Search results for "Compact operator"
showing 10 items of 40 documents
Rolle's Theorem for Polynomials of Degree Four in a Hilbert Space
2002
AbstractIn an infinite-dimensional real Hilbert space, we introduce a class of fourth-degree polynomials which do not satisfy Rolle's Theorem in the unit ball. Extending what happens in the finite-dimensional case, we show that every fourth-degree polynomial defined by a compact operator satisfies Rolle's Theorem.
Pseudodifferential operators on manifolds with a Lie structure at infinity
2003
to appear in Anal. Math.; Several examples of non-compact manifolds $M_0$ whose geometry at infinity is described by Lie algebras of vector fields $V \subset \Gamma(TM)$ (on a compactification of $M_0$ to a manifold with corners $M$) were studied by Melrose and his collaborators. In math.DG/0201202 and math.OA/0211305, the geometry of manifolds described by Lie algebras of vector fields -- baptised "manifolds with a Lie structure at infinity" there -- was studied from an axiomatic point of view. In this paper, we define and study the algebra $\Psi_{1,0,\VV}^\infty(M_0)$, which is an algebra of pseudodifferential operators canonically associated to a manifold $M_0$ with the Lie structure at …
Geometry of spaces of compact operators
2008
We introduce the notion of compactly locally reflexive Banach spaces and show that a Banach space X is compactly locally reflexive if and only if $\mathcal{K}(Y,X^{**})\subseteq\mathcal{K}(Y,X)^{**}$ for all reflexive Banach spaces Y. We show that X * has the approximation property if and only if X has the approximation property and is compactly locally reflexive. The weak metric approximation property was recently introduced by Lima and Oja. We study two natural weak compact versions of this property. If X is compactly locally reflexive then these two properties coincide. We also show how these properties are related to the compact approximation property and the compact approximation prope…
Weyl-Type Theorems on Banach Spaces Under Compact Perturbations
2018
In this paper, we study Browder-type and Weyl-type theorems for operators $$T+K$$ defined on a Banach space X, where K is (a non necessarily commuting) compact operator on X. In the last part, the theory is exemplified in the case of isometries, analytic Toeplitz operators, semi-shift operators, and weighted right shifts.
Lacunary bifurcation for operator equations and nonlinear boundary value problems on ℝN
1991
SynopsisWe consider nonlinear eigenvalue problems of the form Lu + F(u) = λu in a real Hilbert space, where L is a positive self-adjoint linear operator and F is a nonlinearity vanishing to higher order at u = 0. We suppose that there are gaps in the essential spectrum of L and use critical point theory for strongly indefinite functionals to derive conditions for the existence of non-zero solutions for λ belonging to such a gap, and for the bifurcation of such solutions from the line of trivial solutions at the boundary points of a gap. The abstract results are applied to the L2-theory of semilinear elliptic partial differential equations on ℝN. We obtain existence results for the general c…
Some results about operators in nested Hilbert spaces
2005
With the use of interpolation methods we obtain some results about the domain of an operator acting on the nested Hilbert space {ℋf}f∈∑ generated by a self-adjoint operatorA and some estimates of the norms of its representatives. Some consequences in the particular case of the scale of Hilbert spaces are discussed.
The Partial Inner Product Space Method: A Quick Overview
2010
Many families of function spaces play a central role in analysis, in particular, in signal processing (e.g., wavelet or Gabor analysis). Typical are spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases ofpartial inner product spaces(PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, for the whole family, instead o…
Generation of Frames
2004
It is well known that, given a generic frame, there exists a unique frame operator which satisfies, together with its adjoint, a double operator inequality. In this paper we start considering the inverse problem, that is how to associate a frame to certain operators satisfying the same kind of inequality. The main motivation of our analysis is the possibility of using frame theory in the discussion of some aspects of the quantum time evolution, both for open and for closed physical systems.
Some spectral properties for operators acting on Rigged Hilbert spaces
2015
Operators on Rigged Hilbert spaces have been considered from the 80s of the 20th century on as good ones for describing several physical models whose observable set didn’t turn out to be a C∗-algebra.A notion of resolvent set for an operator acting in a rigged Hilbert space \(\mathcal{D}\subset \mathcal{H}\subset \mathcal{D}^{\times }\) is proposed. This set depends on a family of intermediate locally convex spaces living between \(\mathcal{D}\) and \(\mathcal{D}^{\times }\), called interspaces. Some properties of the resolvent set and of the corresponding multivalued resolvent function are derived and some examples are discussed.
Partial Multiplication of Operators in Rigged Hilbert Spaces
2005
The problem of the multiplication of operators acting in rigged Hilbert spaces is considered. This is done, as usual, by constructing certain intermediate spaces through which the product can be factorized. In the special case where the starting space is the set of C∞-vectors of a self-adjoint operator A, a general procedure for constructing a special family of interspaces is given. Their definition closely reminds that of the Bessel potential spaces, to which they reduce when the starting space is the Schwartz space \(\mathcal{S}(\mathbb{R}^n ).\) Some applications are considered.