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We propose a new approach to construct the eigenvalue expansion in a weighted Hilbert space of the solution to the Cauchy problem associated to Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators turn out to be…
We study the spectral convergence of compact, self-adjoint operators on a separable Hilbert space under operator norm perturbations, and derive asymptotic expansions for their eigenvalues and eigenprojections. Our analysis focuses on…
We present a streamlined approach for generalized strong and norm convergence of self-adjoint operators in different Hilbert spaces. In particular, we establish convergence of associated (semi-)groups, (essential) spectra and spectral…
A spectral representation for solutions to linear Hamilton equations with nonnegative energy in Hilbert spaces is obtained. This paper continues our previous work on Hamilton equations with positive definite energy. Our approach is a…
In this work we introduce a new measure for the dispersion of the spectral scale of a Hermitian (self-adjoint) operator acting on a separable infinite dimensional Hilbert space that we call spectral spread. Then, we obtain some…
In this paper, we introduce and study non-local Jacobi operators, which generalize the classical (local) Jacobi operators. We show that these operators extend to generators of ergodic Markov semigroups with unique invariant probability…
We put together a general framework to deal with elliptic and parabolic equations associated with (nonlinear) nonlocal (fractional order) operators. Many well-known nonlocal operators enter into our framework, and in addition one may…
This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is both motivated by…
We study heat and wave type equations on a separable Hilbert space $\mathcal{H}$ by considering non-local operators in time with any positive densely defined linear operator with discrete spectrum. We show the explicit representation of the…
The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of…
We introduce a simple, general, and convergent scheme to compute generalized eigenfunctions of self-adjoint operators with continuous spectra on rigged Hilbert spaces. Our approach does not require prior knowledge about the eigenfunctions,…
We introduce, in the dual Macaev ideal of compact operators of a Hilbert space, the spectral weight $\rho(L)$ of a positive, self-adjoint operator $L$ having discrete spectrum away from zero. We provide criteria for its measurability and…
We establish a spectral representation for solutions to linear Hamilton equations with positive definite energy in a Hilbert space. Our approach is a special version of M. Krein's spectral theory of J-selfadjoint operators is the Hilbert…
Classical Laguerre spectral approximations are highly effective on the half-line when the target function is smooth in the usual polynomial scale. However, their accuracy deteriorates for nonsmooth functions. Such behavior appears naturally…
We construct a functional model for rank one perturbations of compact normal operators acting in a certain Hilbert spaces of entire functions generalizing de Branges spaces. Using this model we study completeness and spectral synthesis…
Covering ill-posed problems with compact and non-compact operators regarding the degree of ill-posedness is a never ending story written by many authors in the inverse problems literature. This paper tries to add a new narrative and some…
In this paper, we provide the spectral decomposition in Hilbert space of the $\mathcal{C}_0$-semigroup $P$ and its adjoint $\hatP$ having as generator, respectively, the Caputo and the right-sided Riemann-Liouville fractional derivatives of…
The aim of this paper is to offer an original and comprehensive spectral theoretical approach to the study of convergence to equilibrium, and in particular of the hypocoercivity phenomenon, for contraction semigroups in Hilbert spaces. Our…
In this paper spectral theorems for not necessarily continuous normal and self-adjoint random operators on a complex separable Hilbert space are proved.
It has been unknown in kinetic theory whether the linearized Boltzmann or Landau equation with soft potentials admits a spectral gap in the spatially inhomogeneous setting. Most of existing works indicate a negative answer because the…