Search results for "Mathematics::Functional Analysis"
showing 10 items of 236 documents
Selected Topics on Banach Space Theory
2019
Basic topics on Banach space theory needed for the text are reviewed. Hahn-Banach theorem, Baire’s theorem, uniform boundedness principle, closed graph theorem, weak topologies, Banach-Alaoglu theorem, unconditional basis, Banach sequence spaces, summing operators, factorable operators, cotype, Kahane inequality.
Critical points for nondifferentiable functions in presence of splitting
2006
A classical critical point theorem in presence of splitting established by Brézis-Nirenberg is extended to functionals which are the sum of a locally Lipschitz continuous term and of a convex, proper, lower semicontinuous function. The obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational-hemivariational eigenvalue problems. © 2005 Elsevier Inc. All rights reserved.
Dimension gap under Sobolev mappings
2015
Abstract We prove an essentially sharp estimate in terms of generalized Hausdorff measures for the images of boundaries of Holder domains under continuous Sobolev mappings, satisfying suitable Orlicz–Sobolev conditions. This estimate marks a dimension gap, which was first observed in [2] for conformal mappings.
Browder-Type Theorems
2018
This chapter may be viewed as the part of the book in which the interaction between local spectral theory and Fredholm theory comes into focus. The greater part of the chapter addresses some classes of operators on Banach spaces that have a very special spectral structure. We have seen that the Weyl spectrum σw(T) is a subset of the Browder spectrum σb(T) and this inclusion may be proper. In this chapter we investigate the class of operators on complex infinite-dimensional Banach spaces for which the Weyl spectrum and the Browder spectrum coincide. These operators are said to satisfy Browder’s theorem. The operators which satisfy Browder’s theorem have a very special spectral structure, ind…
Weyl-Type Theorems
2018
In the previous chapters we introduced several classes of operators which have their origin in Fredholm theory. We also know that the spectrum of a bounded linear operator T on a Banach space X can be split into subsets in many different ways.
Biorthogonal vectors, sesquilinear forms, and some physical operators
2018
Continuing the analysis undertaken in previous articles, we discuss some features of non-self-adjoint operators and sesquilinear forms which are defined starting from two biorthogonal families of vectors, like the so-called generalized Riesz systems, enjoying certain properties. In particular we discuss what happens when they forms two $\D$-quasi bases.
Bounded compositions on scaling invariant Besov spaces
2012
For $0 < s < 1 < q < \infty$, we characterize the homeomorphisms $��: \real^n \to \real^n$ for which the composition operator $f \mapsto f \circ ��$ is bounded on the homogeneous, scaling invariant Besov space $\dot{B}^s_{n/s,q}(\real^n)$, where the emphasis is on the case $q\not=n/s$, left open in the previous literature. We also establish an analogous result for Besov-type function spaces on a wide class of metric measure spaces as well, and make some new remarks considering the scaling invariant Triebel-Lizorkin spaces $\dot{F}^s_{n/s,q}(\real^n)$ with $0 < s < 1$ and $0 < q \leq \infty$.
Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains
2010
We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.
Nonlinear Robin problems with unilateral constraints and dependence on the gradient
2018
We consider a nonlinear Robin problem driven by the p-Laplacian, with unilateral constraints and a reaction term depending also on the gradient (convection term). Using a topological approach based on fixed point theory (the Leray-Schauder alternative principle) and approximating the original problem using the Moreau-Yosida approximations of the subdifferential term, we prove the existence of a smooth solution.
Decompositions and asymptotic limit for bicontractions
2012
The asymptotic limit of a bicontraction T (that is, a pair of commuting contractions) on a Hilbert space H is used to describe a Nagy–Foias–Langer type decomposition of T. This decomposition is refined in the case when the asymptotic limit of T is an orthogonal projection. The case of a bicontraction T consisting of hyponormal (even quasinormal) contractions is also considered, where we have ST∗=S2T∗.