ergodic conditions for T with respect to A. These operators are also employed to investigate the class of orthogonally mean ergodic operators as well as that of A-power bounded operators. Some classes of orthogonally mean ergodic or A-ergodic operators, which come from the theory of generalized Toeplitz operators are considered. In particular, we give an example of an A-ergodic operator (with an injective A) which is not Cesaro ergodic, such that T * is not a quasiaff…
Metric operators, generalized hermiticity and partial inner product spaces
2015
A quasi-Hermitian operator is an operator in a Hilbert space that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Motivated by the recent developments of pseudo-Hermitian quantum mechanics, we analyze the structure of metric operators, bounded or unbounded, in a Hilbert space. We introduce several generalizations of the notion of similarity between operators and explore to what extent they preserve spectral properties. Next we consider canonical lattices of Hilbert spaces generated by unbounded metric operators. Since such lattices constitute the simplest case of a partial inner product space (PIP space), we can exploit the te…
Metric Operators, Generalized Hermiticity and Lattices of Hilbert Spaces
2015
Pseudo-Hermitian quantum mechanics (QM) is a recent, unconventional, approach to QM, based on the use of non-self-adjoint Hamiltonians, whose self-adjointness can be restored by changing the ambient Hilbert space, via a so-called metric operator. The PT-symmetric Hamiltonians are usually pseudo-Hermitian operators, a term introduced a long time ago by Dieudonné for characterizing those bounded operators A that satisfy a relation of the form GA = A G, where G is a metric operator, that is, a strictly positive self-adjoint operator. This chapter explores further the structure of unbounded metric operators, in particular, their incidence on similarity. It examines the notion of similarity betw…
Weyl type theorems for bounded linear operators on Banach spaces
2011
In 1909 H. Weyl [59] studied the spectra of all compact linear perturbations of a self-adjoint operator defined on a Hilbert space and found that their intersection consisted precisely of those points of the spectrum where are not isolated eigenvalues of nite multiplicity. Later, the property established by Weyl for self-adjoint operators has been observed for several other classes of operators, for instance hyponormal operators on Hilbert spaces, Toeplitz operators,convolution operators on group algebras, and many other classes of operators ned on Banach spaces . In the literature, a bounded linear operator defined on a Banach space which satisfies this property is said to satisfy Weyl's t…
Some characterizations of operators satisfying a-Browder's theorem
2005
Abstract We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C . Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λ I − T .
Time operators, innovations and approximations
2003
Abstract We present a new approach to the spectral analysis and prediction of such complex systems for which the time evolution is described by a semigroup of operators. This approach is based on an extended concept of time operator and can be interpreted as a shift representation of dynamical systems. The time operator method includes the multiresolution analysis of wavelets as a particular case but can also be applied for a substantially larger class of dynamical systems. Among the examples where shift representation have been explicitly derived are exact endomorphisms, the diffusion equation, generalized shifts associated with the Haar or Faber–Schauder basis and some classes of stochast…
Kernel methods and their derivatives: Concept and perspectives for the earth system sciences.
2020
Kernel methods are powerful machine learning techniques which implement generic non-linear functions to solve complex tasks in a simple way. They Have a solid mathematical background and exhibit excellent performance in practice. However, kernel machines are still considered black-box models as the feature mapping is not directly accessible and difficult to interpret.The aim of this work is to show that it is indeed possible to interpret the functions learned by various kernel methods is intuitive despite their complexity. Specifically, we show that derivatives of these functions have a simple mathematical formulation, are easy to compute, and can be applied to many different problems. We n…